## PSEB 11th Class Physics Solutions Chapter 9 Mechanical Properties of Solids

Punjab State Board PSEB 11th Class Physics Book Solutions Chapter 9 Mechanical Properties of Solids Textbook Exercise Questions and Answers.

## PSEB Solutions for Class 11 Physics Chapter 9 Mechanical Properties of Solids

PSEB 11th Class Physics Guide Mechanical Properties of Solids Textbook Questions and Answers

Question 1.
A steel wire of length 4.7 m and cross-sectional area 3.0 x 10-5 m2 stretches by the same amount as a copper wire of length 3.5 m and cross-sectional area of 4.0 x 10-5 m2 under a given load. What is the ratio of the Young’s modulus of steel to that of copper?
Solution:
Given, length of the steel wire, L1 = 4.7 m
Area of cross-section of the steel wire,A1 = 3.0 x 10-5 m2
Length of the copper wire, L2 = 3.5 m
Area of cross-section of the copper wire, A2 = 4.0 x 10-5 m2

Change in length = ΔL1 = ΔL2 = ΔL
Force applied in both the cases = F
Young’s modulus of the steel wire,
Y1 = $$\frac{F_{1}}{A_{1}} \times \frac{L_{1}}{\Delta L_{1}}$$
= $$\frac{F \times 4.7}{3.0 \times 10^{-5} \times \Delta L}$$ ………………………………. (i)
Young’s modulus of the copper wire,
Y2 = $$\frac{F_{2}}{A_{2}} \times \frac{L_{2}}{\Delta L_{2}}=\frac{F \times 3.5}{4.0 \times 10^{-5} \times \Delta L}$$ ………………………………. (ii)
Dividing eq. (i) by eq. (ii), we get:
$$\frac{Y_{1}}{Y_{2}}=\frac{4.7 \times 4.0 \times 10^{-5}}{3.0 \times 10^{-5} \times 3.5}$$ = 1.79:1
The ratio of Young’s modulus of steel to that of copper is 1.79: 1.

Question 2.
Figure given below shows the strain-stress curve for a given material. What are (a) Young’s modulus and (b) approximate yield strengths for this material?

Solution:
(a) It is clear from the given graph that for stress 150 x 106 N/m2, strain is 0.002.
∴ Young s modulus, Y = $$\frac{\text { Stress }}{\text { Strain }}$$
= $$\frac{150 \times 10^{6}}{0.002}$$ = 7.5 x 1010 N/m2
Hence, Young’s modulus for the given material is 7.5 x1010 N/m2

(b) The yield strength of a material is the maximum stress that the material can sustain without crossing the elastic limit. It is clear from the given graph that the approximate yield strength of this material is 300 x 106 N/m2 or 3 x 108 N/m2.

Question 3.
The stress-strain graphs for materials A and B are shown in figure given below.

The graphs are drawn to the same scale.
(a) Which of the materials has the greater Young’s modulus?
(b) Which of the two is the stronger material?
Solution:
(a) A, for a given strain, the stress for material A is more than it is for > material B, as shown in the two graphs.
Young’s modulus = $$\frac{\text { Stress }}{\text { Strain }}$$
For a given strain, if the stress for a material is more, then Young’s modulus is also greater for that material. Therefore, Young’s modulus for material A is greater than it is for material B.

(b) A, the amount of stress required for fracturing a material, corresponding to its fracture point, gives the strength of that material. Fracture point is the extreme point in a stress-strain curve. It can be observed that material A can withstand more strain than material B. Hence, material A is stronger than material B.

Question 4.
Read the following two statements below carefully and state, with reasons, if it is true or false.
(a) The Young’s modulus of rubber is greater than that of steel;
(b) The stretching of a coil is determined by its shear modulus.
Solution:
(a) False.
Reason: For a given stress, the strain in rubber is more than it is in steel.
Young’s modulus, Y = $$\frac{\text { Stress }}{\text { Strain }}$$
For a constant stress Y ∝ $$\frac{1}{\text { Strain }}$$
Hence, Young’s modulus for rubber is less than it is for steel.

(b) True.
Reason: Shear modulus is the ratio of the applied stress to the change in the shape of a body. The stretching of a coil changes its shape. Hence, shear modulus of elasticity is involved in this process.

Question 5.
Two wires of diameter 0.25 cm, one made of steel and the other made of brass are loaded as shown in figure given below. The unloaded length of steel wire is 1.5 m and that of brass wire is 1.0 m.
Compute the elongations of the steel and the brass wires.

Solution:
Given, diameter of the wires, d = 0.25 cm
Hence, the radius of the wires, r = $$\frac{d}{2}$$ = 0.125cm = 0.125 x 10-2m
Length of the steel wire, L1 = 1.5 m
Length of the brass wire, L2 = 1.0 m
Total force exerted on the steel wire,
F1 = (4 +6)g= 10×9.8 = 98N .

Young’s modulus for steel
Y1 = $$\frac{\left(\frac{F_{1}}{A_{1}}\right)}{\left(\frac{\Delta L_{1}}{L_{1}}\right)}$$
where, ΔL1 = Change in the length of the steel wire
A1 = Area of cross-section of the steel wire = πr²1

Young’s modulus of steel, Y1 = 2.0 x 1011 Pa
∴ ΔL1 = $$\frac{F_{1} \times L_{1}}{A_{1} \times Y_{1}}=\frac{F_{1} \times L_{1}}{\pi r_{1}^{2} \times Y_{1}}$$
= $$\frac{98 \times 1.5}{3.14\left(0.125 \times 10^{-2}\right)^{2} \times 2 \times 10^{11}}$$
= 1.5 x 10-4 m

Total force on the brass wire
F2 =6 x 9.8=58.8N
Young’s modulus for brass
Y2 = $$\frac{\left(\frac{F_{2}}{A_{2}}\right)}{\left(\frac{\Delta L_{2}}{L_{2}}\right)}$$

where, ΔL2 = Change in length of the steel wire
A2 = Area of cross-section of the brass wire
∴ ΔL2 = $$\frac{F_{2} \times L_{2}}{A_{2} \times Y_{2}}=\frac{F_{2} \times L_{2}}{\pi r_{2}^{2} \times Y_{2}}$$
= $$\frac{58.8 \times 1.0}{3.14 \times\left(0.125 \times 10^{-2}\right)^{2} \times\left(0.91 \times 10^{11}\right)}$$
= 1.3 x 10-4 m
Hence, elongation of the steel wire =1.49 x 10-4 m
and elongation of the brass wire = 1.3 x 10-4 m

Question 6.
The edge of an aluminium cube is 10 cm long. One face of the cube is firmly fixed to a vertical wall. A mass of 100 kg Is the attached to the opposite face of the cube. The shear modulus of aluminium is 25 GPa. What is the vertical deflection of this face?
Solution:
Given, edge of the aluminium cube, L = 10cm = 0.1 m
The mass attached to the cube, m =100 kg
Shear modulus (ri) of aluminium = 25GPa =25 x 109 Pa
Shear modulus, η = $$\frac{\text { Shear stress }}{\text { Shear strain }}=\frac{\frac{F}{A}}{\frac{\Delta L}{L}}$$
where, F = Applied force = mg = 100 x 9.8 = 980 N
A = Area of one of the faces of the cube = 0.1 x 0.1 = 0.01 m2

ΔL = Vertical deflection of the cube
ΔL = $$\frac{F L}{A \eta}=\frac{980 \times 0.1}{10^{-2} \times\left(25 \times 10^{9}\right)}$$
= 3.92 x 10-7 m
The vertical deflection of this face of the cube is 3.92 x 10-7 m.

Question 7.
Four identical tblloW cylindrical columns of mild steel support a big structure of mass’50,000 kg. The inner and outer radii of each column are 30 cm and 60 cm respectively. Assuming the load distribution to be uniform, calculate the compressional strain of each column.
Solution:
Given, mass of the big structure, M = 50000 kg
Inner radius of the column, r = 30 cm = 0.3 m
Outer radius of the column, R = 60 cm = 0.6 m
Young’s modulus of steel, Y = 2 x 1011 Pa
Total force exerted, F =Mg = 50000 x 9.8N
Stress = Force exerted on a single column = $$\frac{50000 \times 9.8}{4}$$ = 122500 N

Young’s modulus, Y = $$\frac{\text { Stress }}{\text { Strain }}$$
Strain = $$\frac{F}{\frac{A}{Y}}$$
where, Area, A = π(R2 – r2) = π[(0.6)2 – (0.3)2]
Strain = $$\frac{122500}{3.14\left[(0.6)^{2}-(0.3)^{2}\right] \times 2 \times 10^{11}}$$ = 7.22 x 10-7
Hence, the compressional strain of each column is 7.22 x 10-7.
∴Compressional strain of all columns is given by
= 7.22 x 10 -7 x 4 = 2.88 x 10-6.

Question 8.
A piece of copper having a rectangular cross-section of 15.2 minxes 19.2 mm is pulled in tension with 44,500 N force, producing only elastic deformation. Calculate the resulting strain?
Solution:
Given, cross-section area.of copper piece (A) = 15.2 mm x 19.1 mm
= (15.2 x 19.1) x 10 -6m2
Force applied (F) = 44500 N
Young’s modulus (Y) =1.1 x 1011 Nm-2

Young s modulus (Y) = $$=\frac{\text { Longitudinal stress }}{\text { Longitudinal strain }}$$
or Longitudinal strain = $$\frac{\text { Longitudinal stress }}{\text { Young’s modulus }}$$
Young’s modulus
= $$\frac{(F / A)}{Y}=\frac{F}{A Y}$$
= $$\frac{44500}{15.2 \times 19.1 \times 10^{-6} \times 1.1 \times 10^{11}}$$
= 0.0013934.

Question 9.
A steel cable with a radius of 1.5 cm supports a chairlift at a ski area. If the maximum stress is not to exceed 108 Nm -2, what is the maximum load the cable can support?
Solution:
Radius of the steel cable, r = 1.5cm = 0.015m
Maximum allowable stress = 108 N m-2
Maximum stress = $$\frac{\text { Maximum force }}{\text { Area of cross – section }}$$
∴ Maximum force = Maximum stress x Area of cross – section
= 108 x π (0.015)2
= 7.065 x 104 N
Hence, the cable can support the maximum load of 7.065 x 104 N.

Question 10.
A rigid bar of mass 15 kg is supported symmetrically by three wires each 2.0 m long. Those at each mid are of copper and the middle one is of iron. Determine the ratio of their diameters if each is to have the same tension.
Solution:
The tension force acting on each wire is the same. Thus, the extension in each case is the same. Since the wires are of the same length, the strain will also be the same.
The relation for Young’s modulus is given as:
Y = $$\frac{\text { Stress }}{\text { Strain }}=\frac{\frac{F}{A}}{\text { Strain }}=\frac{\frac{4 F}{\pi d^{2}}}{\text { Strain }}$$ ……………………………. (i)
where, F = Tension force
A = Area of cross-section
d = Diameter of the wire
It can be inferred from equation (i) that Y ∝ $$\frac{1}{d^{2}}$$
Young’s modulus for iron, Y1 = 190 x 109 Pa
Diameter of the iron wire = d1
Young’s modulus for copper, Y2 = 110 x 109 Pa
Diameter of the copper wire = d2
Therefore, the ratio of their diameters is given as:
$$\frac{d_{2}}{d_{1}}=\sqrt{\frac{Y_{1}}{Y_{2}}}=\sqrt{\frac{190 \times 10^{9}}{110 \times 10^{9}}}=\sqrt{\frac{19}{11}}$$
= 1.31:11.

Question 11.
A 14.5 kg mass, fastened to the end of a steel wire of unstretched length 1.0 m, is whirled in a vertical circle with an angular velocity of 2 rev/s at the bottom of the circle.
The cross-sectional area of the wire is 0.065 cm2. Calculate the elongation of the wire when the mass is at the lowest point of its path.
Solution:
Given, mass, m = 14.5kg
Length of the steel wire, l = 1.0 m
Angular velocity, ω = 2 rev / s
Cross-sectional area of the wire, a = 0.065cm2 = 0.065 x 10-4 m2
Let δl be died elongation of the wire when the mass is at the lowest point of its path.
When the mass is placed at the position of the vertical circle, the total force on the mass is :
F = mg+mlω2 ,
= 14.5 x 9.8 +14.5x 1 x (2)2 = 200.1 N

Young’s modulus = $$\frac{\text { Stress }}{\text { Strain }}$$
Y = $$\frac{\frac{F}{A}}{\frac{\Delta l}{l}}=\frac{F}{A} \cdot \frac{l}{\Delta l}$$
∴ Δl = $$\frac{F l}{A Y}$$

Young’s modulus for steel = 2 x 1011 Pa
∴ Δl = $$\frac{200.1 \times 1}{0.065 \times 10^{-4} \times 2 \times 10^{11}}$$ = 1539.23 x 10-7
= 1.539 x 10-4
Hence, the elongation of the wire is 1.539 x 10-4 m.

Question 12.
Compute the bulk modulus of water from the following data: Initial volume =100.0 litre, Pressure increase =100.0atm (1 atm = 1.013 x 105 Pa), Final volume = 100.5 litre. Compare the bulk modulus of water with that of air (at constant temperature). Explain in simple terms why the ratio is so large.
Solution:
Initial volume, V1 = 100.0 l x 10-3 m3
Final volume, V2 = 100.5l = 100.5 x 10-3 m-3
Increase in volume, V = V2 — V1 = 0.5 x 10-3 m3
Increase in pressure, Δp =100.0 atm = 100 x 1.013 x 105 Pa
Bulk modulus = $$\frac{\Delta p}{\frac{\Delta V}{V_{1}}}=\frac{\Delta p \times V_{1}}{\Delta V}$$
= $$\frac{100 \times 1.013 \times 10^{5} \times 100 \times 10^{-3}}{0.5 \times 10^{-3}}$$ = 2.206 x 109 Pa
Bulk modulus of air = 1.0 x 105 Pa
∴ $$\frac{\text { Bulk modulus of water }}{\text { Bulk modulus of air }}=\frac{2.026 \times 10^{9}}{1.0 \times 10^{5}}$$
= 2.026 x 104
This ratio is very high because air is more compressible than water.

Question 13.
What is the density of water at a depth where pressure is 80.0 atm, given that its density at the surface is 103 x 103 kgm-3?
Solution:
Let the given depth be h.
Pressure at the given depth, p = 80.0 atm = 80 x 1.01 x 105 Pa
Density of water at the surface, ρ1 = 1.03 x 103 kg m-3
Let ρ2 be the density of water at the depth h.
Let V1 be the volume of water of mass m at the surface.
Let V2 be the volumé of water of mass m at the depth h.
Let ΔV be the change in volume.

ΔV = V1 – V2 = $$m\left(\frac{1}{\rho_{1}}-\frac{1}{\rho_{2}}\right)$$
∴ Volumetric strain= $$\frac{\Delta V}{V_{1}}=m\left(\frac{1}{\rho_{1}}-\frac{1}{\rho_{2}}\right) \times \frac{\rho_{1}}{m}$$
∴ $$\frac{\Delta V}{V_{1}}=1-\frac{\rho_{1}}{\rho_{2}}$$ ………………………………. (i)

Bulk modulus, B = $$\frac{p V_{1}}{\Delta V}$$
$$\frac{\Delta V}{V_{1}}=\frac{p}{B}$$
Compressibility of water = $$\frac{1}{B}=45.8 \times 10^{-11} \mathrm{~Pa}^{-1}$$
∴ $$\frac{\Delta V}{V_{1}}=80 \times 1.013 \times 10^{5} \times 45.8 \times 10^{-11}$$ = 3.71 x 10-3 ………….(ii)
From equations (i) and (ii), we get

Therefore, the density of water at the given depth (h) is 1.034 x 103 kg m-3.

Question 14.
Compute the fractional change in volume of a glass slab, when subjected to a hydraulic pressure of 10 atm.
Solution:
Hydraulic pressure exerted on the glass slab, p = 10 atm = 10 x 1.013 x 105 Pa
Bulk modulus of glass, B = 37 x 109 Nm-2
Bulk modulus, B = $$\frac{p}{\frac{\Delta V}{V}}$$
where, $$\frac{\Delta V}{V}$$ = Fractional change in volume
∴ $$\frac{\Delta V}{V}=\frac{p}{B}=\frac{10 \times 1.013 \times 10^{5}}{37 \times 10^{9}}$$
= 2.73 x 10-5
Hence, the fractional change in the volume of the glass slab is
2.73 x 10-5 = 2.73 x 10-3% = 0.0027%

Question 15.
Determine the volume contraction of a solid copper cube, 10 cm on an edge, when subjected to a hydraulic pressure of 7.0x 106 Pa.
Solution:
Length of an edge of the solid copper cube, l =10 cm = 0.1 m
Hydraulic pressure, p = 7.0 x 106 Pa
Bulk modulus of copper, B = 140 x 109 Pa
Bulk modulus, B = $$\frac{P}{\frac{P}{\Delta V}}$$
where, $$\frac{\Delta V}{V}$$ = Volumetric strain
ΔV = Change in volume
V =,Original volume
ΔV = $$\frac{p V}{B}$$
Original volume of the cube, V = l3

Therefore, the volume contraction of the solid copper cube is 0.05 cm3.

Question 16.
How much should the pressure on a litre of water be changed to compress by 0.10%?
Solution:
Volume of water, V =1 L
It is given that water is to be compressed by 0.10%.
∴ Fractional change, $$\frac{\Delta V}{V}=\frac{0.1}{100 \times 1}=10^{-3}$$
Bulk modulus, B = $$\frac{p}{\frac{\Delta V}{V}}$$
p = B x $$\frac{\Delta V}{V}$$

Bulk modulus of water, B = 2.2 x 109 Nm-2
p = 22 x 109 x 10-3
=2.2 x 106 Nm-2
Therefore, the pressure on water should be 2.2 x 106 Nm-2.

Question 17.
Anvils made of single crystals of diamond, with the shape as shown in figure given below, are used to investigate behaviour of materials under very high pressures. Flat faces at the narrow end of the anvil have a diameter of 0.50 mm, and the wide ends are subjected to a compressional force of 50,000 N. What is the pressure at the tip of the anvil?

Solution:
Diameter of the cones at the narrow ends, d = 0.50 mm = 0.5 x 10-3m
radius, r = $$\frac{d}{2}$$ = 0.25 x 10-3 m
Compressional force, F = 50000 N
Pressure at the tip of the anvil,
p = $$\frac{\text { Force }}{\text { Area }}=\frac{F}{\pi r^{2}}$$
= $$\frac{50000}{3.14 \times\left(0.25 \times 10^{-3}\right)^{2}}$$
= 2.55 x 1011 Pa
Therefore, the pressure at the tip of the anvil is 2.55 x 1011 Pa.

Question 18.
A rod of length 1.05 m having negligible mass is supported at its ends by two wires of steel (wire A) and aluminium (wire B) of equal lengths as shown in figure given below. The cross-sectional areas of wires A and B are 1.0 mm2 and 2.0mm2, respectively. At what point along the rod should a mass m he suspended in order to produce (a) equal stresses and (b) equal strains in both steel and aluminium wires.

Solution:
Given, cross-sectional area of wire A, a1 = 1.0 mm2 = 1.0 x 10-6 m2
Cross-sectional area of wire B, a2 = 2.0 mm2 = 2.0 x 10-6 m2
Young’s modulus for steel, Y1 = 2 x 1011 Nm-2
Young’s modulus for aluminium, Y2 = 7.0 x 1010 Nm-2

Let a small mass m be suspended to the rod at a distance y from the end where wire A is attached.
Stress in the wire = $$\frac{\text { Force }}{\text { Area }}=\frac{F}{a}$$
If the two wires have equal stresses, then,
$$\frac{F_{1}}{a_{1}}=\frac{F_{2}}{a_{2}}$$
where, F1 = Force exerted on the steel wire

F2 = Force exerted on the aluminium wire
$$\frac{F_{1}}{F_{2}}=\frac{a_{1}}{a_{2}}=\frac{1}{2}$$ …………………………. (i)
The situation is shown in the following figure.

Taking torque about the point of suspension, we have
F1y = F2(1.05— y)
$$\frac{F_{1}}{F_{2}}=\frac{(1.05-y)}{y}$$ ……………………….(ii)
Using equations (i) and (ii), we can write

In order to produce an equal stress in the two wires, the mass should be suspended at a distance of 0.7 m from the end where wire A is attached.

(b) Young s modulus = $$\frac{\text { Stress }}{\text { Strain }}$$
Strain = $$\frac{\text { Stress }}{\text { Young’s modulus }}=\frac{\frac{F}{a}}{Y}$$
If the strain in the two wires is equal, then,

Taking torque about the point where mass m, is suspended at a distance y1 from the side where wire A attached, we get
F1y1 =F2(1.05-y1)
$$\frac{F_{1}}{F_{2}}=\frac{\left(1.05-y_{1}\right)}{y_{1}}$$ ……………………………. (iv)
Using equations (iii) and (iv), we get
$$\frac{\left(1.05-y_{1}\right)}{y_{1}}=\frac{10}{7}$$
7(1.05 – y1) = 10 y1
⇒ 17 y1 = 7.35
y1 = 0.432 m
In order to produce an equal strain in the two wires, the mass should be suspended at a distance of 0.432 m from the end where wire A is attached.

Question 19.
A mild steel wire of length 1.0 m and cross-sectional area 0.50 x 10-2 cm2 is stretched, well within its elastic limit, horizontally between two pillars. A mass of 100 g is suspended from the mid-point of the wire. Calculate the depression at the mid-point.
Solution:

Length of the steel wire = 1.0 m
Area of cross-section, A = 0.50 x 10-2 cm2 = 0.50 x 10-6 m2
A mass 100 g is suspended from its mid-point.
m = 100 g = 0.1kg
Hence, the wire dips, as shown in the given figure.

Original length = XZ
Depression = l
The length after mass m, is attached to the wire = XO +OZ
Increase in the length of the wire:
Δl = (XO + OZ)-XZ
Where

Expanding and neglecting higher terms, we get:
Δl = $$\frac{l^{2}}{0.5}$$
Strain = $$\frac{\text { Increase in length }}{\text { Original length }}$$
Let T be the tension in the wire.

∴ mg = 2Tcos θ
Using the figure, it can be written as
Cos θ = $$\frac{l}{\left[(0.5)^{2}+l^{2}\right]^{\frac{1}{2}}}=\frac{l}{(0.5)\left[1+\left(\frac{l}{0.5}\right)^{2}\right]^{\frac{1}{2}}}$$
Expanding the expression and eliminating the higher terms, we get
Cos θ = $$\frac{l}{(0.5)\left(1+\frac{l^{2}}{2(0.5)^{2}}\right)}$$
$$\left(1+\frac{l^{2}}{2(0.5)^{2}}\right)$$ ≈ 1 for small l

Hence, the depression at the mid-point is 0.0107 m.

Question 20.
Two strips of metal are riveted together at their ends by four rivets, each of diameter 6.0 mm. What is the maximum tension’ that can be exerted by the riveted strip if the shearing stress on the rivet is not to exceed 6.9 x 107 Pa?
Assume that each rivet is to carry one-quarter of the load.
Solution:
Diameter of the metal strip, d = 6.0 mm = 6.0 x 10-3 m
Radius, r = $$\frac{d}{2}$$ = 3.0 x 10-3 m
Maximum shearing stress = 6.9 x 107 Pa
Maximum stress = $$\frac{\text { Maximum load or force }}{\text { Area }}$$
Maximum force = Maximum stress x Area
= 6.9 x 107 x π x (r)2
= 6.9 x 107 x 3.14 x (3 x10-3)2
= 1949.94 N
Each rivet carries one quarter of the load.
∴ Maximum tension on each rivet = 4 x 1949.94 = 7799.76 N.

Question 21.
The Marina trench is located in the Pacific Ocean, and at one place it is nearly eleven km beneath the surface of the water. The water pressure at the bottom of the trench is about 1.1 x 108 Pa. A steel ball of initial volume 0.32m is dropped into the ocean and falls to the bottom of the trench. What is the change in the volume of the ball when it reaches to the bottom?
Solution:
Water pressure at the bottom, p = 1.1 x 108 Pa
Initial volume of the steel ball, V = 0.32 m3
Bulk modulus of steel, B = 1.6 x 1011 Nm-2
The ball falls at the bottom of the Pacific Ocean, which is 11 km beneath the surface.
Let the change in the volume of the ball on reaching the bottom of the trench be ΔV.
Bulk modulus, B = $$\frac{p}{\frac{\Delta V}{V}}$$
ΔV = $$\frac{p V}{B}=\frac{1.1 \times 10^{8} \times 0.32}{1.6 \times 10^{11}}$$
= 2.2 x 10-4 m3
Therefore, the change in volume of the ball on reaching the bottom of the trench is 2.2 x 10-4 m3.

## PSEB 11th Class Physics Solutions Chapter 6 Work, Energy and Power

Punjab State Board PSEB 11th Class Physics Book Solutions Chapter 6 Work, Energy and Power Textbook Exercise Questions and Answers.

## PSEB Solutions for Class 11 Physics Chapter 6 Work, Energy and Power

PSEB 11th Class Physics Guide Work, Energy and Power Textbook Questions and Answers

Question 1.
The sign of work done by a force on a body is important to understand. State carefully if the following quantities are positive or negative:
(a) work done by a man in lifting a bucket out of a well by means of a rope tied to the bucket.
(b) work done by gravitational force in the above case,
(c) work done by friction on a body sliding down an inclined plane,
(d) work done by an applied force on a body moving on a rough horizontal plane with uniform velocity,
(e) work done by the resistive force of air on a vibrating pendulum in bringing it to rest.
Solution:
(a) Positive
In the given case, force and displacement are in the same direction. Hence, the sign of work done is positive. In this case, the work is done on the bucket.

(b) Negative
In the given case, the direction of force (vertically downward) and displacement (vertically upward) are opposite to each other. Hence, the sign of work done is negative.

(c) Negative
Since the direction of frictional force is opposite to the direction of motion, the work done by frictional force is negative in this case.

(d) Positive
Here the body is moving on a rough horizontal plane. Frictional force opposes the motion of the body. Therefore, in order to maintain a uniform velocity, a uniform force must be applied to the body. Since the applied force acts in the direction of motion of the body, the work done is positive.

(e) Negative
The resistive force of air acts in the direction opposite to the direction of motion of the pendulum. Hence, the work done is negative in this case.

Question 2.
A body of mass 2 kg initially at rest moves under the action of an applied horizontal force of 7 N on a table with coefficient of kinetic friction = 0.1.
Compute the
(a) work done by the applied force in 10 s,
(b) work done by friction in 10 s,
(c) work done by the net force on the body in 10 s,
(d) change in kinetic energy of the body in 10 s, and interpret your results.
Solution:
Mass of the body, m = 2 kg
Applied force, F = 7 N
Coefficient of kinetic friction, μ = 0.1
Initial velocity, u = 0
Time, t = 10 s
The acceleration produced in the body by the applied force is given by Newton’s second law of motion as:
a’ = $$\frac{F}{m}$$ = $$\frac{7}{2}$$ = 3.5 m/s2
Frictional force is given as:
f = μ mg = 0.1 × 2 × 9.8 = -1.96 N
The acceleration produced by the frictional force:
a” = –$$\frac{1.96}{2}$$ = -0.98 m/s2
Total acceleration of the body:
a = a’ + a”
= 3.5 + (-0.98) = 2.52 m/s2
The distance travelled by the body is given by the equation of motion:
s = ut + $$\frac{1}{2}$$ at2
= 0 + $$\frac{1}{2}$$ × 2.52 × (10)2 = 126 m
(a) Work done by the applied force, Wa = F × s = 7x 126 = 882 J
(b) Work done by the frictional force, Wf = f × s = -1.96 × 126 = -247 J
(c) Net force = 7 + (-1.96) = 5.04 N
Work done by the net force, Wnet = 5.04 × 126 = 635 J
(d) From the first equation of motion, final velocity can be calculated as:
υ = u + at
= 0 + 2.52 × 10 = 25.2 m/s
Change in kinetic energy = $$\frac{1}{2}$$ mυ 2 – $$\frac{1}{2}$$ mu2
= $$\frac{1}{2}$$ × 2(υ2 – u2) = (25.2)2 – 02 = 635 J

Question 3.
Given in figure below are examples of some potential energy functions in one dimension. The total energy of the particle is indicated by a cross on the ordinate axis. In each case, specify the regions, if any, in which the particle cannot be found for the given energy. Also, indicate the minimum total energy the particle muqt have in each case. Think of simple physical contexts for which these potential energy shapes are relevant.

Solution:
Total energy of a system is given by the relation:
E = P.E. + K.E.
> K.E. = E – P.E.
Kinetic energy of a body is a positive quantity. It cannot be negative. Therefore, the particle will not exist in a region where K.E. becomes negative.
(i) In the given case, the potential energy (V0) of the particle becomes greater than total energy (E) for x > a. Hence, kinetic energy becomes negative in this region. Therefore, the particle will not exist in this region. The minimum total energy of the particle is zero.

(ii) In the given case, the potential energy (V0) is greater than total energy (E) in all regions. Hence, the particle will not exist in this region.

(iii) In the given case, the condition regarding the positivity of K.E. is satisfied only in the region between x > a and x<b.
The minimum potential energy in this case is -V1. Therefore, K.E. = E-(-V1) = E + V1.
Therefore, for the positivity of the kinetic energy, the total energy of the particle must be greater than -Vj. So, the minimum total energy the particle must have is -V1.

(iv) In the given case, the potential energy (V0) of the particle becomes greater than the total energy (E) for –$$\frac{b}{2}$$< x <$$\frac{b}{2}$$ and –$$\frac{a}{2}$$< x <$$\frac{a}{2}$$.
Therefore, the particle will not exist in these regions.
The minimum potential energy in this case is -V1. Therefore, K.E. = E – (-V1 ) = E + V1. Therefore, for the positivity of the kinetic energy, the total energy of the particle must be greater than -V1. So, the minimum total energy the particle must have is -V1.

Question 4.
The potential energy function for a particle executing linear simple harmonic motion is given by V (x) = kx2 / 2, where k is the force constant of the oscillator. For k = 0.5 Nm-1, the graph of V (x) versus x is shown in figure below. Show that a particle of total energy 1 J moving under this potential must ‘turn back’ when it reaches x = ±2m.

Solution:
Total energy of the particle, E = 1 J
Force constant, k = 0.5 Nm-1
Kinetic energy of the particle, K = $$\frac{1}{2}$$mυ2
According to the conservation law:
E = V + K
1 = $$\frac{1}{2}$$kx2 + $$\frac{1}{2}$$mυ 2
At the moment of ‘turn back’, velocity (and hence K) becomes zero.
> 1 = $$\frac{1}{2}$$kx2
$$\frac{1}{2}$$ × 0.5 ×2 = 1
x2 = 4
x = ±2
Hence, the particle turns back when it reaches x = ±2 m.

Question 5.
(a) The casing of a rocket in flight burns up due to friction. At whose expense is the heat energy required for burning obtained? The rocket or the atmosphere?
(b) Comets move around the sun in highly elliptical orbits. The gravitational force on the comet due to the sun is not normal to the comet’s velocity in general. Yet the work done by the gravitational force over every complete orbit of the comet is zero. Why?
(c) An artificial satellite orbiting the earth in very thin atmosphere loses its energy gradually due to dissipation against atmospheric resistance, however small. Why then does its speed increase progressively as it comes closer and closer to the earth?
(d) In fig. (i) the man walks 2 m carrying a mass of 15 kg on his hands. In fig. (ii), he walks the same distance pulling the rope behind him. The rope goes over a pulley, and a mass of 15 kg hangs at its other end. In which case is the work done greater?

Solution:
(a) The burning of the casing of a rocket in flight (due to friction) results in the reduction of the mass of the rocket.
According to the conservation of energy:
Total Energy (T.E.) = Potential energy (P.E.) + Kinetic energy (K.E.)
= mgh + $$\frac{1}{2}$$ mυ2
The reduction in the rocket’s mass causes a drop in the total energy. Therefore, the heat energy required for the burning is obtained from the rocket.

(b) Gravitational force is a conservative force. Since the work done by a conservative force over a closed path is zero, the work done by the gravitational force over every complete orbit of a comet is zero.

(c) When an artificial satellite, orbiting around earth, moves closer to earth, its potential energy decreases because of the reduction in the height. Since the total energy of the system remains constant, the reduction in P.E. results in an increase in K.E. Hence, the velocity of the satellite increases. However, due to atmospheric friction, the total energy of the satellite decreases by a small amount.

(d) Case (i)
Mass, m = 15 kg
Displacement, s = 2 m
Work done, W = Fs cosθ
where, θ = Angle between force and displacement
= mgs cosθ = 15 × 2 × 9.8 cos 90°
= 0 ( cos90° = 0)

Case (ii)
Mass, m = 15 kg Displacement, s = 2 m
Here, the direction of the force applied on the rope and the direction of the displacement of the rope are same.
Therefore, the angle between them, θ =0°
Since, cos0° = 1
Work done, W = Fs cosθ = mgs
= 15 × 9.8 × 2 = 294J
Hence, more work is done in the second case.

Question 6.
Underline the correct alternative:
(a) When a conservative force does positive work on a body, the potential energy of the body increases/decreases/ remains unaltered.
(b) Work done by a body against friction always results in a loss of its kinetic/potential energy.
(c) The rate of change of total momentum of a many-particle system is proportional to the external force/sum of the internal forces on the system.
(d) In an inelastic collision of two bodies, the quantities which do not change after the collision are the total kinetic energy/total linear momentum/total energy of the system of two bodies.
Solution:
(a) Decreases, A conservative force does a positive work on a body when it displaces the body in the direction of force. As a result, the body advances toward the centre of force. It decreases the separation between the two, thereby decreasing the potential energy of the body.

(b) Kinetic energy, The work done against the direction of friction reduces the velocity of a body. Hence, there is a loss of kinetic energy of the body.

(c) External force, Internal forces, irrespective of their direction, cannot produce any change in the total momentum of a body. Hence, the total momentum of a many-particle system is proportional to the external forces acting on the system.

(d) Total linear momentum, The total linear momentum always remains conserved whether it is an elastic collision or an inelastic collision.

Question 7.
State if each of the following statements is true or false. Give reasons for your answer.
(a) In an elastic collision of two bodies, the momentum and energy of each body is conserved.
(b) Total energy of a system is always conserved, no matter what internal and external forces on the body are present.
(c) Work done in the motion of a body over a closed loop is zero for every force in nature.
(d) In an inelastic collision, the final kinetic energy is always less than the initial kinetic energy of the system.
Solution:
(a) False, In an elastic collision, the total energy and momentum of both the bodies, and not of each individual body, is conserved.

(b) False, Although internal forces are balanced, they cause no work to be done on a body. It is the external forces that have the ability to do work. Hence, external forces are able to change the energy of a system.

(c) False, The work done in the motion of a body over a closed loop is zero for a conservation force only.

(d) True, In an inelastic collision, the final kinetic energy is always less than the initial kinetic energy of the system. This is because in such collisions, there is always a loss of energy in the form of heat, sound, etc.

Question 8.
(a) In an elastic collision of two billiard balls, is the total kinetic energy conserved during the short time of collision of the balls (i. e., when they are in contact)?
(b) Is the total linear momentum conserved during the short time of an elastic collision of two balls?
(c) What are the answers to (a) and (b) for an inelastic collision?
(d) If the potential energy of two billiard balls depends only on the separation distance between their centres, is the collision elastic or inelastic? (Note, we are talking here of potential energy corresponding to the force during collision, not gravitational potential energy).
Solution:
(a) No, In an elastic collision, the total initial kinetic eneigy of the bails will be equal to the total final kinetic energy of the balls. This kinetic energy is not conserved at the instant the two balls are in contact with each other. In fact, at the time of collision, the kinetic energy of the balls will get converted into potential energy.

(b) Yes, In an elastic collision, the total linear momentum of the system always remains conserved.

(c) No, In an inelastic collision, there is always a loss of kinetic energy, i. e., the total kinetic energy of the billiard balls before collision will always be greater than that after collision.
Yes, The total linear momentum of the system of billiards balls will remain conserved even in the case of an inelastic collision.

(d) Elastic, In the given case, the forces involved are conservation. This is because they depend on the separation between the centres of the billiard balls. Hence, the collision is elastic.

Question 9.
A body is initially at rest. It undergoes one-dimensional motion with constant acceleration. The power delivered to it at time t is proportional to
(i) $$t^{\frac{1}{2}}$$
(ii) t
(iii) $$t^{\frac{3}{2}}$$
(iv) t2
Solution:
(ii) t
Let a body of mass m which is initially at rest undergoes one-dimensional motion under a constant force F with a constant acceleration a.
Acceleration (a) = $$\frac{F}{m}$$ …………. (i)
Using equation of motion, υ = u + at
⇒ υ = 0 + $$\frac{F}{m}$$.t …………. (∵ u = 0)
⇒ υ = $$\frac{F}{m}$$t …………… (ii)
Power delivered (P) = Fυ
Substituting the value from eq. (ii), we get
⇒ P = F × $$\frac{F}{m}$$ × t
⇒ P = $$\frac{F^{2}}{m}$$t
Dividing and multiplying by m in R.H.S.
P = $$\frac{F^{2}}{m^{2}}$$ × mt= a2mt [Using eq.(i)]
As mass m and acceleration a are constant.
∴ P ∝ t

Question 10.
A body is moving unidirectionally under the influence of a source of constant power. Its displacement in time t is proportional to
(i) $$t^{\frac{1}{2}}$$
(ii) t
(iii) $$t^{\frac{3}{2}}$$
(iv) t2
Solution:
(iii) $$t^{\frac{3}{2}}$$
Power is given by the relation:
P = Fυ
= maυ = mυ $$\frac{d v}{d t}$$ = Constant (say,k )
υ dv = $$\frac{k}{m}$$ dt
Integrating both sides:
$$\frac{v^{2}}{2}=\frac{k}{m}$$t
υ = $$\sqrt{\frac{2 k t}{m}}$$
For displacement x of the body, we have:
υ = $$\frac{d x}{d t}=\sqrt{\frac{2 k}{m}} t^{\frac{1}{2}}$$
dx = k’$$t^{\frac{1}{2}}$$dt
where, k’ = $$\sqrt{\frac{2 k}{3}}$$ = New constant
On integrating both sides, we get:
x = $$\frac{2}{3} k^{\prime} t^{\frac{3}{2}}$$
x ∝ $$t^{\frac{3}{2}}$$

Question 11.
A body constrained to move along the z-axis of a coordinate system is subject to a constant force F given by F = -î + 2ĵ + 3k̂N
where î, ĵ, k̂ are unit vectors along the x-, y- and z-axis of the system respectively. What is the work done by this force in moving the body a distance of 4 m along the z-axis?
Solution:
Force exerted on the body, F = -î + 2ĵ + 3k̂N
Displacement, s = 4 k̂ m
Work done, W = F.s
= (-î + 2ĵ + 3k̂).(4k̂)
= 0 + 0 + 3 × 4 = 12 J
Hence, 12 J of work is done by the force on the body.

Question 12.
An electron and a proton are detected in a cosmic ray experiment, the first with kinetic energy 10 keV, and the second with 100 keV. Which is faster, the electron or the proton? Obtain the ratio of their speeds, (electron mass = 9.11 × 10-31 kg, proton mass = 1.67 × 10-27 kg, 1 eV = 1.60 × 10-19 J).
Solution:
Mass of the electron, me = 9.11 × 10-31 kg
Mass of the proton, mp =1.67 × 10-27 kg
Kinetic energy of the electron, EKe =10 keV = 104 eV
= 104 × 1.60 × 10-19
1.60 × 10-15 J
Kinetic energy of the proton, EKp = 100 keV = 105 eV = 1.60 × 10-14 J
For the velocity of an electron ve, its kinetic energy is given by the relation:

Question 13.
A rain drop of radius 2 mm falls from a height of500 m above the ground. It falls with decreasing acceleration (due to viscous resistance of the air) until at half its original height, it attains its maximum (terminal) speed, and moves with uniform speed thereafter. What is the work done hy the gravitational force on the drop in the first and second half of its journey? What is the work done by the resistive force in the entire journey if its speed on reaching the ground is 10 ms-1?
Solution:
Radius of the rain drop, r = 2 mm = 2 × 10 -3 m
Volume of the rain drop, V = $$\frac{4}{3}$$ πr3
= $$\frac{4}{3}$$ × 3.14 × (2 × 10-3)3 m-3
Density of water, ρ = 103 kgm-3
Mass of the rain drop, m = ρV
= $$\frac{4}{3}$$ × 3.14 × (2 × 10-3)3 × 103
Gravitational, F = mg
= $$\frac{4}{3}$$ × 3.14 × (2 × 10-3)3 × 103 9.8N
The work done by the gravitational force on the drop in the first half of its journey.
W1 = Fs
= $$\frac{4}{3}$$ × 3.14 × (2 × 10-3 )3 × 103 × 9.8 × 250
= 0.082 J

This amount of work is equal to the work done by the gravitational force on the drop in the second half of its journey, i. e., WII, = 0.082 J.
As per the law of conservation of energy, if no resistive force is present, then the total energy of the rain drop will remain the same.
∴ Total energy at the top:
ET = mgh + 0
= $$\frac{4}{3}$$ × 3.14 × (2 × 10-3)3 × 103 × 9.8 × 500
= 0.164 J
Due to the presence of a resistive force, the drop hits the ground with a velocity of 10 m/s.
∴ Total energy at the ground:
Eg = $$\frac{1}{2}$$ mυ2 + 0
= $$\frac{1}{2}$$ × $$\frac{4}{3}$$ × 3.14 × (2 × 10-3)3 × 103 × 9.8 × (10)2
= 1.675 × 10-3 J = 0.001675
∴ Resistive force = Eg – Et = 0.001675 – 0.164 = -0.162 J

Question 14.
A molecule in a gas container hits a horizontal wall with speed 200 m s-1 and angle 30° with the normal, and rebounds with the same speed. Is momentum conserved in the collision? Is the collision elastic or inelastic?
Solution:
Yes; Collision is elastic
The momentum of the gas molecule remains conserved Whether the collision is elastic or inelastic.
The gas molecule moves with a velocity of 200 m/s and strikes the stationary wall of the container, rebounding with the same speed.
It shows that the rebound velocity of the wall remains zero. Hence, the total kinetic energy of the molecule remains conserved during the collision. The given collision is an example of an elastic collision.

Question 15.
A pump on the ground floor of a building can pump up water to fill a tank of volume 30 m3 in 15 min. If the tank is 40 m above the ground, and the efficiency of the pump is 30%, how much electric power is consumed by the pump?
Solution:
Volume of the tank, V = 30 m3
Time of operation, t = 15 min = 15 × 60 = 900 s
Height of the tank, h = 40 m
Efficiency of the pump, η = 30%
Density of water, ρ = 103 kg/m3
Mass of water, m = ρ V = 30 × 103 kg
Output power can be obtained as:
P0 = $$\frac{\text { Work done }}{\text { Time }}=\frac{m g h}{t}$$
= $$\frac{30 \times 10^{3} \times 9.8 \times 40}{900}$$ = 13.067 × 103 W
For input power Pi efficiency η, is given by the relation :
η = $$\frac{P_{o}}{P_{i}}$$ = 30%
Pi = $$\frac{13.067}{30}$$ × 100 103
= 0.436 × 102 W = 43.6 kW

Question 16.
Two identical ball bearings in contact with each other and resting on a frictionless table are hit head-on by another ball bearing of the same mass moving initially with a speed V. If the collision is elastic, which of the following figure is a possible result after collision?

Solution:
It can be observed that the total momentum before and after collision in each case is constant.
For an elastic collision, the total kinetic energy of a system remains conserved before and after collision.
For mass of each ball bearing m, we can write:
Total kinetic energy of the system before collision:
= $$\frac{1}{2}$$mV2 + $$\frac{1}{2}$$(2m)0
= $$\frac{1}{2}$$mV2

Case (i)
Total kinetic energy of the system after collision:
= $$\frac{1}{2}$$m × 0 + $$\frac{1}{2}$$(2m) ($$\frac{V}{2}$$)2
= $$\frac{1}{4}$$mV2
Hence, the kinetic energy of the system is not conserved in case (i).

Case (ii)
Total kinetic energy of the system after collision:
= $$\frac{1}{2}$$ (2m) × 0 + $$\frac{1}{2}$$mV2
= $$\frac{1}{2}$$ mV2
Hence, the kinetic energy of the system is conserved in case (ii).

Case (iii)
Total kinetic energy of the system after collision:
= $$\frac{1}{2}$$(3m) ($$\frac{V}{3}$$)2
= $$\frac{1}{6}$$ mV2
Hence, the kinetic energy of the system is not conserved in case (iii).

Question 17.
The bob A of a pendulum released from 30° to the vertical hits another hob B of the same mass at rest on a table as shown in’ figure given below. How high does the bob A rise after the collision? Neglect the size of the bobs and assume the collision to be elastic.

Solution:
Bob A will not rise at all
In an elastic collision between two equal masses in which one is stationary, while the other is moving with some velocity, the stationary mass acquires the same velocity, while the moving mass immediately comes to rest after collision. In this case, a complete transfer of momentum takes place from the moving mass to the stationary mass. Hence, bob A of mass m, after colliding with bob B of equal mass, will come to rest, while bob B will move with the velocity of bob A at the instant of collision.

Question 18.
The bob of a pendulum is released from a horizontal position. If the length of the pendulum is 1.5 m, what is the speed with which the bob arrives at the lowermost-point, given that it dissipated 5% of its initial energy against air resistance?
Solution:
Length of the pendulum, l = 1.5 m
Mass of the bob = m
Energy dissipated = 5%
According to the law of conservation of energy, the total energy of the system remains constant.

At the horizontal position:
Potential energy of the bob, EP = mgl
Kinetic energy of the bob, EK = 0
Total energy = mgl …………. (i)

At the lowermost point (mean position):
Potential energy of the bob, EP = 0
Kinetic energy of the bob, EK = $$\frac{1}{2}$$mυ2
Total energy, Ex = $$\frac{1}{2}$$mυ2 ………….. (ii)
As the bob moves from the horizontal position to the lowermost point, 5% of its energy gets dissipated.
The total energy at the lowermost point is equal to 95% of the total energy at the horizontal point, i. e.,
$$\frac{1}{2}$$mυ2 = $$\frac{95}{100}$$ × mgl
υ = $$\sqrt{\frac{2 \times 95 \times 1.5 \times 9.8}{100}}$$ = 5.28 m/s

Question 19.
A trolley of mass 300 kg carrying a sandbag of 25 kg is moving uniformly with a speed of 27 km/h on a frictionless track. After a while, sand starts leaking out of ahole on the floor of the trolley at the rate of 0.05 kg s-1 . What is the speed of the trolley after the entire sand bag is empty?
Solution:
The sand bag is placed on a trolley that is moving with a uniform speed of 27 km/h. The external forces acting on the system of the sand bag and the trolley is zero. When the sand starts leaking from the bag, there will be no change in the velocity of the trolley. This is because the leaking action does not produce any external force on the system. This is in accordance with Newton’s first law of motion. Hence, die speed of the trolley will remain 27 km/h.

Question 20.
A body of mass 0.5 kg travels in a straight line with velocity υ = ax3/2 where a = 5 m-1/2s-1. What is the work done by the net force during its displacement from x = 0 to x = 2 m?
Solution:
Mass of the body, m = 0.5 kg
Velocity of the body is governed by the equation, υ = a $$x^{\frac{3}{2}}$$ where,
a = 5m-1/2s-1
Initial velocity, u (at x = 0) = 0
Final velocity, υ (at x = 2 m) = 5 × (2)3/2 m/s = 10√2 m/s
Work done, W = Change in kinetic energy
= $$\frac{1}{2}$$m(υ2 – u2)
= $$\frac{1}{2}$$ × 0.5[(10√2)2 – (0)2]
= $$\frac{1}{2}$$ × 0.5 × 10 × 10 × 2
= 50 J

Question 21.
The blades of a windmill sweep out a circle of area A. (a) If the wind flows at a velocity v perpendicular to the circle, what is the mass of the air passing through it in time t? (b) What is the kinetic energy of the air? (c) Assume that the windmill converts 25% of the wind’s energy into electrical energy, and that A = 30 m2, υ = 36km/h and the density of air is 1.2 kg m-3. What is the electrical power produced?
Solution:
(a) Area of the circle swept by the windmill = A
Velocity of the wind = υ
Density of air = ρ
Volume of the wind flowing through the windmill per sec = Aυ
Mass of the wind flowing through the windmill per sec = ρ Aυ
Mass m, of the wind flowing through the windmill in time t = ρ A υ t

(b) Kinetic energy of air = $$\frac{1}{2}$$ mυ2
= $$\frac{1}{2}$$ (ρAυt)v2 = $$\frac{1}{2}$$ρ Aυ3t

(c) Area of the circle swept by the windmill, A = 30 m2
Velocity of the wind, υ =36 km/h = 36 × $$\frac{5}{18}$$ m/s
= 10 m/s [1 km/s = $$\frac{5}{18}$$ m/s]
Density of air, ρ = 1.2 kg m-3
Electric energy produced = 25% of the wind energy
= $$\frac{25}{100}$$ ×Kinetic energy of air
= $$\frac{1}{8}$$ρAυ3t

 = $$\frac{1}{8}$$ρAυ3
= $$\frac{1}{8}$$ × 1.2 × 30 × (10)3
= 4.5 × 103 W = 4.5 kW

Question 22.
A person trying to lose weight (dieter) lifts a 10 kg mass, one thousand times, to a height of 0.5 m each time. Assume that the potential energy lost each time she lowers the mass is dissipated, (a) How much work does she do against the gravitational force? (b) Fat supplies 3.8 × 107 J of energy per kilogram which is converted to mechanical energy with a 20% efficiency rate. How much fat will the dieter use up?
Solution:
(a) Mass of the weight, m = 10 kg
Height to which the person lifts the weight, h = 0.5
m Number of times the weight is lifted, n = 1000
∴ Work done against gravitational force:
= n(mgh)
=1000 × 10 × 9.8 × 0.5
= 49 × 103 J = 49 kJ

(b) Energy equivalent of 1 kg of fat = 3.8 × 107 J
Efficiency rate = 20%
Mechanical energy supplied by the person’s body
= $$\frac{20}{100}$$ × 3.8 × 107 J
= $$\frac{1}{5}$$ × 3.8 × 107 J 5
Equivalent mass of fat lost by the dieter
= $$\frac{1}{\frac{1}{5} \times 3.8 \times 10^{7}}$$ × 49 × 3
= $$\frac{245}{3.8}$$× 3.8 × 107
= × 10-4
= 6.45 × 10-3 kg

Question 23.
A family uses 8 kW of power, (a) Direct solar energy is incident on the horizontal surface at an average rate of 200 W per square meter. If 20% of this energy can be converted to useful electrical energy, how large an area is needed to supply 8 kW? (b) Compare this area to that of the roof of a typical house.
Solution:
(a) Power used by the family, P = 8 kW = 8 × 103 W
Solar energy received per square meter = 200 W
Efficiency of conversion from solar to electrical energy = 20%
Area required to generate the desired electricity = A
As per the information given in the question, we have
8 × 103 = 20% × (A × 200)
= $$\frac{20}{100}$$ × A × 200
> A = $$\frac{8 \times 10^{3}}{40}$$ = 200m2

(b) In order to compare this area to that of the roof of a typical house, let ‘a’ be the side of the roof
∴ area of roof =a × a = a2
Thus a2 = 200 m2
or a = $$\sqrt{200 \mathrm{~m}^{2}}$$ = 14.14 m
∴ area of roof = 14.14 × 14.14 m2
Thus 200 m2 is comparable to the roof of a typical house of dimensions 14.14 m × 14.14 m = 14 m × 14 m.

Question 24.
A bullet of mass 0.012 kg and horizontal speed 70 ms-1 strikes a block of wood of mass 0.4 kg and instantly comes to rest with respect to the block. The block is suspended from the ceiling by means of thin wires. Calculate the height to which the block rises. Also, estimate the amount of heat produced in the block.
Solution:
Mass of the bullet, m = 0.012 kg
Initial speed of the bullet, ub = 70 m/s
Mass of the wooden block, M = 0.4 kg
Initial speed of the wooden block, uB = 0
Final speed of the system of the bullet and the block = υ
Applying the law of conservation of momentum,
mub +MUB = (m + M)υ
012 × 70 + 0.4 × 0 = (0.012 + 0.4)υ
∴ υ = $$\frac{0.84}{0.412}$$ = 2.04 m/s

For the system of the bullet and the wooden block,
Mass of the system, m’ = 0.412 kg
Velocity of the system = 2.04 m/s
Height up to which the system rises = h
Applying the law of conservation of energy to this system,
Potential energy at the highest point
= Kinetic energy at the lowest point
m’ gh = $$\frac{1}{2}$$m’ υ2
h = $$\frac{1}{2}$$($$\frac{v^{2}}{g}$$)
= $$\frac{1}{2}$$ ($$\frac{(2.04)^{2}}{9.8}$$) = 0.2123m
The wooden block will rise to a height of 0.2123 m.
Heat produced = Kinetic energy of the bullet
– Kinetic energy of the system
= $$\frac{1}{2}$$ mub– $$\frac{1}{2}$$m’ υ2
= $$\frac{1}{2}$$ × 0.012 × (70)2 – $$\frac{1}{2}$$ × 0.412 × (2.04)2
= 29.4 – 0.857 = 28.54 J

Question 25.
Two inclined frictionless tracks, one gradual and the other steep meet at A from where two stones are allowed to slide down from rest, one on each track (see figure). Will the stones reach the bottom at the same time? Will they reach there with the same speed? Explain. Given θ1 = 30°, θ2 = 60°and h = 10m, what are the speeds and times taken by the two stones?

Solution:
No; the stone moving down the steep plane will reach the bottom first Yes; the stones will reach the bottom with the same speed The given situation can be shown as in the following figure:

Here, the initial height (AD) for both the stones is the same (h).
Hence, both will have the same potential energy at point A.
As per the law of conservation of energy, the kinetic energy of the stones at points B and C will also be the same, i. e.,
$$\frac{1}{2}$$ mυ12 = $$\frac{1}{2}$$mυ22
υ1 = υ1 = υ, say
where,
m = Mass of each stone
υ = Speed of each stone at points B and C
Hence, both stones will reach the bottom with the same speed, υ.

For stone I: Net force acting on this stone is given by:
Fnet = ma1 = mgsinθ1
a1 = gsinθ1

For stone II: a2 = gsinθ2
∵ θ2 > θ1
∴ sinθ2 > sinθ1
∴ a2 > a1
Using the first equation of motion, the time of slide can be obtained as:
υ = u + at
∴ t = $$\frac{v}{a}$$ (∵ u = 0)
For stone I: t1 = $$\frac{v}{a_{1}}$$
For stone II: t2 = $$\frac{v}{a_{2}}$$
∵ a2 > a1
∴ t2 < t1
Hence, the stone moving down the steep plane will reach the bottom first. The speed (υ) of each stone at points B and C is given by the relation obtained from the law of conservation of energy.

Question 26.
A1 kg block situated on a rough incline is connected to a spring of spring constant 100 Nm-1 as shown in figure given below. The block is released from rest with the spring in the unstretched position. The block moves 10 cm down the incline before coming to rest. Find the coefficient of friction between the block and the incline. Assume that the spring has a negligible mass and the pulley is frictionless.

Solution:
Mass of the block, m = 1 kg
Spring constant, k = 100 N m-1
Displacement in the block, x = 10 cm = 0.1 m
The given situation can be shown as in the following figure.

At equilibrium:
Normal reaction, R = mg cos 37°
Frictional force, f = μR = p mg cos 37°
where, μ is the coefficient of friction
Net force acting on the block = mg sin 3 7° – f
= mgsin37° – μmgcos37°
= mg (sin37° – μcos37°)
At equilibrium, the work done by the block is equal to the potential energy of the spring, i.e.,
mg (sin37° – μcos37°) x = $$\frac{1}{2}$$kx 2
1 × 9.8 (sin 37° – μcos37°) = $$\frac{1}{2}$$ × 100 × 0.1
0.6018 – μ × 0.7986 = 0.5102
0.7986 μ = 0.6018 – 0.5102 = 0.0916
∴ μ = $$\frac{0.0916}{0.7986}$$ = 0.115

Question 27.
A bolt of mass 0.3 kg falls from the ceiling of an elevator moving down with an uniform speed of 7 m-1. It hits the floor of the elevator (length of the elevator = 3m) and does not rebound. What is the heat produced by the impact? Would your answer be different if the elevator were stationary?
Solution:
Mass of the bolt, m = 0.3 kg
Speed of the elevator = 7 m/s
Height, h = 3 m
Since the relative velocity of the bolt with respect to the lift is zero, at the time of impact, potential energy gets converted into heat energy.
Heat produced = Loss of potential energy
= mgh = 0.3 × 9.8 × 3 = 8.82 J
The heat produced will remain the same even if the lift is stationary. This is because of the fact that the relative velocity of the bolt with respect to the lift will remain zero.

Question 28.
A trolley of mass 200 kg moves with a uniform speed of 36 km/h on a frictionless track. A child of mass 20 kg runs on the trolley from one end to the other (10 m away) with a speed of 4 m s-1 relative to the trolley in a direction opposite to the its motion, and jumps out of the trolley. What is the final speed of the trolley? How much has the trolley moved from the time the child begins to run?
Solution:
Mass of the trolley, M = 200 kg
Speed of the trolley, υ = 36 km/h = 36 × $$\frac{5}{18}$$m/s = 10 m/s
Mass of the boy, m = 20 kg
Initial momentum of the system of the boy and the trolley
= (M + m)υ
= (200 + 20) × 10
= 2200 kg-m/s
Let v’ be the final velocity of the trolley with respect to the ground.
Final velocity of the boy with respect to the ground = υ’ – 4
Final momentum = Mυ’ + m(υ’ – 4)
= 200 υ’ + 20υ’ – 80
= 220 υ’ – 80
As per the law of conservation of momentum,
Initial momentum = Final momentum
2200 =220 υ’ – 80
∴ υ’ = $$\frac{2280}{220}$$ = 10.36 m/s
Length of the trolley, l = 10 m
Speed of the boy, υ” = 4 m/s
Time taken by the boy to run, t = $$\frac{10}{4}$$ = 2.5 s
∴ Distance moved by the trolley = υ’ × t= 10.36 × 2.5 = 25.9 m

Question 29.
Which of the following potential energy curves in the given figure cannot possibly describe the elastic collision of two billiard balls? Here r is the distance between centres of the balls.

Solution:
(i), (ii), (iii), (iv), and (vi)
The potential energy of a system of two masses is inversely proportional to the separation between them. In the given case, the potential energy of the system of the two balls will decrease as they come closer to each other. It will become zero [i.e., V (r) = 0] when the two balls touch each other, i.e., at r = 2R, where R is the radius of each billiard ball. The potential energy curves given in figures (i), (ii), (iii), (iv), and (vi) do not satisfy these two conditions. Hence, they do not describe the elastic collisions between them.

Question 30.
Consider the decay of a free neutron at rest: n → p + e. Show that the two-body decay of this type must necessarily give an electron of fixed energy and, therefore, cannot account for the observed continuous energy distribution in the β-decay of a neutron or a nucleus (see figure).

[Note: The simple result of this exercise was one among the several arguments advanced by W. Pauli to predict the existence of a third particle in the decay products of P-decay. This particle is known as
neutrino. We now know that it is a particle of intrinsic spin $$\frac{1}{2}$$ (like e, p or n), but is neutral, and either massless or having an extremely small mass
(compared to the mass of electron) and which interacts very weakly with matter. The correct decay process of neutron is: n → p + e + υ]
Solution:
The decay process of free neutron at rest is given as:
n → p + e
From Einstein’s mass-energy relation, we have the energy of electron as
Δ𝜏 mc2
where,
Δ𝜏 m = Mass defect
= Mass of neutron – (Mass of proton + Mass of electron)
c = Speed of light
Δ𝜏 m and c are constants. Hence, the given two-body decay is unable to explain the continuous energy distribution in the p-decay of a neutron or a nucleus. The presence of neutrino v on the LHS of the decay correctly explains the continuous energy distribution.
Here, Δ𝜏 m= mass of neutron – (mass of proton + mass of electron)
= 1.6747 × 10-24 -(1.6724 × 10-24 + 9.11 × 10-28)
= (1.6747 – 1.6733) × 10-24
= 0.0014 × 10-24 gms ‘
∴ E = 0.0014 × 10-24 × (3 × 1010)2 ergs
= 0.0126 × 10-4 ergs
= [Latex]\frac{0.0126 \times 10^{-4}}{1.6 \times 10^{-12}}[/Latex]
( ∵ 1 eV = 1.6 × 10-19 J = 1.6 × 10-19 × 107 ergs)
= 0.007875 × 10 8 eV
= 0.7875 × 106 eV
= 0.79 MeV

A positron has the same mass as an electron but an opposite charge of+e. When an electron and a positron come close to each other, they destroy each other. Their masses, are converted into energy according to Einstein’s relation and the energy so obtained is released in the form of y-rays and is given by
E’ = mc2 = 2 × 9.1 × 10-31 kgx (3 × 108 ms-1)2
= 1.638 × 10-13 J
= $$\frac{1.638 \times 10^{-13}}{1.6 \times 10^{-13}}$$ MeV
= 1.02MeV (∵1 MeV = 1.6 × 10-13 eV)
= Minimum energy a photon must possess for pair production.

Alternate method
Decay of a free neutron at rest,
n → p + e
Let Δ m be the mass defect during this process.
∴ Mass defect (Δ m) = (Mass of neutron) – (Mass of proton and electron) This mass defect is fixed and hence electron of fixed energy should be produced. Therefore, two-body decay of this type cannot explain the observed continuous energy distribution in the (3-decay of a neutron in a nucleus.

## PSEB 11th Class Physics Solutions Chapter 12 Thermodynamics

Punjab State Board PSEB 11th Class Physics Book Solutions Chapter 12 Thermodynamics Textbook Exercise Questions and Answers.

## PSEB Solutions for Class 11 Physics Chapter 12 Thermodynamics

PSEB 11th Class Physics Guide Thermodynamics Textbook Questions and Answers

Question 1.
A geyser heats water flowing at the rate of 3.0 liters per minute from 27°C to 77°C. If the geyser operates on a gas burner, what is the rate of consumption of the fuel if its heat of combustion is 4.0 x 104 J/g?
Solution:
Water is flowing at a rate of 3.0 liter/min.
The geyser heats the water, raising the temperature from 27°C to 77°C.
Initial temperature, T1 = 27°C
Final temperature, T2 = 77°C
Rise in temperature, ΔT = T2 -T1
= 77-27 = 50°C

Heat of combustion = 4 x 104 J/g
Specific heat of water, C = 4.2 J g-1 °C-1
Mass of flowing water, m = 3.0 litre/min = 3000 g/min
Total heat used, ΔQ = mC ΔT
= 3000 x 4.2 x 50
= 6.3 x 105 J/min
∴ Rate of consumption = $$\frac{6.3 \times 10^{5}}{4 \times 10^{4}}$$ = 15.75 g/min

Question 2.
What amount of heat must be supplied to 2.0 x 10-2 kg of nitrogen (at room temperature) to raise its temperature by 45°C at constant pressure? (Molecular mass of N2 = 28; R = 8.3 J mol-1K-1).
Solution:
Mass of nitrogen, m = 2.0 x 10-2 kg = 20g
Rise in temperature, ΔT = 45°C
Molecular mass of N2,M =28
Universal gas constant, R = 8.3 J mol-1K-1
Number of moles, n = $$\frac{m}{M}$$
= $$\frac{2.0 \times 10^{-2} \times 10^{3}}{28}$$ = 0.714
Molar specific heat at constant pressure for nitrogen,
Cp = $$\frac{7}{2}$$ R = $$\frac{7}{2}$$ x 8.3
= 29.05J mol-1 K-1

The total amount of heat to be supplied is given by the relation
ΔQ = nCpΔT
= 0.714 x 29.05 x 45 = 933.38 J
Therefore, the amount of heat to be supplied is 933.38 J

Question 3.
Explain why
(a) Two bodies at different temperatures T1 and T2 if brought in thermal contact do not necessarily settle to the mean temperature (T1 + T2) / 2.
(b) The coolant in a chemical or a nuclear plant (i.e., the liquid used to prevent the different parts of a plant from getting too hot) should have high specific heat.
(c) Air pressure in a car tyre increases during driving.
(d) The climate of a harbour town is more temperate than that of a town in a desert at the same latitude.
(a) When two bodies at different temperatures T1 and T2 are brought in thermal contact, heat flows from the body at the higher temperature to the body at the lower temperature till equilibrium is achieved, i.e., the temperatures of both the bodies become equal. The equilibrium temperature is equal to the mean temperature (T1 + T2) / 2 only when the thermal capacities of both the bodies are equal.

(b) The coolant in a chemical or nuclear plant should have a high specific heat. This is because higher the specific heat of the coolant, higher is its heat-absorbing capacity and vice versa. Hence, a liquid having a high specific heat is the best coolant to be used in a nuclear or chemical plant. This would prevent different parts of the plant from getting too hot.

(c) When a car is in motion, the air temperature inside the car increases because of the motion of the air molecules. According to Charles’ law, temperature is directly proportional to pressure. Hence, if the temperature inside a tyre increases, then the air pressure in it will also increase.

(d) A harbour town has a more temperate climate (i.e., without the extremes of heat or cold) than a town located in a desert at the same latitude. This is because the relative humidity in a harbour town is more than it is in a desert town.

Question 4.
A cylinder with a movable piston contains 3 moles of hydrogen at standard temperature and pressure. The walls of the cylinder are made of a heat insulator, and the piston is insulated by having a pile of sand on it. By what factor does the pressure of the gas increase if the gas is compressed to half its original volume?
Solution:
The cylinder is completely insulated from its surroundings. As a result, no heat is exchanged between the system (cylinder) and its surroundings. Thus, the process is adiabatic.
Initial pressure inside the cylinder = P1
Final pressure inside the cylinder = P2
Initial volume inside the cylinder = V1
Final volume inside the cylinder = V2
Ratio of specific heats, γ = 1.4 For an adiabatic process,
we have
P1V1γ = P2V2γ
The final volume is compressed to half of its initial volume.

Hence, the pressure increases by a factor of 2.639.

Question 5.
In changing the state of a gas adiabatically from an equilibrium state A to another equilibrium state B, an amount of work equal to 22.3 J is done on the system. If the gas is taken from state A to B via a process in which the net heat absorbed by the system is 9.35 cal, how much is the net work done by the system in the latter case? (Take 1 cal = 4.19 J)
Solution:
The work done (W) on the system while the gas changes from state A to state Bis 22.3 J.
This is an adiabatic process. Hence, change in heat is zero.
ΔQ = 0
ΔW = -22.3 J (since the work is done on the system)
From the first law of thermodynamics, we have:
ΔQ = ΔU + ΔW

where, ΔU = Change in the internal energy of the gas .
ΔU = ΔQ – ΔW = -(-22.3 J)
ΔU = +22.3 J
When the gas goes from state A to state B via a process, the net heat absorbed by the system is
ΔQ =9.35cal = 9.35 x 4.19 = 39.1765J
Heat absorbed, ΔQ = ΔU + ΔW
ΔW = ΔQ – ΔU = 39.1765 – 22.3 – = 16.8765J
Therefore, 16.88 J of work is done by the system.

Question 6.
Two cylinders A and B of equal capacity are connected to each other via a stopcock.
A contains a gas at standard temperature and pressure.
B is completely evacuated. The entire system is thermally insulated.
The stopcock is suddenly opened. Answer the following:
(a) What is the final pressure of the gas in A and B?
(b) What is the change in internal energy of the gas?
(c) What is the change in the temperature of the gas?
(d) Do the intermediate states of the system (before settling to the final equilibrium state) lie on its P -V – T surface?
Solution:
(a) 0.5 atm
The volume available to the gas is doused as soon as the stopcock between cylinders A and B is opened? Since volume is inversely proportional to pressure, the pressure will decrease to one-half of the original value. Since the initial pressure of the gas is 1 atm, the pressure in each cylinder will be 0.5atm.

(b) Zero
The internal energy of the gas can change only when work is done by or on the gas. Since in this case no work is done by or on the gas, the internal energy of the gas will not change.

(c) Zero
Since no work is being done by the gas during the expansion of the gas, the temperature of the gas will not change at all.

(d) No
The given process is a case of free expansion. It is rapid and cannot be controlled. The intermediate states do not satisfy the gas equation and since they are in non-equilibrium states, they do not lie on the P-V-T surface of the system.

Question 7.
A steam engine delivers 5.4 x 108 J of work per minute and services 3.6 x 109 J of heat per minute from its boiler. What is the efficiency of the engine? How much heat is wasted per minute?
Solution:
Work done by the steam engine per minute, W = 5.4 x 108 J
Heat supplied from the boiler, H = 3.6 x 109 J
Efficiency of the engine = $$\frac{\text { Output energy }}{\text { Input energy }}$$
∴ η = $$\frac{W}{H}=\frac{5.4 \times 10^{8}}{3.6 \times 10^{9}}$$ = 0.15
Hence, the percentage efficiency of the engine is 15%.
Amount of heat wasted= 3.6 x 109 – 5.4 x 108
= 30.6 x 108 = 3.06 x 109 J
Therefore, the amount of heat wasted per minute is 3.06 x 109J.

Question 8.
An electric heater supplies heat to a system at a rate of 100 W. If system performs work at a rate of 75 joules per second. At what rate is the internal energy increasing?
Solution:
Heat is supplied to the system at a rate of 100 W.
∴ Heat supplied, ΔQ = 100 J/s
The system performs at a rate of 75 J/s.
∴ Work done, ΔW = 75 J/s

From the first law of thermodynamics, we have
ΔQ = ΔU + ΔW
where ΔU = Rate of change in internal energy
ΔU = ΔQ – ΔW = 100 – 75 = 25 J/s = 25W
Therefore, the internal energy of the given electric heater increases at a rate of 25 W.

Question 9.
A thermodynamic system is taken from an original state to an intermediate state by the linear process shown in figure given below.

Its volume is then reduced to the original value from E to F by an isobaric process. Calculate the total work done by the gas from D to E to F.
Solution:
Total work done by the gas from D to E to F = Area of ΔDEF
Area of ΔDEF = $$\frac{1}{2}$$ DF x EF
where, DF = Change in pressure
=600 N/m2
= 300N/m2 = 300N/m2
FE = Change in volume
5.0 m3 – 2.0 m3 = 3.0m3
Area of ADEF = $$\frac{1}{2}$$ x 300 x 3 = 450 J
Therefore, the total work done by the gas from D to E to F is 450 J.

Question 10.
A refrigerator is to maintain eatables kept inside at 9°C. If the room temperature is 36°C, Calculate the coefficient of performance.
Solution:
Temperature inside the refrigerator, T1 = 9°C = 273 + 9 = 282 K
Room temperature, T2 = 36°C = 273+36
Coefficient of performance = $$\frac{T_{1}}{T_{2}-T_{1}}$$
= $$\frac{282}{309-282}=\frac{282}{27}$$
309-282 = 10.44
Therefore, the coefficient of performance of the given refrigerator is 10.44.

## PSEB 11th Class Physics Solutions Chapter 11 Thermal Properties of Matter

Punjab State Board PSEB 11th Class Physics Book Solutions Chapter 11 Thermal Properties of Matter Textbook Exercise Questions and Answers.

## PSEB Solutions for Class 11 Physics Chapter 11 Thermal Properties of Matter

PSEB 11th Class Physics Guide Thermal Properties of Matter Textbook Questions and Answers

Question 1.
The triple points of neon and carbon dioxide are 24.57 K and 216.55K respectively. Express these temperatures on the Celsius and Fahrenheit scales.
Solution:
Kelvin and Celsius’s scales are related as
TC =TK -273.15
Celsius and Fahrenheit’s scales are related as …(i)
TF = $$\frac{9}{5}$$ TC +32 ………………………….. (ii)
For neon:
TK = 24.57K
∴ TG = 24.57 – 273.15 = -284.58°C
TF =$$\frac{9}{5}$$ TC +32
= $$\frac{9}{5}$$ (-248.58) + 32
=-415.44° F

For carbon dioxide:
TK = 216.55K
∴ TC =216.55-273.15 = -56.60°C
TF =$$\frac{9}{5}$$ TC +32
= $$\frac{9}{5}$$ (-56.60) + 32 = -69.88°C

Question 2.
Two absolute scales A and B have triple points of water defined to be 200A and 350B.
What is the relation between TAand TB?
Solution:
Triple point of water on absolute scale A, T1 = 200 A
Triple point of water on absolute scale B, T2 = 350 B
Triple point of water on Kelvin scale, TK = 273.15K
The temperature 273.15K on Kelvein scale is equivalent to 200 A on absolute scale A.
T1 =TK
200 A = 273.15 K .
∴ A= $$\frac{273.15}{200}$$

The temperature 273.15 K on Kelvin scale is equivalent to 350 B on absolute scale B.
T2=TK
350 B = 273.15
∴ B = $$\frac{273.15}{350}$$
TA is triple point of water on scale A.
TB is triple point of water on scale B.
∴ $$\frac{273.15}{200} \times T_{A}=\frac{273.15}{350} \times T_{B}$$
TA = $$\frac{200}{350} T_{B}$$
TA = $$\frac{4}{7} T_{B}$$
Therefore, the ratioTA:TB is given as 4 : 7.

Question 3.
The electrical resistance in ohms of a certain thermometer varies with temperature according to the approximate law:
R = R0[l + α (T-T0)]
The resistance is 101.6Ω at the triple-point of water 273.16 K, and 165.50 at the normal melting point of lead (600.5 K). What is the temperature when the resistance is 123.4Ω?
Solution:
It is given that:
R = R0[l + α (T-T0)] ………………………….(i)
where R0 and T0 are the initial resistance and temperature respectively R and T are the final resistance and temperature respectively α is a constant At the triple point of water, T0 = 273.16 K
Resistance of lead, R0 =101.6 Ω

At normal melting point of lead, T = 600.5 K
Resistance of lead, R = 165.5 Ω
Substituting these values in equation (i), we get:
R=R0[l + α (T-T0)]
165.5 = 101.6[1 + α (600.5-273.16)]
1.629 = 1+α (327.34)
∴ α = $$\frac{0.629}{327.34}$$ = 1.92 x10-3K-1
For resistance, R1 = 123.4Ω
R1 =R0[l + α (T-T0)]

where, T is the temperature when the resistance of lead is 123.4Ω
123.4 =101.6[1 +1.92 x 10-3(T-273.16)]
1.214 =1+1.92 x 10-3(T- 273.16)
$$\frac{0.214}{1.92 \times 10^{-3}}$$ = T -273.16
111.46 = T-273.16
⇒ T =111.46 +273.16
∴ T = 384.62 K

Question 4.
(a) The triple-point of water is a standard fixed point in modern thermometry. Why? What is wrong in taking the melting point of ice and the boiling point of water as standard fixed points (as was originally done in the Celsius scale)?

(b) There were two fixed points in the original Celsius scale as mentioned above which were assigned the numbers 0°C and 100°C respectively. On the absolute scale, one of the fixed points is the triple-point of water, which on the Kelvin absolute scale is assigned the number 273.16 K. What is the other fixed point on this (Kelvin) scale?

(c) The absolute temperature (Kelvin scale) T is related to the temperature tc on the Celsius scale by
tc = T – 273.15
Why do we have 273.15 in this relation, and not 273.16?

(d) What is the temperature of the triple-point of water on an absolute scale whose unit interval size is equal to that of the Fahrenheit scale?
Solution:
(a) The triple point of water has a unique value of 273.16 K. At particular values of volume and pressure, the triple point of water is always 273.16 K. The melting point of ice and boiling point of water do not have particular values because these points depend on pressure and temperature.

(b) The absolute zero or 0 K is the other fixed point on the Kelvin absolute scale.

(c) The temperature 273.16 K is the triple point of water. It is not the melting point of ice. The temperature 0°C on Celsius scale is the melting point of ice. Its corresponding value on Kelvin scale is 273.15 K. Hence, absolute temperature (Kelvin scafe) T, is related to temperature tc, on Celsius scale as:
tc =T -273.15

(d) Let TF be the temperature on Fahrenheit scale and TK be the temperature on absolute scale. Both the temperatures can be related as
$$\frac{T_{F}-32}{180}=\frac{T_{K}-273.15}{100}$$ ………………………….. (i)
Let TF1 be the temperature on Fahrenheit scale and TK1 be the temperature on absolute scale. Both the temperatures can be relates as
$$\frac{T_{F 1}-32}{180}=\frac{T_{K 1}-273.15}{100}$$ ……………………….. (ii)
It is given that:
TK1-TK= 1K

Subtracting equation (i) from equation (ii), we get
$$\frac{T_{F 1}-T_{F}}{180}=\frac{T_{K 1}-T_{K}}{100}$$
$$T_{F 1}-T_{F}=\frac{1 \times 180}{100}=\frac{9}{5}$$
Triple point of water = 273.16 K
∴ Triple point of water on absolute scale = 273.16 x $$\frac{9}{5}$$ = 491.69

Question 5.
Two ideal gas thermometers A and B use oxygen and hydrogen respectively. The following observations are made:

 Temperature Pressure thermometer A Pressure thermometer B Triple-point of water 1.250 x 105 Pa 0.200 x 105 Pa Normal melting point of sulphur 1.797 x 105 Pa 0.287 x 105 Pa

(a) What is the absolute temperature of normal melting point of sulphur as read by thermometers A and B?
(b) What do you think is the reason behind the slight difference in answers of thermometers A and B? (The thermometers are not faulty). What further procedure is needed in the experiment to reduce the discrepancy between the two readings?
Solution:
Triple point of water, T = 273.16 K.
At this temperature, pressure in thermometer A, PA = 1.250 x 105 Pa
Let T1 be the normal melting point of sulphur.
At this temperature, pressure in thermometer A,P1 = 1.797 x 105 Pa
According to Charles’ law, we have the relation
$$\frac{P_{A}}{T}=\frac{P_{1}}{T_{1}}$$
∴ T1 = $$\frac{P_{1} T}{P_{A}}=\frac{1.797 \times 10^{5} \times 273.16}{1.250 \times 10^{5}}$$
= 392.69 K

Therefore, the absolute temperature of the normal melting point of sulphur as read by thermometer A is 392.69 K.
At triple point 237.16 K, the pressure in thermometer B,
PB =0.200 x 105 Pa
At temperature T2, the pressure in thermometer B, P2 = 0.287 x 105 Pa
According to Charles’ law, we can write the relation
$$\frac{P_{B}}{T}=\frac{P_{2}}{T_{2}}$$
$$\frac{0.200 \times 10^{5}}{273.16}=\frac{0.287 \times 10^{5}}{T_{2}}$$
∴ T2 = $$\frac{0.287 \times 10^{5}}{0.200 \times 10^{5}} \times 273.16$$ = 391.98 K
Therefore, the absolute temperature of the normal melting point of sulphur as read by thermometer B is 391.98 K

(b) The oxygen and hydrogen gas present in thermometers A and B respectively are not perfect ideal gases. Hence, there is a slight difference between the readings of thermometers A and B. To reduce the discrepancy between the two readings, the experiment should be carried under low-pressure conditions. At low pressure, these gases behave as perfect ideal gases.

Question 6.
A steel tape lm long is correctly calibrated for a temperature of 27.0°C. The length of a steel rod measured by this tape is found to be 63.0 cmon a hot day when the temperature is 45°C. What is the actual length of the steel rod on that day? What is the length of the same steel rod on a day when the temperature is 27.0°C? Coefficient of linear expansion of steel = 1.20 x 10-5 K-1.
Solution:
Length of the steel tape at temperature T = 27°C, l = 1 m = 100 cm
At temperature T1 = 45°C,
the length of the steel rod, l1 = 63 cm
Coefficient of linear expansion of steel, α = 1.20 x 10-5K-1

Let l2 be the actual length of the steel rod and l’ be the length of the steel tape at 45°C.
l’ =l+αl(T1 -T)
∴ l’ = 100 +1.20 x 10-5x 100(45 -27)
= 100,0216 cm

Hence, the actual length of the steel rod measured by the steel tape at 45° C can be calculated as
l2 = $$\frac{100.0216}{100} \times 63$$ = 63.0136cm
Therefore, the actual length of the rod at 45°C is 63.0136 cm. Its length at 27.0 °C is 63.0 cm

Question 7.
A large steel wheel is to be fitted onto a shaft of the same material. At 27°C, the outer diameter of the shaft is 8.70 cm, and the diameter of the central hole in the wheel is 8.69cm. The shaft is cooled using ‘dry ice’. At what temperature of the shaft does the wheel slip on the shaft? Assume the coefficient of linear expansion of the steel to be constant over the required temperature range: αsteel = 1.20 x 10-5 K-1.
Solution:
The given temperature, T = 27°C can be written in Kelvin as
27 + 273 =300K
Outer diameter of the steel shaft at T, d1 = 8.70 cm
Diameter of the central hole in the wheel at T, d2 = 8.69 cm
Coefficient of linear expansion of steel, a steel = 1.20 x 10-5 K -1
After the shaft is cooled using ‘dry ice’, its temperature becomes T1.
The wheel will slip on the shaft if the change in diameter,
Δd = 8.69-8.70 =-0.01 cm

Temperature T1; can be calculated from the relation
Δd = d1αsteel(T1 -T)
-0.01 =8.70 x 1.20 x 10-5(T1 -300)
(T1 – 300) = $$\frac{-0.01}{8.70 \times 1.20 \times 10^{-5}}$$
(T1 -300) = -95.78
∴ T1 = 300 – 95.78 = 204.22 K
= (204.22-273)°C
= -68.78°C ≈ 69°C
Therefore, the wheel will slip on the shaft when the temperature of the shaft is -69°C.

Question 8.
A hole is drilled in a copper sheet. The diameter of the hole is 4.24 cm at 27.0°C. What is the change in the diameter of the hole when the sheet is heated to 227°C? Coefficient of linear expansion of copper = 1.70 x 10-5 K -1.
solution:
Initial temperature, T1 = 27.0°C
Diameter of the hole at T1, d1 = 4.24 cm
Final temperature, T2 = 227°C
Let, diameter of the hole at T2=d2
Coefficient of linear expansion of copper, αCu = 1.70 x 10-5 K-1
For coefficient of superficial expansion β, and change in temperature ΔT, we have the relation:
$$\frac{\text { Change in area }(\Delta \mathrm{A})}{\text { Original area }(\mathrm{A})}=\beta \Delta \mathrm{T}$$

Change in diameter = d2 – d1 = 4.2544 – 4.24 = 0.0144 cm
Hence, the diameter increases by 1.44 x 10-2 cm.

Question 9.
A brass wire 1.8m long at 27°C is held taut with little tension between two rigid supports. If the wire is cooled to a temperature of-39°C, what is the tension developed in the wire, if its diameter is 2.0 mm? Coefficient of linear expansion of brass = 2.0 x 10-5K-1; Young’s modulus of brass = 0.91 x 1011 Pa.
Solution:
Initial temperature, T1 = 27°C
Length of the brass wire at T1,l = 1.8 m
Final temperature, T2 = -39 °C
Diameter of the wire, d = 2.0 mm = 2 x 10-3 m
Let, tension developed in the wire = F
Coefficient of linear expansion of brass, a = 2.0 x 10-5 K-1

Young’s modulus of brass, Y = 0.91 x 1011Pa
Young’s modulus is given by the relation ‘
Y = $$\frac{\text { Stress }}{\text { Strain }}=\frac{\frac{F}{A}}{\frac{\Delta L}{L}}$$
ΔL = $$\frac{F \times L}{A \times Y}$$ ……………………………….. (i)
where, F = Tension developed in the wire
A = Area of cross-section of the wire.
ΔL = Change in the length, given by the relation
ΔL = αL(T2 -T1) …………………………… (ii)

Equating equations (i) and (ii), we get

= -3.8 x 102 N
(The negative sign indicates that the tension is directed inward.) Hence, the tension developed in the wire is 3.8 x 102 N.

Question 10.
A brass rod of length 50 cm and diameter 3.0 mm is joined to a steel rod of the same length and diameter. What is the change in length of the combined rod at 250°C, if the original lengths are at 40.0°C? Is there a ‘thermal stress’ developed at the junction? The ends of the rod are free to expand (Coefficient of linear expansion of brass = 2.0 x 10-5 K-1, steel = 1.2 x 10-5K-1).
Solution:
Initial temperature, T1 = 40°C
Final temperature, T2 = 250 °C
Change in temperature, ΔT = T2 – T1 = 210°C
Length of the brass rod at T1,l1 = 50 cm
Diameter of the brass rod at T2, d1 = 3.0 mm
Length of the steel rod at T2,l2 = 50 cm
Diameter of the steel rod at T2,d2 = 3.0 mm
Coefficient of linear expansion of brass, α1 = 2.0 x 10-5 K-1
Coefficient of linear expansion of steel, α2 = 1.2 x 10-5 K-1

For the expansion in the brass rod, we have
$$\frac{\text { Change in length }\left(\Delta l_{1}\right)}{\text { Original length }\left(l_{1}\right)}=\alpha_{1} \Delta T$$
∴ Δl1 = 50 x (2.1 x 10-5)X210
= 0.2205cm
For the expansion in the steel rod, we have Change in length
$$\frac{\text { Change in length }\left(\Delta l_{2}\right)}{\text { Original length }\left(l_{2}\right)}=\alpha_{2} \Delta T$$
∴ Δl2 =50 x (1.2 x 10-5)x 210
= 0.126 cm

Total change in the lengths of brass and steel,
Δl = Δl1 + Δl2
=0.2205 + 0.126
= 0.346 cm
Total change in the length of the combined rod = 0.346 cm Since the rod expands freely from both ends, no thermal stress is developed at the junction.

Question 11.
The coefficient of volume expansion of glycerin is 49 x 10-5K-1.
What is the fractional change in its density for a 30°C rise in temperature?
Solution:
Coefficient of volume expansion of glycerin, αv = 49 x 10-5 K-1
Rise in temperature, ΔT = 30°C
Fractional change in its volume = $$\frac{\Delta V}{V}$$

where, m = Mass of glycerine
PT1 = Initial density at T1
PT2 = Final density at T2
$$\frac{\rho_{T_{1}}-\rho_{T_{2}}}{\rho_{T_{2}}}=\alpha_{\mathrm{V}} \Delta T$$
Where, $$\frac{\rho_{T_{1}}-\rho_{T_{2}}}{\rho_{T_{2}}}$$ = Fractional change in the density
∴ Fractional change in the density of glycerin
= 49 x 10-5 x 30 = 1.47 x 10-2

Question 12.
A 10 kW drilling machine is used to drill a bore in a small aluminium block of mass 8.0 kg. How much is the rise in temperature of the block in 2.5 minutes, assuming 50% of power is used up in heating the machine itself or lost to the surroundings? Specific heat of aluminium = 0.91 Jg-1K-1.
Solution:
Power of the drilling machine, P =10 kW = 10 x 103 W
Mass of the aluminum block, m = 8.0 kg = 8 x 103 g
Time for which the machine is used, t = 2.5min = 2.5x 60 = 150 s
Specific heat of aluminium, C = 0.91 J g-1K-1
Rise in the temperature of the block after drilling = ΔT

Total energy of the drilling machine = Pt
= 10 x 103 x 150 = 1.5 x 106 J

It is given that only 50% of the power is useful.

Useful energy, ΔQ = $$\frac{50}{100} \times 1.5 \times 10^{6}$$ = 7.5×105J
But ΔQ = mCΔT
∴ ΔT = $$\frac{\Delta Q}{m C}=\frac{7.5 \times 10^{5}}{8 \times 10^{3} \times 0.91}$$
=103°C
Therefore, in 2.5 minutes of drilling, the rise in the temperature of the block is 103°C.

Question 13.
A copper block of mass 2.5 kg is heated in a furnace to a temperature of 500°C and then placed on a large ice block. What is the maximum amount of ice that can melt? (Specific heat of copper = 0.39 Jg-1K-1; heat of fusion of water = 335Jg-1).
Solution:
Mass of the copper block, m = 2.5kg = 2500 g
Rise in the temperature of the copper block, Δθ = 500 °C
Specific heat of copper, C = 0.39 J g-1 °C-1
The heat of fusion of water, L = 335 J g-1

The maximum heat the copper block can lose, Q = mCΔθ
= 2500×0.39×500 =487500J
Let m1 g be the amount of ice that melts when the copper block is placed on the ice block.

The heat gained by the melted ice, Q = m1L
∴ m1 = $$\frac{Q}{L}=\frac{487500}{335}$$ =1455.22g
Hence, the maximum amount of ice that can melt is 1.45 kg.

Question 14.
In an experiment on the specific heat of a metal, a 0.20kg block of the metal at 150°C is dropped in a copper calorimeter (of water equivalent 0.025 kg) containing 150 cm3 of water at 27°C. The final temperature is 40° C. Compute the specific heat of the metal. If heat losses to the surroundings are not negligible, is your answer greater or smaller than the actual value for specific heat of the metal?
Solution:
Mass of the metal, m = 0.20 kg = 200 g
Initial temperature of the metal, T1 = 150°C
Final temperature of the metal, T2 = 40°C
Calorimeter has water equivalent of mass, m’ = 0.025kg = 25g
Volume of water, V = 150 cm3

Mass (M) of water at temperature T = 27°C,
150×1 =150g
Fall in the temperature of the metal,
ΔT =T1 -T2 =150-40 =110°C
Specific heat of water, Cw = 4.186 J/g/K
Specific heat of metal = C
Heat lost by the metal, Q = mCΔT …………………………….. (i)

Rise in the temperature of the water and calorimeter system,
ΔT’ = 40 -27 = 13°C
Heat gained by the water and calorimeter system,
ΔQ’ = m1CwΔT’
= (M + m’)CwΔT’ ……………………………… (ii)

Heat lost by the metal = Heat gained by the water and colourimeter system
mCΔT =(M + m’)CwΔT’
200 xC x 110 = (150+25) x 4.186 x 13
∴ C = $$\frac{175 \times 4.186 \times 13}{110 \times 200}$$ = 0.43 Jg-1K-1
If some heat is lost to the surroundings, then the value of C will be smaller than the actual value.

Question 15.
Given below are observations on molar specific heats at room temperature of some common gases.

 Gas Molar specific heat (Cv) (cal mol-1K-1) Hydrogen 4.87 Nitrogen 4.97 Oxygen 5.02 Nitric oxide 4.99 Carbon monoxide 5.01 Chlorine 6.17

The measured molar specific heats of these gases are markedly different from those for monatomic gases. Typically, molar specific heat of a monatomic gas is 2.92 cal/mol K. Explain this difference. What can you infer from the somewhat larger (than the rest) value for chlorine?
Solution:
The gases listed in the given table are diatomic. Besides the translational degree of freedom, they have other degrees of freedom (modes of motion). Heat must be supplied to increase the temperature of these gases. This increases the average energy of all the modes of motion.

Hence, the molar specific heat of diatomic gases is more than that of monatomic gases. If only rotational mode of motion is considered, then the molar specific heat of a diatomic gas = $$\frac{5}{2} R=\frac{5}{2} \times 1.98$$ = 4.95 cal mol-1K-1 With the exception of chlorine, all the observations in the given table agree with($$\frac{5}{2} R$$) This is because at room temperature, chlorine also has vibrational modes of motion besides rotational and translational modes of motion.

Question 16.
Answer the following questions based on the P-T phase diagram of carbon dioxide:
(a) At what temperature and pressure can the solid, liquid and vapour phases of CO2 co-exist in equilibrium?
(b)What is the effect of decrease of pressure on the fusion and boiling point of CO2?
(c) What are the critical temperature and pressure for CO2? What is their significance?
(d) Is CO2 solid, liquid or gas at
(a) -70°C under 1 atm,
(b) -60°C under 10 atm,
(c) 15°C under 56 atm?
Solution:
The P-T phase diagram for CO2 is shown in the following figure.

(a) C is the triple point of the CO2 phase diagram. This means that at the temperature and pressure corresponding to this point (i.e., at -56.6°C and 5.11 atm), the solid, liquid, and vaporous phases of CO2 co-exist in equilibrium.
(b) The fusion and boiling point of CO2 decrease with a decrease in pressure.
(c) The critical temperature and critical pressure of CO2 are 31.1°C and 73 atm respectively.
Even if it is compressed to a pressure greater than 73 atm, CO2 will not liquefy above the critical temperature.
(d) It can be concluded from the P-T phase diagram of CO2 that:
(i) CO2 is gaseous at -70 °C, under 1 atm pressure
(ii) CO2 is solid at -60 °C, under 10 atm pressure
(iii) CO2 is liquid at 15°C, under 56 atm pressure.

Question 17.
Answer the following questions based on the P-T phase diagram of CO2:
(a) CO2 at 1 atm pressure and temperature -60°C is compressed isothermally. Does it go through a liquid phase?
(b) What happens when C02 at 4 atm pressure is cooled from room temperature at constant pressure?
(c) Describe qualitatively the changes in a given mass of solid CO2 at 10 atm pressure and temperature -65°C as it is heated up to room temperature at constant pressure.
(d) CO2 is heated to a temperature 70°C and compressed isothermally. What changes in its properties do you expect to observe?
The P-T phase diagram for CO2 is shown in the following figure:

Solution:
(a) No
Explanation:
At 1 atm pressure and at -60°C, CO2 lies to the left of -56.6°C (triple point C). Hence, it lies in the region of vaporous and solid phases. Thus, CO2 condenses into the solid-state directly, without going through the liquid state.
(b) It condenses to solid directly.
Explanation:
At 4 atm pressure, C02 lies below 5.11 atm (triple point C). Hence, it lies in the region of vaporous and solid phases. Thus, it condenses into the solid-state directly, without passing through the liquid state.

(c) The fusion and boiling points are given by the intersection point where this parallel line cuts the fusion and vaporisation curves.
Explanation:
When the temperature of a mass of solid CO2 (at 10 atm pressure and at -65° C) is increased, it changes to the liquid phase and then to the vaporous phase. It forms a line parallel to the temperature axis at 10 atm. The fusion and boiling points are given by the intersection point where this parallel line cuts the fusion and vaporisation curves.

(d) It departs from ideal gas behaviour as pressure increases.
Explanation:
If CO2 is heated to 70 °C and compressed isothermally, then it will not exhibit any transition to the liquid state. This is because 70 °C is higher than the critical temperature of CO2. It will remain in the vapour state but will depart from its ideal behaviour as pressure increases.

Question 18.
A child running a temperature of 101°F is given an antipyrin (i. e., a medicine that lowers fever) which causes an increase in the rate of evaporation of sweat from his body. If the fever is brought down to 98°F in 20 min, what is the average rate of extra evaporation caused, by the drug? Assume the evaporation mechanism to be the only way by which heat is lost. The mass of the child is 30 kg. The specific heat of human body is approximately the same as that of water, and latent heat of evaporation of water at that temperature is about 580 cal g-1.
Solution:
Initial temperature of the body of the child, T1 = 101°F
Final temperature of the body of the child, T2 = 98°F
Change in temperature, ΔT = $$\left[(101-98) \times \frac{5}{9}\right]$$°c
Time taken to reduce the temperature, t = 20 min
Mass of the child, m = 30 kg = 30 x 103 g

Specific heat of the human body = Specific heat of water = C
= 1000 cal/kg/°C
Latent heat of evaporation of water, L = 580 cal g-1
The heat lost by the child is given as:
ΔQ = mCΔT
= 30 x 1000 x (101-98)x $$\frac{5}{9}$$ = 50000cal

Let m1 be the mass of the water evaporated from the child’s body in 20 min.
Loss of heat through water is given by
ΔQ = m1L
∴ m1 = $$\frac{\Delta Q}{L}=\frac{50000}{580}$$ = 86.2g
∴ Average rate of extra evaporation caused by the drug = $$\frac{m_{1}}{t}$$
= $$\frac{86.2}{20}$$ = 4.3 g/mm.

Question 19.
A ‘thermal’ icebox is a cheap and efficient method for storing small quantities of cooked food in summer in particular. A cubical icebox of side 30 cm has a thickness of 5.0 cm. If 4.0 kg of ice is put in the box, estimate the amount of ice remaining after 6 h. The outside temperature is 45°C, and coefficient of thermal conductivity of thermal is 0.01 J s-1m-1 K-1.
[Heat of fusion of water = 335 x 103 Jkg -1 ]
Solution:
Side of the given cubical icebox, s = 30 cm = 0.3 m
Thickness of the icebox, l = 5.0 cm = 0.05 m
Mass of ice kept in the icebox, m = 4 kg
Time gap, t=6h = 6x 60 x 60 s
Outside temperature, T = 45°C

Coefficient of thermal conductivity of thermacole,
K =0.01 Js-1 m-1 K-1
Heat of fusion of water, L = 335 x 103 J kg-1
Let m be the total amount of ice that melts in 6 h.
The amount of heat lost by the food,
Q = $$\frac{K A(T-0) t}{l}$$

where, A = Surface area of the box = 6s2 =6 x (0.3)2 = 0.54 m2
Q = $$\frac{0.01 \times 0.54 \times(45) \times 6 \times 60 \times 60}{0.05}$$ = 104976 J
But Q=m’L
∴ m’ = $$\frac{Q}{L}=\frac{104976}{335 \times 10^{3}}$$ = 0.313 kg
Mass of ice left = 4-0.313 = 3.687kg .
Hence, the amount of ice remaining after 6 h is 3.687 kg.

Question 20.
A brass boiler has a base area of 0.15m2 and a thickness 1.0 cm. It boils water at the rate of 6.0 kg/min when placed on a gas stove. Estimate the temperature of the part of the flame in contact with the boiler. Thermal conductivity of brass = 109 Js-1m-1 K-1; Heat of vaporisation of water = 2256 x 103 Jkg-1.
Solution:
Base area of the boiler, A = 0.15 m2
Thickness of the boiler,l = 1.0 cm = 0.01 m
Roiling rate of water, R = 6.0 kg/min
Mass, m = 6 kg
Time, t = 1 min = 60 s

Thermal conductivity of brass, K = 109 J s-1 m-1 K-1
Heat of vaporisation, L = 2256 x103 J kg-1
The amount of heat flowing into water through the brass base of the boiler is given by
Q = $$\frac{K A\left(T_{1}-T_{2}\right) t}{l}$$ ………………………. (i)
where, T1 = Temperature of the flame in contact with the boiler
T2 = Boiling point of water = 100 °C
Heat required for boiling the water
Q = mL …………………………………. (ii)

Equating equations (i) and (ii), we get
∴ mL = $$\frac{K A\left(T_{1}-T_{2}\right) t}{l}$$
T1 -T2= $$\frac{m L l}{K A t}$$
= $$\frac{6 \times 2256 \times 10^{3} \times 0.01}{109 \times 0.15 \times 60}$$ = 137.98°C
T1 -T2 = 137.98°C
∴ T1=137.98+100 = 237.98°C
Therefore, the temperature of the part of the flame in contact with the boiler is 237.98°C.

Question 21.
Explain why:
(a) a body with large reflectivity is a poor emitter
(b) a brass tumbler feels much colder than a wooden tray on a chilly day.
(c) an optical pyrometer (for measuring high temperatures) calibrated for an ideal black body radiation gives too low a value for the temperature of a red hot iron piece in the open but gives a correct value for the temperature when the same piece is in the furnace
(d) the earth without its atmosphere would be inhospitably cold
(e) heating systems based on circulation of steam are more efficient in warming a building than those based on circulation of hot water
Solution:
(a) A body with a large reflectivity is a poor absorber of light radiations. A poor absorber will in turn be a poor emitter of radiations. Hence, a body with a large reflectivity is a poor emitter.

(b) Brass is a good conductor of heat. When one touches a brass tumbler, heat is conducted from the body to the brass tumbler easily. Hence, the temperature of the body reduces to a lower value and one feels cooler. Wood is a poor conductor of heat. When one touches a wooden tray, very little heat is conducted from the body to the wooden tray. Hence, there is only a negligible drop in the temperature of the body and one does not feel cool. •
Thus, a brass tumbler feels colder than a wooden tray on a chilly day.

(c) An optical pyrometer calibrated for an ideal black body radiation gives too low a value for temperature of a red hot iron piece kept in the open. Black body radiation equation is given by
E = σ(T4 -T04) where, E – Energy radiation
T = Temperature of optical pyrometer
T0 = Temperature of open space
σ = Constant
Hence, an increase in the temperature of open space reduces the radiation energy.
When the same piece of iron is placed in a furnace, the radiation energy, E =σT4

(d) Without its atmosphere, earth would be inhospitably cold. In the absence of atmospheric gases, no extra heat will be .trapped. All the heat would be radiated back from earth’s surface.

(e) A heating system based on the circulation of steam is more efficient in warming a building than that based on the circulation of hot water. This is because steam contains surplus heat in the form of latent heat (540 cal/g).

Question 22.
A body cools from 80°C to 50°C in 5 minutes. Calculate the time it takes to cool from 60°C to 30°C. The temperature of the surroundings is 20 C.
Solution:
According to Newton’s law of cooling, the rate of cooling cc difference in temperature.
Here, average of 80 °C and 50 °C, T = $$\frac{T_{1}+T_{2}}{2}=\frac{80+50}{2}$$ = 65°C
Temperature of surroundings, T0 = 20 °C .
∴Difference, ΔT = T – T0 = 65 – 20 = 45°C
Under these conditions, the body cools 30 °C in time 5 minutes.
∴
or $$\frac{30}{5}$$ = k x 45 ……………………………….. (i)
The average of 60°C and 30 °C is 45°C which is 25°C (45 – 20) above the room temperature and the body cools by 30 °C (60 – 30) in a time t (say).
∴$$\frac{30}{t}$$ = k x 25 ……………………………….. (ii)

where k is same for this situation as for the original.
Dividing eq. (i) from eq. (ii), we get
$$\frac{30 / 5}{30 / t}=\frac{k \times 45}{k \times 25}$$
or $$\frac{t}{5}=\frac{9}{5}$$
or t = 9 min

## PSEB 11th Class Physics Solutions Chapter 10 Mechanical Properties of Fluids

Punjab State Board PSEB 11th Class Physics Book Solutions Chapter 10 Mechanical Properties of Fluids Textbook Exercise Questions and Answers.

## PSEB Solutions for Class 11 Physics Chapter 10 Mechanical Properties of Fluids

PSEB 11th Class Physics Guide Mechanical Properties of Fluids Textbook Questions and Answers

Question 1.
Explain why
(a) The blood pressure in humans is greater at the feet than at the brain
(b) Atmospheric pressure at a height of about 6 km decreases to nearly half of its value at the sea level, though the height of the atmosphere is more than 100 km
(c) Hydrostatic pressure is a scalar quantity even though pressure is force divided by area.
(a) The pressure of a liquid is given by the relation
P =hρg
where, P = Pressure
h = Height of the liquid column
ρ = Density of the liquid ‘ .
g = Acceleration due to the gravity

It can be inferred that pressure is directly proportional to height. Hence, the blood pressure in human vessels depends on the height of the blood column in the body. The height of the blood column is more at the feet than it is at the brain. Hence, jthe blood pressure at the feet is more than it is at the brain.

(b) Density of air is the maximum near the sea level. Density of air decreases with increase in height from the surface. At a height of about 6 km, density decreases to nearly half of its value at the sea level. Atmospheric pressure is proportional to density. Hence, at a height of 6 km from the surface, it decreases to nearly half of its value at the sea level.

(c) When force is applied on a liquid, the pressure in the liquid is transmitted in all directions. Hence, hydrostatic pressure does not have a fixed direction and it is a scalar physical quantity.

Question 2.
Explain why
(a) The angle of contact of mercury with glass is obtuse, while that of water with glass is acute.
(b) Water on a clean glass surface tends to spread out while mercury on the same surface tends to form drops. (Put differently, water wets glass while mercury does not.)
(c) Surface tension of a liquid is independent of the area of the surface
(d) Water with detergent dissolved in it should have small angles of contact.
(e) A drop of liquid under no external forces is always spherical in shape
Solution:
(a) The angle between the tangent to the liquid surface at the point of contact and the surface inside the liquid is called the angle of contact (0), as shown in the given figure.

Sla, Ssa, and Ssl are the respective interfacial tensions between the liquid-air, solid-air, and solid-liquid interfaces. At the line of contact, the surface forces between the three media must be in equilibrium, i. e.,
cos θ = $$\frac{S_{s a}-S_{s l}}{S_{l a}}$$
The angle of contact 0, is obtuse if Ssa < Sla (as in the case of mercury on glass). This angle is acute if Ss < Sa (as in the case of water on glass).

(b) Mercury molecules (which make an obtuse angle with glass) have a strong force of attraction between themselves and a weak force of attraction toward solids. Hence, they tend to form drops. On the other hand, water molecules make acute angles with glass. They have a weak force of attraction between themselves and a strong force of attraction toward solids. Hence, they tend to spread out.

(c) Surface tension is the force acting per unit length at the interface between the plane of a liquid and any other surface. This force is independent of the area of the liquid surface. Hence, surface tension is also independent of the area of the liquid surface.

(d) Water with detergent dissolved in it has small angles of contact (0). This is because for a small 0, there is a fast capillary rise of the detergent in the cloth. The capillary rise of a liquid is directly proportional to the cosine of the angle of contact (0). If 0 is small, then cos 0 will be large and the rise of the detergent water in the cloth will be fast.

(e) A liquid tends to acquire the minimum surface area because of the presence of surface tension. The surface area of a sphere is the minimum for a given volume. Hence, under no external forces, liquid drops always take spherical shape.

Question 3.
Fill in the blanks using the word(s) from the list appended with each statement:
(a) Surface tension of liquids generally…with temperatures (increases/ decreases)
(b) Viscosity of gases …………………. with temperature, whereas viscosity of liquids ………………………… with temperature (increases/decreases)
(c) For solids with elastic modulus of rigidity, the shearing force is proportional to …………………………… while for fluids it is proportional to ………………………………… (shear strain/rate of shear strain)
(d) For a fluid in a steady flow, the increase in flow speed at a constriction follows (conservation of mass / Bernoulli’s principle)
(e) For the model of a plane in a wind tunnel, turbulence occurs at a ………………………….. speed for turbulence for an actual plane (greater /smaller)
Solution:
(a) decreases
The surface tension of a liquid is inversely proportional to temperature.

(b) increases; decreases
Most fluids offer resistance to their motion. This is like internal mechanical friction, known as viscosity. Viscosity of gases increases with temperature, while viscosity of liquids decreases with temperature.

(c) shear strain; rate of shear strain
With reference to the elastic modulus of rigidity for solids, the shearing force is proportional to the shear strain. With reference to the elastic modulus of rigidity for fluids, the shearing force is proportional to the rate of shear strain.

(d) conservation of mass/Bernoulli’s principle
For a steady-flowing fluid, an increase in its flow speed at a constriction follows the conservation of mass/Bemoulli’s principle.

(e) greater
For the model of a plane in a wind tunnel, turbulence occurs at a greater speed than it does for an actual plane. This follows from Bernoulli’s principle and different Reynolds numbers are associated with the motions of the two planes. ,

Question 4.
Explain why
(a) To keep a piece of paper horizontal, you should blow over, not under, it
(b) When we try to close a water tap with our fingers, fast jets of water gush through the openings between our fingers
(c) The size of the needle of a syringe controls flow rate better than the thumb pressure exerted by a doctor while administering an injection
(d) A fluid flowing out of a small hole in a vessel results in a backward thrust on the vessel
(e) A spinning cricket ball in air does not follow a parabolic trajectory
(a) When air is blown under a paper, the velocity of air is greater under the paper than it is above it. As per Bernoulli’s principle, atmospheric pressure reduces under the paper. This makes the paper fall. To keep a piece of paper horizontal, one should blow over it. This increases the velocity of air above the paper. As per Bernoulli’s principle, atmospheric pressure reduces above the paper and the paper remains horizontal.

(b) According to the equation of continuity,
Area x Velocity = Constant
For a smaller opening, the velocity of flow of a fluid is greater than it is when the opening is bigger. When we try to close a tap of water with our fingers, fast jets of water gush through the openings between our fingers. This is because very small openings are left for the water to flow out of the pipe. Hence, area and velocity are inversely proportional to each other.

(c) The small opening of a syringe needle controls the velocity of the blood flowing out. This is because of the equation of continuity. At the constriction point of the syringe system, the flow rate suddenly increases to a high value for a constant thumb pressure applied.

(d) When a fluid flows out from a small hole in a vessel, the vessel receives a backward thrust. A fluid flowing out from a small hole has a large velocity according to the equation of continuity,
Area x Velocity = Constant
According to the law of conservation of momentum, the vessel attains a backward velocity because there are no external forces acting on the system.

(e) A spinning cricket ball has two simultaneous motions-rotatory and linear. These two types of motion oppose the effect of each other. This decreases the velocity of air flowing below the ball. Hence, the pressure on the upper side of the ball becomes lesser than that on the lower side. An upward force acts upon the ball. Therefore, the ball takes a curved path. It does not follow a parabolic path.

Question 5.
A 50 kg girl wearing high heel shoes balances on a single heel. The heel is circular with a diameter 1.0 cm. What is the pressure exerted by the heel on the horizontal floor?
Solution:
Mass of the girl, m = 50 kg
Diameter of the heel, d = 1 cm = 0.01 m
Radius of the heel, r = $$\frac{d}{2}$$ = 0.005 m
Area of the heel = πr²
= 3.14 x (0.005)2
= 7.85 x 10-5 m2

Force exerted by the heel on the floor,
F = mg
= 50 x 9.8 = 490 N
Pressure exerted by the heel on the floor,
P = $$\frac{\text { Force }}{\text { Area }}$$
= $$\frac{490}{7.85 \times 10^{-5}}$$ = 6.24 x 106Nm-2
Therefore, the pressure exerted by the heel on the horizontal floor is 6.24 x 106Nm-2 .

Question 6.
Torieelli’s barometer used mercury. Pascal duplicated it using French wine of density 984kg m3. Determine the height of the wine column for normal atmospheric pressure.
Solution:
Density of mercury, ρ1 = 13.6 x 103 kg / m3
Height of the mercury column, h1 = 0.76 m
Density of French wine, ρ2 = 984 kg / m3
Height of the French wine column = h2
Acceleration due to gravity, g = 9.8 m / s2

The pressure in both the columns is equal, i. e.,
Pressure in the mercury column = Pressure in the French wine column
ρ1h1g = ρ2h2g
h2 = $$\frac{\rho_{1} h_{1}}{\rho_{2}}$$
= $$\frac{13.6 \times 10^{3} \times 0.76}{984}$$
= 10.5m
Hence, the height of the French wine column for normal atmospheric pressure is 10.5 m.

Question 7.
A vertical off-shore structure is built to withstand a maximum stress of 109 Pa. Is the structure suitable for putting up on top of an oil well in the ocean? Take the depth of the ocean to bet roughly 3 km, and ignore ocean currents.
Solution:
Yes The maximum allowable stress for the structure, P = 109Pa
Depth of the ocean, d = 3 km = 3 x 103 m
Density of water, ρ = 103 kg / m3
Acceleration due to gravity, g = 9.8 m / s2

The pressure exerted because of the sea water at depth, d = ρdg
= 3 x 103 x 103 x 9.8 = 2.94 x 107 Pa
The maximum allowable stress for the structure (109 Pa) is greater than the pressure of the seawater (2.94 x 107 Pa). The pressure exerted by the ocean is less than the pressure that the structure can withstand. Hence, the structure is suitable for putting up on top of an oil well in the ocean.

Question 8.
A hydraulic automobile lift is designed to lift cars with a maximum mass of 3000 kg. The area of cross-section of the piston carrying the load is 425cm2. What maximum pressure would the smaller piston have to bear?
Solution:
The maximum mass of a car that can be lifted, m = 3000 kg
Area of cross-section of the load-carrying piston, A = 425 cm2 = 425x 10-4m2
The maximum force exerted by the load, F = mg
= 3000 x 9.8 = 29400N
The maximum pressure exerted on the load-carrying piston, P = $$\frac{F}{A}$$
= $$\frac{29400}{425 \times 10^{-4}}$$
= 6.917 x 105 Pa
Pressure is transmitted equally in all directions in a liquid. Therefore, the maximum pressure that the smaller piston would have to bear is 6.917 x 105Pa.

Question 9.
A U-tube contains water and methylated spirit separated by mercury. The mercury columns in the two arms are in level with 10.0 cm of water in one arm and 12.5 cm of spirit in the other. What is the specific gravity of spirit?
Solution:
The given system of water, mercury, and methylated spirit is shown as follows:

Height of the spirit column, h1 = 12.5cm = 0.125m
Height of the water column, h2 = 10 cm = 0.1 m
P0 = Atmospheric pressure
ρ1 = Density of spirit
ρ2 = Density of water
Pressure at point B = P0 + h1ρ1g
Pressure at point D = P0 + h2ρ2g
Pressure at points B and D is the same.

Therefore, the specific gravity of spirit is 0.8.

Question 10.
In the previous problem, if 15.0 cm of water and spirit each are further poured into the respective arms of the tube, what is the difference in the levels of mercury in the two arms? (Specific gravity of mercury = 13.6)
Solution:
Height of the water column, h1 =10+15 = 25cm
Height of the spirit column, h2 = 12.5 +15 = 27.5cm
Density of water, ρ1 = 1 g cm-3
Density of spirit, ρ2 = 0.8 g cm-3
Density of mercury = 13.6 g cm-3

Let h be the difference between the levels of mercury in the two arms. Pressure exerted by height h, of the mercury column:
= hρg = h x 13.6g ……………………………….. (i)
Difference between the pressures exerted by water and spirit
h1ρ1g – h2ρ1g
= g (25 x 1 – 27.5 x 0.8) = 3g ……………………………….. (ii)
Equating equations (i) and (ii), we get
13.6 hg = 3g
h = 0.220588 ≈ 0.221 cm .
Hence, the difference between the levels of mercury in the two arms is 0.221 cm.

Question 11.
Can Bernoulli’s equation be used to describe the flow of water through a rapid in a river? Explain.
No Explanation: Bernoulli’s equation cannot be used to describe the flow of water through a rapid in a river because of the turbulent flow of water. This principle can only be applied to a streamlined flow.

Question 12.
Does it matter if one uses gauge instead of absolute pressures in applying Bernoulli’s equation? Explain.
No Explanation: It does not matter if one uses gauge pressure instead of absolute pressure while applying Bernoulli’s equation. The two points where Bernoulli’s equation is applied should have significantly different atmospheric pressures.

Question 13.
Glycerine flows steadily through a horizontal tube of length 1.5 m and radius 1.0 cm. If the amount of glycerine collected per second at one end is 4.0 x 10-3 kgs-1, what is the pressure difference between the two ends of the tube?(Density of glycerine = 1.3 x 103 kg m -3 and viscosity of glycerine = 0.83Pas). [You may also like to check if the assumption of laminar flow in the tube is correct].
Solution:
Length of the horizontal tube, l = 1.5m
Radius of the tube, r = 1 cm = 0.01 m
Diameter of the tube, d = 2r = 0.02 m
Glycerine is flowing at a rate of 4.0 x 10 kgs .
M = 4.0 x 10-3 kgs-1
Density of glycerine, ρ = 1.3 x 10-3 kg m-3
Viscosity of glycerine, η = 0.83Pas
Volume of glycerine flowing per sec,
V = $$\frac{M}{\rho}=\frac{4.0 \times 10^{-3}}{1.3 \times 10^{3}}$$
= 3.08 x 10-6 m3 s-1
According to Poisevelle’s formula, we have the relation for the rate of flow,
V = $$\frac{\pi p r^{4}}{8 \eta l}$$
where, p is the pressure difference between the two ends of the tube
∴ p = $$\frac{V 8 \eta l}{\pi r^{4}}$$
= $$\frac{3.08 \times 10^{-6} \times 8 \times 0.83 \times 1.5}{3.14 \times(0.01)^{4}}$$
= 9.8 x 102 Pa
Reynold’s number is given by the relation,
R = $$\frac{4 \rho V}{\pi d \eta}=\frac{4 \times 1.3 \times 10^{3} \times 3.08 \times 10^{-6}}{3.14 \times(0.02) \times 0.83}$$
= 0.3
Reynold’s number is about 0.3. Hence, the flow is laminar.

Question 14.
In a test experiment on a model aeroplane in a wind tunnel, the flow speeds on the upper and lower surfaces of the wing are 70ms-1 and 63 ms-1 respectively. What is the lift on the wing if its area is 2.5m2? Take the density of air to be 1.3 kg m-3.
Solution:
Speed of wind on the upper surface of the wing, V1 = 70 m/s
Speed of wind on the lower surface of the wing, V2 = 63 m/s
Area of the wing, A = 2.5 m2
Density of air, ρ = 1.3 kg m
According to Bernoulli’s theorem, we have the relation:

where, P1 = Pressure on the upper surface of the wing
P2 = Pressure on the lower surface of the wing
The pressure difference between the upper and lower surfaces of the wing provides lift to the aeroplane.
Lift on the wing = (P2 – P1 )A

Therefore, the lift on the wing of the aeroplane is 1.51 x 103N.

Question 15.
Figures (a) and (b) refer to the steady flow of a (non-viscous) liquid. Which of the two figures is incorrect? Why?

Solution:
Figure (a) is incorrect.
Take the case given in figure (b).

where, A1 = Area of pipe 1
A2 = Area of pipe 2
V1 = Speed of the fluid in pipe 1
V2 = Speed of the fluid in pipe 2
From the law of continuity, we have
A1V1 = A2V2
When the area of a cross-section in the middle of the venturi meter is small, the speed of the flow of liquid through this part is more. According to Bernoulli’s principle, if speed is more, then pressure is less. Pressure is directly proportional to height. Hence, the level of water in pipe 2 is less. Therefore, figure (a) is not possible.

Question 16.
The cylindrical tube of a spray pump has a cross-section of 8.0 cm2 one end of which has 40 fine holes each of diameter 1.0 mm. If the liquid flow inside the tube is 1.5 m min-1, what is the speed of ejection of the liquid through the holes?
Solution:
Area of cross-section of the spray pump. A = 8 cm2 = 8 x 10-4 m2
number of holes, n = 40
Diameter of each hole, d = 1 mm = 1 x 10-3 m
Radius of each hole,r = d/2 = 0.5 x 10-3 m
Area of cross-section of each hole, a = πr² = π(0.5 x 10-3)2m2
Total area of 40 holes, A2 = n x a
= 40 x 3.14 x (0.5 x 10-3)2 m2
= 31.41 x 10-6 m2

Speed of flow of liquid inside the tube, V1 = 1.5 m/min = 0.025m/s
Speed of ejection of liquid through the holes = V2
According to the law of continuity, we have A1V1 = A2V2
V2 = $$\frac{A_{1} V_{1}}{A_{2}}=\frac{8 \times 10^{-4} \times 0.025}{31.41 \times 10^{-6}}$$
= 0.636 m/s
Therefore, the speed of ejection of the liquid through the holes is 0.636 m/s.

Question 17.
A U-shaped wire is dipped in a soap solution and removed. The thin soap film formed between the wire and the light slider supports a weight of 1.5 x 10-2 N (which includes the small weight of the slider). The length of the slider is 30 cm. What is the surface tension of the film?
Solution:
The weight that the soap film supports, W = 1.5 x 10-2 N
Length of the slider, l = 30 cm = 0.3 m
A soap film has two free surfaces.
∴ Total length = 2l = 2 x 0.3 = 0.6 m
Surface tension, T = $$\frac{\text { Force or Weight }}{2 l}$$
= $$\frac{1.5 \times 10^{-2}}{0.6}$$ =  2.5 x10-2  N/m
Therefore, the surface tension of the film is 2.5 x10-2Nm-1.

Question 18.
Figure (a) shows a thin liquid film supporting a small weight = 4.5 x 10-2 N.
What is the weight supported by a film of the same liquid at the same temperature in fig. (b) and (c)? Explain your answer physically.

Solution:
Take case (a):
The length of the liquid film supported by the weight, l = 40 cm = 0.4 m
The weight supported by the film, W = 4.5 x 10-2 N
A liquid film has two free surfaces.
∴ Surface tension = $$\frac{W}{2 l}=\frac{4.5 \times 10^{-2}}{2 \times 0.4}$$ = 5.625 x 10-2 Nm-1
In all the three figures, the liquid is the same. Temperature is also the same for each case. Hence, the surface tension in figure (b) and figure (c) is the same as in figure (a), i.e., 5.625x 10 ~2Nm-1.
Since the length of the film in all the cases is 40 cm, the weight supported in each case is 4.5 x 10-2 N.

Question 19.
What is the pressure inside the drop of mercury of radius 3.00 mm at room temperature? Surface tension of mercury at that temperature (20°C) is 4.65 x 10-1 Nm-1. The atmospheric pressure is 1.01 x 105 Pa. Also give the excess pressure inside the drop.
Solution:
Radius of the mercury drop, r = 3.00 mm = 3 x 10-3 m
Surface tension of mercury, T = 4.65 x 10-1 N m-1
Atmospheric pressure, P0 = 1.01 x 105 Pa
Total pressure inside the mercury drop = Excess pressure inside mercury + Atmospheric pressure
= $$\frac{2 T}{r}+P_{0}$$

Excess pressure = $$\frac{2 T}{r}=\frac{2 \times 4.65 \times 10^{-1}}{3 \times 10^{-3}}$$ = 310 Pa

Question 20.
What is the excess pressure inside a bubble of soap solution of ‘ radius 5.00 mm, given that the surface tension of soap solution at the temperature (20°C) is 2.50 x 10-2 Nm-1? If an air bubble of the same dimension were formed at depth of 40.0 cm inside a container containing the soap solution (of relative density 1.20), what would be the pressure inside the bubble? (1 atmospheric pressure is 1.01 x 105 Pa).
Solution:
Soap bubble is of radius, r = 5.00 mm = 5 x 10-3 m
Surface tension of the soap solution, T = 2.50 x 10-2 Nm-1
Relative density of the soap solution = 1.20
∴ Density of the soap solution, ρ = 1.2 x 103 kg/m3
Air bubble formed at a depth, h = 40 cm = 0.4 m
Radius of the air bubble, r = 5 mm = 5 x 10-3 m
1 atmospheric pressure = 1.01 x 105Pa

Acceleration due to gravity, g = 9.8 m/s2
Hence, the excess pressure inside the soap bubble is given by the relation
P = $$\frac{4 T}{r}=\frac{4 \times 2.5 \times 10^{-2}}{5 \times 10^{-3}}$$ = 20 Pa
Therefore, the excess pressure inside the soap bubble is 20 Pa.
The excess pressure inside the air bubble is given by the relation
P’ = $$\frac{2 T}{r}$$
= $$\frac{2 \times 2.5 \times 10^{-2}}{5 \times 10^{-3}}$$
=10 Pa

Therefore, the excess pressure inside the air bubble is 10 Pa.
At a depth of 0.4 m, the total pressure inside the air bubble =Atmospheric pressure + hρg + P’
= 1.01 x 105 + 0.4 x 1.2 x 103 x 9.8 + 10 ,
= 1.057 x 105 Pa = 1.06 x 105 Pa
Therefore, the pressure inside the air bubble is 1.06 x 105 Pa.

Question 21.
A tank with a square base of area 1.0 m2 is divided by a vertical partition in the middle. The bottom of the partition has a small-hinged door of area 20 cm2. The tank is filled with water in one compartment, and an acid (of relative density 1.7) in the other, both to a height of 4.0 m. compute the force necessary to keep the door close.
Solution:
Base area of the given tank, A = 1.0 m2
Area of the hinged door, a = 20 cm2 = 20 x 10-4 m2
Density of water, ρ1 = 103 kg/m3
Density of acid, ρ2 = 1.7 x 103 kg/m3
Height of the water column, h1 = 4 m
Height of the acid column, h2 = 4 m
Acceleration due to gravity, g = 9.8 m/s2
Pressure due to water is given as
P1 =h1ρ1g = 4 x 103 x 9.8 = 3.92 x 104Pa
Pressure due to acid is given as, P2 = h2ρ2g
= 4 x 1.7 x103 x 9.8
= 6.664 x 104 Pa

Pressure difference between the water and acid columns,
ΔP=P2– P1
= 6.664 x 104 -3.92 x104
= 2.744 x104 Pa
Hence, the force exerted on the door = ΔP x a
= 2.744 x 104 x 20 x 10-4 = 54.88N
Therefore, the force necessary to keep the door closed is 54.88N.

Question 22.
A manometer reads the pressure of a gas in an enclosure as shown in figure (a). When a pump removes some of the gas, the manometer reads as in figure (b). The liquid used in the manometers is mercury and the atmospheric pressure is 76 cm of mercury.
(a) Give the absolute and gauge pressure of the gas in the enclosure for cases (a) and (b), in units of cm of mercury.
(b) How would the levels change in case (b) if 13.6 cm of water (immiscible with mercury) are poured into the right limb of the manometer? (Ignore the small change in the volume of the gas).

Solution:
(a) For figure (a)
Atmospheric pressure, P0 = 76 cm of Hg
The difference between the levels of mercury in the two limbs gives gauge pressure Hence, gauge pressure is 20 cm of Hg.
Absolute pressure = Atmospheric pressure + Gauge pressure
= 76+20 =96 cm of Hg

For figure (b)
Difference between the levels of mercury in the two limbs = -18 cm Hence, gauge pressure is -18 cm of Hg.
Absolute pressure = Atmospheric pressure + Gauge pressure
= 76 cm-18cm = 58 cm

(b) 13.6 cm of water is poured into the right limb of figure (b).
Relative density of mercury = 13.6
Hence, a column of 13.6 cm of water is equivalent to 1 cm of mercury. Let h be the difference between the levels of mercury in the two limbs. The pressure in the right limb is given as,
PR = Atmospheric pressure + 1 cm of Hg
= 76+1 = 77 cm of Hg …………………………. (i)
The mercury column will rise in the left limb.
Hence, pressure in the left limb,
PL = 58 + h ……………………………. (ii)
Equating equations (i) and (ii), we get
77 = 58 + h
h = 19 cm
Hence, the difference between the levels of mercury in the two limbs will be 19 cm.

Question 23.
Two vessels have the same base area but different shapes. The first vessel takes twice the volume of water that the second vessel requires to fill upto a particular common height. Is the force exerted by the water on the base of the vessel the same in the two cases? If so, why do the vessels filled with water to that same height give different readings on a weighing scale?
Yes.
Two vessels having the same base area have identical force and equal pressure acting on their common base area. Since the shapes of the two vessels are different, the force exerted on the sides of the vessels has non-zero vertical components. When these vertical components are added, the total force on one vessel comes out to be greater than that on the other vessel. Hence, when these vessels are filled with water to the same height, they give different readings on a weighing scale.

Question 24.
During blood transfusion the needle is inserted in a vein where the gauge pressure is 2000Pa. At what height must the blood container be placed so that blood may just enter the vein? [Take the density of whole blood = 1.06 x 103 kg m-3 ].
Solution:
Given, gauge pressure, P = 2000 Pa
Density of whole blood, p = 1.06 x 103 kg m-3
Acceleration due to gravity, g = 9.8 m/s2
Height of the blood container = h
Pressure of the blood container, P = hρg
h = $$\frac{P}{\rho g}=\frac{2000}{1.06 \times 10^{3} \times 9.8}$$
= 0.1925 m
The blood may enter the vein if the blood container is kept at a height greater than 0.1925m, i. e., about 0.2 m.

Question 25.
In deriving Bernoulli’s equation, we equated the work done on the fluid in the tube to its change in the potential and kinetic energy,
(a) What is the largest average velocity of blood flow in an artery of diameter 2 x 10-3 m if the flow must remain laminar?
(b) Do the dissipative forces become more important as the fluid velocity increases? Discuss qualitatively.
Solution:
(a) Diameter of the artery, d = 2×10-3 m
Viscosity of blood, η = 2.084 x 10-3 Pas
Density of blood, ρ = 1.06 x 103 kg/m3
Reynolds’ number for laminar flow, NR = 2000
The largest average velocity of blood is given as
Vavg = $$\frac{N_{R} \eta}{\rho d}$$
= $$\frac{2000 \times 2.084 \times 10^{-3}}{1.06 \times 10^{3} \times 2 \times 10^{-3}}$$
= 1.966 m/s
Therefore, the largest average velocity of blood is 1.966 m/s
(b) Yes, as the fluid velocity increases, the dissipative forces become more important. This is because of the rise of turbulence. Turbulent flow causes dissipative loss in a fluid.

Question 26.
(a) What is the largest average velocity of blood flow in an artery of radius 2 x 10-3 m if the flow must remain laminar?
(b) What is the corresponding flow rate? (Take viscosity of blood to be 2.084 x 10-3 Pas).
Solution:
(a) Radius of the artery, r = 2 x 10-3 m
Diameter of the artery, d=2 x 2x 10-3 m = 4 x 10-3m
Viscosity of blood, η = 2.084 x 10-3 Pa s
Density of blood, ρ = 1.06 x 103 kg/m3
Reynolds’ number for laminar flow, NR = 2000
The largest average velocity of blood is given by the relation
VAvg = $$\frac{N_{R} \eta}{\rho d}=\frac{2000 \times 2.084 \times 10^{-3}}{1.06 \times 10^{3} \times 4 \times 10^{-3}}$$
= 0.983 m/s
Therefore, the largest average velocity of blood is 0.98,3 m/s.

(b) Flow rate is given by the relation
R = πr² Vavg
= 3.14 x (2 x 10-3)2 x 0.983
= 1.235 x 10-5m3s-1
Therefore, the corresponding flow rate is 1.235 x 10-5m3s-1.

Question 27.
A plane is in level flight at constant speed and each of its two wings has an area of 25m2. If the speed of the air is 180km/h over the lower wing and 234 km/h over the upper wing surface, determine the plane’s mass. (Take air density to be 1kg m-3).
Solution:
The area of the wings of the plane, A = 2 x 25 = 50 m2
Speed of air over the lower wing,
V1 = 180 km/h = 180 x $$\frac{5}{18}$$ m/s = 50 m/s
Speed of air over the upper wing,
V2 = 234 km/h = 234 x $$\frac{5}{18}$$ m/s = 65 m/s
Density of air, ρ = 1 kg m-3
Pressure of air over the lower wing = P1
Pressure of air over the upper wing = P2
The upward force on the plane can be obtained using Bernoulli’s equation as

The upward force (F) on the plane can be calculated as

Using Newton’s force equation, we can obtain the mass (m) of the plane as
F = mg
m = $$\frac{43125}{9.8}$$
= 4400.51 kg ≈ 4400 kg
Hence, the mass of the plane is about 4400 kg.

Question 28.
In Millikan’s oil drop experiment, what is the terminal speed of an uncharged drop of radius 2.0 x 10-5 m and density 1.2 x 10-5 kg m?
Take the viscosity of air at the temperature of the experiment to be 1.8x 105 Pas. How much is the viscous force on the drop at that speed? Neglect buoyancy of the drop due to air.
Solution:
Terminal speed = 5.8cm/s; Viscous force = 3.9 x 10-10 N
Radius of the given uncharged drop, r = 2.0 x 10-5 m
Density of the uncharged drop, ρ = 1.2 x 103 kg m-3
Viscosity of air, η = 1.8 x 10-5 Pa s
Density of air (ρ0) can be taken as zero in order to neglect buoyancy of air.
Acceleration due to gravity, g = 9.8 m/s2
Terminal velocity (ν) is given by the relation
ν = $$\frac{2 r^{2} \times\left(\rho-\rho_{0}\right) g}{9 \eta}$$
= $$\frac{2 \times\left(2.0 \times 10^{-5}\right)^{2}\left(1.2 \times 10^{3}-0\right) \times 9.8}{9 \times 1.8 \times 10^{-5}}$$
= 5.807 x 10-2ms-1
= 5.8 cm s-1
Hence, the terminal speed of the drop is 5.8 cms-1.
The viscous force on the drop is given by:
F = 6πηrν
∴ F = 6 x 3.14 x 1.8 x 10-5 x 2.0 x 10-5 x 5.8 x 10-2
= 3.9 x 10-10N
Hence, the viscous force on the drop is 3.9 x 10-10N.

Question 29.
Mercury has an angle of contact equal to 140° with soda-lime glass. A narrow tube of radius 1.00 mm made of this glass is dipped in a trough containing mercury. By what amount does the mercury dip down in the tube relative to the liquid surface outside? Surface tension of mercury at the temperature of the experiment is 0.465 Nm-1. Density of mercury = 13.6 x 103 kgm-3.
Solution:
Angle of contact between mercury and soda-lime glass, θ = 140°
Radius of the narrow tube, r = 1 mm = 1 x 10-3 m
Surface tension of mercury at the given temperature, T = 0.465N m-1
Density of mercury, ρ = 13.6 x 103 kg/m3
Dip in the height of mercury = h

Acceleration due to gravity, g = 9.8 m/s2
Surface tension is related with the angle of contact and the dip in the height as
T = $$\frac{h \rho g r}{2 \cos \theta}$$
∴ h = $$\frac{2 T \cos \theta}{r \rho g}$$

= -5.34 mm
Here, the negative sign shows the decreasing level of mercury. Hence, the mercury level dips by 5.34 mm

Question 30.
Two narrow bores of diameters 3.0mm and 6.0mm are joined together to form a U-tube open at both ends. If the U-tube contains water, what is the difference in its levels in the two limbs of the tube?
Surface tension of water at the temperature of the experiment is 7.3 x 10-2Nm-1.
Take the angle of contact to be zero and density of water to be 1.0x 103 kg m-3 (g = 9.8ms-2).
Solution:
Diameter of the first bore, d1 = 3.0 mm = 3 x 10-3 m
Hence, the radius of the first bore, r1 = $$\frac{d_{1}}{2}$$ =1.5 x 10-3m
Diameter of the second bore, d2 =6.0 mm
Hence, the radius of the second bore, r2 = $$\frac{d_{2}}{2}$$ = 3 x 10-3 m
Surface tension of water, T = 7.3 x 10-2 N m-1
Angle of contact between the bore surface and water, θ=0
Density of water, ρ = 1.0 x 103 kg/m-3
Acceleration due to gravity, g =9.8 m/s2
Let h1 and h2 be the heights to which water rises in the first and second tubes respectively.

These heights are given by the relations
h1 = $$\frac{2 T \cos \theta}{r_{1} \rho g}$$ …………………..(i)
h2 = $$\frac{2 T \cos \theta}{r_{2} \rho g}$$ …………………… (ii)
The difference between the levels of water in the two limbs of the tube can be calculated as
= h1 – h2
= $$\frac{2 T \cos \theta}{r_{1} \rho g}-\frac{2 T \cos \theta}{r_{2} \rho g}$$

= 4.966 x 10-3m = 4.97 mm
Hence, the difference between levels of water in the two bores is 4.97 mm.

Question 31.
(a) It is known that density p of air decreases with height y as ρ = ρoe-y/yo
where ρo = 1.25kg m-3 is the density at sea level, and a constant. This density variation is called the law of atmospheres. Obtain this law assuming that the temperature of atmosphere remains a constant (isothermal conditions). Also assume that the value of g remains constant.

(b) A large He balloon of volume 1425m3 is used to lift a payload of 400 kg. Assume that the balloon maintains constant radius as it rises. How high does it rise?
[Take y0 =8000m and ρHe = 018 kg m-3]
Solution:
Volume of the balloon, V = 1425m3
Mass of the payload, m = 400 kg
Acceleration due to gravity, g = 9.8 m/s2
yo =8000m
ρHe =0.18kgm-3
ρo =1.25kg/m3

Density of the balloon = ρ
Height to which the balloon rises = y
Density (ρ) of air decreases with height (y) as
ρ = ρ0e-y/yo
$$\frac{\rho}{\rho_{0}}=e^{-y / y_{0}}$$ …………………………… (i)

This density variation is called the law of atmospheres.
It can be inferred from equation (i) that the rate of decrease of density with height is directly proportional to ρ, i. e.,

where, k is the constant of proportionality
Height changes from 0 to y, while density changes from ρo to ρ).
Integrating the sides between these limits, we get

Comparing equations (i) and (ii) we get
y0 = $$\frac{1}{k}$$
k = $$\frac{1}{y_{0}}$$ ……………………………………. (iii)

From equations (ii) and (iii), we get
ρ = ρ0e-y/yo
(b) Density,
ρ = $$\frac{\text { Mass }}{\text { Volume }}$$

= 0.46 kg/m3
From equations (ii) arid (iii), we can obtain y as
ρ = ρ0e-y/yo
log e$$\frac{\rho}{\rho_{0}}=-\frac{y}{y_{0}}$$
∴ y =-8000 x loge $$\frac{0.46}{1.25}$$
=-8000 x-1=8000m8 km
Hence, the balloon will rise to a height of 8 km.

## PSEB 11th Class Physics Solutions Chapter 5 Laws of Motion

Punjab State Board PSEB 11th Class Physics Book Solutions Chapter 5 Laws of Motion Textbook Exercise Questions and Answers.

## PSEB Solutions for Class 11 Physics Chapter 5 Laws of Motion

PSEB 11th Class Physics Guide Laws of Motion Textbook Questions and Answers

Question 1.
Give the magnitude and direction of the net force acting on
(a) a drop of rain falling down with a constant speed,
(b) a cork of mass 10 g floating on water,
(c) a kite skillfully held stationary in the sky,
(d) a car moving with a constant velocity of 30 km/h on a rough road,
(e) a high-speed electron in space far from all material objects, and free of electric and magnetic fields.
(a) Zero net force
The rain drop is falling with a constant speed. Hence, its acceleration is zero. As per Newton’s second law of motion, the net force acting on the rain drop is zero.

(b) Zero net force
The weight of the cork is acting downward. It is balanced by the buoyant force exerted by the water in the upward direction. Hence, no net force is acting on the floating cork.

(c) Zero net force
The kite is stationary in the sky, i. e., it is not moving at all. Hence, as per Newton’s first law of motion, no net force is acting on the kite.

(d) Zero net force
The car is moving on a rough road with a constant velocity. Hence, its acceleration is zero. As per Newton’s second law of motion, no net force is acting on the car.

(e) Zero net force
The high speed electron is free from the influence of all fields. Hence, no net force is acting on the electron.

Question 2.
A pebble of mass 0.05 kg is thrown vertically upwards. Give the direction and magnitude of the net force on the pebble,
(a) during its upward motion,
(b) during its downward motion,
(c) at the highest point where it is momentarily at rest. Do your answers change if the pebble was thrown at an angle of 45° with the horizontal direction?
Ignore air resistance.
Solution:
Acceleration due to gravity, irrespective of the direction of motion of an object, always acts downward. The gravitational force is the only force that acts on the pebble in all three cases. Its magnitude is given by Newton’s second law of motion as:
F = m × a
Where,
F = Net force
m = Mass of the pebble = 0.05 kg
a = g =10 m/s2
F =0.05 × 10 =0.5 N
The net force on the pebble in all three cases is 0.5 N and this force acts in the downward direction.

If the pebble is thrown at an angle of 45° with the horizontal, it will have both the horizontal and vertical components of velocity. At the highest point, only the vertical component of velocity becomes zero. However, the pebble will have the horizontal component of velocity throughout its motion. This component of velocity produces no effect on the net force acting on the pebble.

Question 3.
Give the magnitude and direction of the net force acting on a stone of mass 0.1 kg,
(a) just after it is dropped from the window of a stationary train,
(b) just after it is dropped from the window of a train running at a constant velocity of 36 km/h,
(c) just after it is dropped from the window of a train accelerating with 1 ms-2,
(d) lying on the floor of a train which is accelerating with 1 ms-2, the stone being at rest relative to the train.
Neglect air resistance throughout.
Solution:
(a) Mass of the stone, m = 0.1 kg
Acceleration of the stone, a = g = 10 m/s2
As per Newton’s second law of motion, the net force acting on the stone,
F = ma = mg = 0.1 × 10 = 1 N
Acceleration due to gravity always acts in the downward direction.
The net force acting on the stone is because of acceleration due to gravity and it always acts vertically downward. The magnitude of this force is 1 N.

(b) The train is moving with a constant velocity. Hence, its acceleration is zero in the direction of its motion, i.e., in the horizontal direction. Hence, no force is acting on the stone in the horizontal direction.

(c) It is given that the train is accelerating at the rate of 1 m/s2 .
Therefore, the net force acting on the stone, F’ = ma = 0.1 × 1 = 0.1 N
This force is acting in the horizontal direction. Now, when the stone is dropped, the horizontal force F’, stops acting on the stone. This is because of the fact that the force acting on a body at an instant depends on the situation at that instant and not on earlier situations.

Therefore, the net force acting on the stone is given only by acceleration due to gravity.
F = mg = 1 N
This force acts vertically downward.

(d) The weight of the stone is balanced by the normal reaction of the floor. The only acceleration is provided by the horizontal motion of the train.
Acceleration of the train, a = 0.1 m/s2
The net force acting on the stone will be in the direction of motion of the train. Its magnitude is given by:
F = ma
= 0.1 × 1 = 0.1 N

Question 4.
One end of a string of length l is connected to a particle of mass m and the other to a small peg on a smooth horizontal table. If the particle moves in a circle with speed υ the net force on the
particle (directed towards the centre) is:
(i) T,
(ii) T – $$\frac{m v^{2}}{l}$$,
(iii) T + $$\frac{m v^{2}}{l}$$,
(iv) 0
T is the tension in the string. [Choose the correct alternative].
Solution:
(i) When a particle connected to a string revolves in a circular path around a centre, the centripetal force is provided by the tension produced , in the string. Hence, in the given case, the net force on the particle is the tension T, i. e.,
F = T = $$\frac{m v^{2}}{l}$$
where F is the net force acting on the particle.

Question 5.
A constant retarding force of 50 N is applied to a body of mass 20 kg moving initially with a speed of 15 ms-1. How long does the body take to stop?
Solution:
Retarding force, F = -50 N
Mass of the body, m = 20 kg
Initial velocity of the body, u = 15 m/s
Final velocity of the body, υ = 0
Using Newton’s second law of motion, the acceleration (a) produced in the body can be calculated as:
F = ma
-50 = 20 × a
∴ a = $$\frac{-50}{20}$$ = -2.5 m/s2
20
Using the first equation of motion, the time (t) taken by the body to come to rest can be calculated as:
υ = u + at
t = $$\frac{-u}{a}=\frac{-15}{-2.5}$$ = 6s

Question 6.
A constant force acting on a body of mass 3.0 kg changes its speed from 2.0 ms-1 to 3.5 ms-1 in 25 s. The direction of the motion of the body remains unchanged. What is the magnitude and direction of the force?
Solution:
Mass of the body, m = 3 kg
Initial speed of the body, u = 2 m/s
Final speed of the body, υ = 3.5 m/s Time,
Time t = 25 s
Using the first equation of motion, the acceleration (a) produced in the body can be calculated as:
υ = u + at
∴ a = $$\frac{v-u}{t}$$
= $$\frac{3.5-2}{25}=\frac{1.5}{25}$$ = 0.06 m/s2
As per Newton’s second law of motion, force is given as:
F = ma
= 3 × 0.06 = 0.18 N
Since the application of force does not change the direction of the body, the net force acting on the body is in the direction of its motion.

Question 7.
A body of mass 5 kg is acted upon by two perpendicular forces 8 N and 6 N. Give the magnitude and direction of the acceleration of the body.
Solution:
Mass of the body, m = 5 kg
The given situation can be represented as follows:

The resultant of two forces is given as:
R = $$\sqrt{(8)^{2}+(-6)^{2}}=\sqrt{64+36}$$ = 10N
θ is the angle made by R with the force of 8 N
∴ θ = tan-1 ($$\frac{-6}{8}$$) = -36.87°
The negative sign indicates that 0 is in the clockwise direction with respect to the force of magnitude 8 N.
Hence, the magnitude of the acceleration is 2 m/s2, at an angle of 37° with a force of 8 N.
As per Newton’s second law of motion, the acceleration (a) of the body is given as :
F = ma
a = $$\frac{F}{m}=\frac{10}{5}$$ = 2m/s2
Hence, the magnitude of the acceleration is 2 m/s2, at an angle of 37° with a force of 8 N.

Question 8.
The driver of a three-wheeler moving with a speed of 36 km/h sees a child standing in the middle of the road and brings his vehicle to rest in 4.0 s just in time to save the child.
What is the average retarding force on the vehicle? The mass of the three-wheeler is 400 kg and the mass of the driver is 65 kg.
Solution:
Initial speed of the three-wheeler, u = 36 km/h
Final speed of the three-wheeler, υ = 10 m/s
Time, t = 4s
Mass of the three-wheeler, m = 400 kg
Mass of the driver, = m’ = 65 kg
Total mass of the system, M = 400 + 65 = 465 kg

Using the first law of motion, the acceleration (a) of the three-wheeler can be calculated as:
= u + at
a = $$\frac{v-u}{t}=\frac{0-10}{4}$$ = -2.5 m/s2
The negative sign indicates that the velocity of the three-wheeler is decreasing with time.
Using Newton’s second law of motion, the net force acting on the three-wheeler can be calculated as:
F = Ma
= 465 × (-2.5) = -1162.5 N
= -1.2 × 103 N
The negative sign indicates that the force is acting against the direction of motion of the three-wheeler.

Question 9.
A rocket with a lift-off mass 20,000 kg is blasted upwards with an initial acceleration of 5.0 ms-2. Calculate the initial thrust (force) of the blast.
Solution:
Mass of the rocket, m = 20,000 kg
Initial acceleration, a = 5 m/s2
Acceleration due to gravity, g = 10 m/s2
Using Newton’s second law of motion, the net force (thrust) acting on the rocket is given by the relation:
F – mg = ma
F = m(g + a)
= 20000 × (10 + 5)
= 20000 × 15 = 3 × 105 N

Question 10.
A body of mass 0.40 kg moving initially with a constant speed of 10 m s-1 to the north is subject to a constant force of 8.0 N directed towards the south for 30 s. Take the instant the force is applied to be t = 0, the position of the body at that time to be t = 0, and predict its position at t = -5 s, 25 s, 100 s.
Solution:
Mass of the body, m = 0.40 kg
Initial speed of the body, u = 10 m/s due north
Force acting on the body, F = -8.0 N
Acceleration produced in the body, a = $$\frac{F}{m}=\frac{-8.0}{0.40}$$ = -20 m/s2
At t = -5 s
Acceleration, a’ = 0 and u = 10 m/s
s = ut + $$\frac{1}{2}$$ a’t2
= 10 × (-5) + 0
= -50 m

At t = 258
Acceleration, a” = -20 m/s2
and u = 10 m/s
s’ =ut + $$\frac{1}{2}$$ a” t2
= 10 × 25 + $$\frac{1}{2}$$ × (-20) × (25)2
= 250 – 6250 = -6000 m

At t = 100 s
For 0 ≤ t ≤ 30 s
a = -20 m/s2
u = 10 m/s
s1 = ut + $$\frac{1}{2}$$ a”t2
= 10 × 30 + $$\frac{1}{2}$$ × (-20) × (30)2
= 300 – 9000
= -8700 m
For 30 < t ≤ 100 s
As per the first equation of motion, for t = 30 s, final velocity is given as:
υ = u + at
= 10 + (-20) × 30 =-590 m/s
Velocity of the body after 30 s = -590 m/s
Distance travelled in time interval from t = 30 s to t =100 s
s2 = υt
= -590 × 70 = -41300 m
.’.Total distance, s” = s1 + s2 = -8700 – 41300 = -50000 m = -50 km

Question 11.
A truck starts from rest and accelerates uniformly at 2.0 ms-2. At t = 10 s, a stone is dropped by a person standing on the top of the truck (6 m high from the ground). What are the (a) velocity, and (b) acceleration of the stone at t = 11 s? (Neglect air resistance.)
Solution:
(a) Initial velocity of the truck, u = 0
Acceleration, a = 2 m/s2
Time, t = 10 s
As per the first equation of motion, final velocity is given as:
υ = u + at
= 0 + 2 × 10 =20 m/s
The final velocity of the truck and hence, of the stone is 20 m/s.
At t = 11 s, the horizontal component (υx ) of velocity, in the absence of air resistance, remains unchanged, i.e.,
υx = 20 m/s
The vertical component (υy) of velocity of the stone is given by the first equation of motion as :
υy = u + ayδt
where, δt = 11 – 10 = 1 s
and ay = g = 10 m/s2
υy = 0 + 10 × 1 =10 m/s
The resultant Velocity (υ) of the stone is given as:

υ = $$\sqrt{v_{x}^{2}+v_{y}^{2}}$$
= $$\sqrt{20^{2}+10^{2}}=\sqrt{400+100}$$
= $$\sqrt{500}$$ = 22.36 m/s

Hence, velocity is 22.36 mIs, at an angle of 26.57° with the motion of the truck.

b) Let θ be the angle made by the resultant velocity with the horizontal component of velocity, υx
∴ tanθ = ($$\frac{v_{y}}{v_{x}}$$)
θ = tan-1($$\frac{10}{20}$$)
= tan-1 (0.5)
= 26.57°
When the stone is dropped from the truck, the horizontal force acting on it becomes zero. However, the stone continues to move under the influence of gravity. Hence, the acceleration of the stone is 10 m/s2 and it acts vertically downward.

Question 12.
A bob of mass 0.1 kg hung from the ceiling of a room by a string 2 m long is set into oscillation. The speed of the bob at its mean position is 1 ms-1. What is the trajectory of the bob if the String is cut when the bob is (a) at one of its extreme positions, (b) at its mean position.
Solution:
(a) At the extreme position, the velocity of the bob becomes zero. If the string is cut at this moment, then the bob will fall vertically on the ground.

(b) At the mean position, the velocity of the bob is 1 m/s. The direction of this velocity is tangential to the arc formed by the oscillating bob. If the bob is cut at the mean position, then it will trace a projectile path having the horizontal component of velocity only. Hence, it will follow a parabolic path.

Question 13.
A man of mass 70 kg stands on a weighing scale in a lift which is moving
(a) upwards with a uniform speed of 10 ms-1,
(b) downwards with a uniform acceleration of 5 m s-2,
(c) upwards with a uniform acceleration of 5ms-2.
What would be the readings on the scale in each case?
(d) What would be the reading if the lift mechanism failed and it hurtled down freely under gravity?
Solution:
(a) Mass of the man, m = 70 kg
Acceleration, a = 0
Using Newton’s second law of motion, we can write the equation of motion as:
R – mg = ma
where, ma is the net force acting on the man.
As the lift is moving at a uniform speed, acceleration a = 0
∴ R = mg
= 70 × 10= 700 N
∴ Reading on the weighing scale = $$\frac{700}{g}=\frac{700}{10}$$ 70 kg

(b) Mass of the man, m = 70 kg
Acceleration, a = 5 m/s2 downward
Using Newton’s second law of motion, we can write the equation of motion as:
R + mg = ma
R = m(g – a)
= 70 (10 – 5) = 70 × 5
= 350 N
Reading on the weighing scale = $$\frac{350}{g}=\frac{350}{10}$$ = 35 kg

(c) Mass of the man, m = 70 kg
Acceleration, a = 5 m/s2 upward
Using,Newton’s second law of motion, we can write the equation of motion as:
R – mg = ma
R = m(g + a)
= 70 (10 + 5)
= 70 × 15 = 1050 N
∴ Reading on the weighing scale = $$\frac{1050}{g}=\frac{1050}{10}$$ = 105 kg

(d) When the lift moves freely under gravity, acceleration a = g
Using Newton’s second law of motion, we can write the equation of motion as:
R + mg = ma
R = m(g – a)
= m(g – g) = 0
∴ Reading on the weighing scale = $$\frac{0}{g}$$ = 0 kg
The man will be in a state of weightlessness.

Question 14.
Following figure shows the position-time graph of a particle of mass 4 kg. What is the (a) Force on the particle for t< 0, t > 4 s, 0< t< 4s? (b) impulse at f = 0 and f = 4s? (Consider one-dimensional motion only).

Solution:
(a) For t < 0 It can be observed from the given graph that the position of the particle is coincident with the time axis. It indicates that the displacement of the particle in this time interval is zero. Hence, the force acting on the particle is zero. For t > 4 s
It can be observed from the given graph that the position of the particle is parallel to the time axis. It indicates that the particle is at rest at a distance of 3 m from the origin. Hence, no force is acting on the particle.
For 0 < t < 4
It can be observed that the given position-time graph has a constant slope. Hence, the acceleration produced in the particle is zero. Therefore, the force acting on the particle is zero.

(b) At t = 0
Impulse = Change in momentum
= mυ – mu
Mass of the particle, m = 4 kg
Initial velocity of the particle, u = 0
Final velocity of the particle, υ = $$\frac{3}{4}$$ m/s
∴ Impulse = ($$\frac{3}{4}$$ – 0) = 3 kg m/s
At t = 4s
Initial velocity of the particle, u = $$\frac{3}{4}$$ m/s
Final velocity of the particle, υ = 0
∴ Impulse = 4(0 – $$\frac{3}{4}$$) = -3 kg m/s

Question 15.
Two bodies of masses 10 kg and 20 kg respectively kept on a smooth, horizontal surface are tied to the ends of a light string. A horizontal force F = 600 N is applied to (i) A, (ii) B, along the direction of string. What is the tension in the string in each case?
Solution:
Horizontal force, F = 600 N
Mass of body A, m1 = 10 kg
Mass of body B, m2 = 20 kg
Total mass of the system, m = m1 + m2 = 30 kg
Using Newton’s second law of motion, the acceleration (a) produced in the system can be calculated as :
F = ma
∴ a = $$\frac{F}{m}=\frac{600}{30}$$ = 20 m/s2
When force F is applied on body A:

The equation of motion can be written as:
F – T = m1a
∴ T = F – m1a
= 600 – 10 × 20 =400 N
When force F is applied on body B:

The equation of motion can be written as:
F – T = m2a
T = F – m2a
∴ T =600 – 20 × 20 = 200 N

Question 16.
Two masses 8 kg and 12 kg are connected at the two ends of a light inextensible string that goes over a frictionless pulley. Find the acceleration of the masses, and the tension in the string when the masses are released.
Solution:
The given system of two masses and a pulley can be represented as shown in the following figure:

Smaller mass, m1 = 8 kg
Larger mass, m2 = 12 kg
Tension in the string = T
Mass m2, owing to its weight, moves downward with acceleration a and mass m1 moves upward.
Applying Newton’s second law of motion to the system of each mass:
For mass m1:
The equation of motion can be written as:
T – m1g = ma ……………. (i)

For mass m2:
m2g – T = m2 a ………………. (ii)
Adding equations (i) and (ii),we get:
(m2 – m1)g = (m1 + m2)a
∴ a = [Latex](\frac{m_{2}-m_{1}}{m_{1}+m_{2}}[/Latex]) g
= ($$\frac{12-8}{12+8}$$) × 10 = $$\frac{4}{20}$$ × 10 = 2m/s2
Therefore, the acceleration of the masses is 2 m/s2 .
Substituting the value of a in equation (ii), we get:

Therefore, the tension in the string is 96 N.

Question 17.
A nucleus is at rest in the laboratory frame of reference. Show that if it disintegrates into two smaller nuclei the products must move in opposite directions.
Solution:
Let m, m1 and m2 be the respective masses of the parent nucleus and the two daughter nuclei. The parent nucleus is at rest.
Initial momentum of the system (parent nucleus) = 0
Let υ1 and υ2 be the respective velocities of the daughter nuclei having masses m1 and m2.
Total linear momentum of the system after disintegration
= m1 υ1 + m2υ2
According to the law of conservation of momentum,
Total initial momentum = Total final momentum
0 = m1υ1+ m2υ2
υ1 = $$\frac{-m_{2} v_{2}}{m_{1}}$$
Here, the negative sign indicates that the fragments of the parent nucleus move in directions opposite to each other.

Question 18.
Two billiard balls each of mass 0.05 kg moving in opposite directions with speed 6 ms-1 collide and rebound with the same speed. What is the impulse imparted to each hall due to the other?
Solution:
Mass of each ball = 0.05 kg
Initial velocity of each ball = 6 m/s
Magnitude of the initial momentum of each ball, pi = 0.3 kg m/s
After collision, the balls change their directions of motion without changing the magnitudes of their velocity.
Final momentum of each ball, pf = -0.3 kg m/s
Impulse imparted to each ball = Change in the momentum of the system
= Pf – Pi
= -0.3 -0.3 = -0.6 kg m/s
The negative sign indicates that the impulses imparted to the balls are opposite in direction.

Question 19.
A shell of mass 0.020 kg is fired by a gun of mass 100 kg. If the muzzle speed of the shell is 80 m s-1, what is the recoil speed of the gun?
Solution:
Mass of the gun, M = 100 kg
Mass of the shell, m = 0.020 kg
Muzzle speed of the shell, υ = 80 m/s
Recoil speed of the gun = V
Both the gun and the shell are at rest initially.
Initial momentum of the system = 0
Final momentum of the system = mυ – MV
Here, the negative sign appears because the directions of the shell and the gun are opposite to each other.
According to the law of conservation of momentum,
Final momentum = Initial momentum
mυ – MV = 0
∴ V = $$\frac{m v}{M}$$
= $$\frac{0.020 \times 80}{100}$$ = 0.016 M/S
= 1.6 cm/s

Question 20.
A batsman deflects a ball by an angle of 45° without changing its initial speed which is equal to 54 km/h. What is the impulse imparted to the ball? (Mass of the ball is 0.15 kg.)
Solution:
The given situation can be represented as shown in the following figure.

where,
AO = Incident path of the ball
OB = Path followed by the ball after deflection
∠AOB = Angle between the incident and deflected paths of the ball = 45°
∠AOP = ∠BOP = 22.5° = θ
Initial and final velocities of the ball = υ
Horizontal component of the initial velocity = υcosθ along RO
Vertical component of the initial velocity = υ sinθ along PO
Horizontal component of the final velocity = υ cosθ along OS
Vertical component of the final velocity = υ sinθ along OP
The horizontal components of velocities suffer no change. The vertical components of velocities are in the opposite directions.
∴ Impulse imparted to the ball
= Change in the linear momentum of the ball
= m υcosθ – (-mυ cosθ)
= 2mυ cosθ
Mass of the ball, m = 0.15 kg
Velocity of the ball, υ = 54 km/h = 54 × $$\frac{5}{18}$$ m/s = 15 m/s
∴ Impulse = 2 × 0.15 × 15 cos 22.5° = 4.5 × 0.9239 = 4.16 kg m/s

Question 21.
A stone of mass 0.25 kg tied to the end of a string is whirled round in a circle of radius 1.5 m with a speed of 40 rev./min in a horizontal plane. What is the tension in the string? What is the maximum speed with which the stone can be whirled around if the string can withstand a maximum tension of 200 N?
Solution:
Mass of the stone, m = 0.25 kg
Radius of the circle, r = 1.5 m
Number of revolution per second, n = $$\frac{40}{60}=\frac{2}{3}$$ rps

Angular velocity, ω = $$\frac{v}{r}$$ = 2πn ……………… (i)
The centripetal force for the stone is provided by the tension T, in the string, i.e.,

Therefore, the maximum speed of the stone is 34.64 m/s.

Question 22.
If, in question 21, the speed of the stone is increased beyond the maximum permissible value, and the string breaks suddenly, which of the following correctly describes the trajectory of the stone after the string breaks:
(a) the stone moves radially outwards,
(b) the stone flies off tangentially from the instant the string breaks,
(c) the stone flies off at an angle with the tangent whose magnitude depends on the speed of the particle?
Solution:
(b) When the string breaks, the stone will move in the direction of the velocity at that instant. According to the first law of motion, the direction of velocity vector is tangential to the path of the stone at that instant. Hence, the stone will fly off tangentially from the instant the string breaks.

Question 23.
Explain why
(a) a horse cannot pull a cart and run in empty space,
(b) passengers are thrown forward from their seats when a speeding bus stops suddenly,
(c) it is easier to pull a lawn mower than to push it,
(d) a cricketer moves his hands backwards while holding a catch.
(a) In order to pull a cart, a horse pushes the ground backward with some force. The ground in turn exerts an equal and opposite reaction force upon the feet of the horse. This reaction force causes the horse to move forward.
An empty space is devoid of any such reaction force. Therefore, a horse cannot pull a cart and run in empty space.

(b) When a speeding bus stops suddenly, the lower portion of a passenger’s body, which is in contact with the seat, suddenly comes to rest. However, the upper portion tends to remain in motion (as per the first law of motion). As a result, the passenger’s upper body is thrown forward in the direction in which the bus was moving.

(c) While pulling a lawn mower, a force at ah angle θ is applied on it, as shown in the following figure.

The vertical component of this applied force acts upward. This reduces the effective weight of the mower.
On the other hand, while pushing a lawn mower, a force at an angle θ is applied on it, as shown in the following figure.

In this case, the vertical component of the applied fords acts in the direction of the weight of the mower. This increases the effective weight of the mower.
Since the effective weight of the lawn mower is lesser in the first case, pulling the lawn mower is easier than pushing it.

(d) According to Newton’s second law of motion, we have the equation of motion:
F = ma = m$$\frac{\Delta v}{\Delta t}$$ ……………. (i)
where,
F = Stopping force experienced by the cricketer as he catches the ball m = Mass of the ball
∆t = Time of impact of the ball with the hand It can be inferred from equation (i) that the impact force is inversely proportional to the impact time, i.e.,
f ∝ $$\frac{1}{\Delta t}$$ ………….. (ii)
Equation (ii) shows that the force experienced by the cricketer decreases if the time of impact increases and vice versa.
While taking a catch, a cricketer moves his hand backward so as to increase the time of impact (∆t). This is turn results in the decrease in the stopping force, thereby preventing the hands of the cricketer from getting hurt.

Question 24.
Figure below shows the position-time graph of a body of mass 0.04 kg. Suggest a suitable physical context for this motion. What is the time between two consecutive impulses received by the body? What is the magnitude of each impulse?

Solution:
A ball rebounding between two walls located between at x = 0 and x = 2 cm; after every 2 s, the bah receives an impulse of magnitude 0.08 × 10-2 kg-m/s from the walls.
The given graph shows that a body changes its direction of motion after every 2 s.
Physically, this situation can be visualized as a ball rebounding to and fro between two stationary walls situated between positions x = 0 and x = 2 cm. Since the slope of the x – t graph reverses after every 2 s, the ball collides with a wall after every 2 s. Therefore, ball receives an impulse after every 2 s.
Mass of the ball, m = 0.04 kg
The slope of the graph gives the velocity of the ball. Using the graph, we can calculate initial velocity (u) as:
u = $$\frac{(2-0) \times 10^{-2}}{(2-0)}$$ = 10-2 m/s
Velocity of the ball before collision, u = 10-2 m/s
Velocity of the ball after collision, υ = -10-2 m/s
(Here, the negative sign arises as the ball reverses its direction of motion.) Magnitude of impulse = Change in momentum
= | mυ – mu | = 10.04 (υ – u) |
= | 0.04 (-10-2 – 10 -2 ) |
= 0.08 × 10-2 kg-m/s
= 8 × 10-4 kg-ms-1

Question 25.
Figure below shows a man standing stationary with respect to a horizontal conveyor belt that is accelerating with 1 m s-2. What is the net force on the man? If the coefficient of static friction between the man’s shoes and the belt is 0.2, up to what acceleration of the belt can the man continue to be stationary relative to the belt? (Mass of the man = 65 kg.)

Solution:
Mass of the man, m = 65 kg
Acceleration of the belt, a = 1 m/s2
Coefficient of static friction, μ = 0.2
The net force F, acting on the man is given by Newton’s second law of motion as:
Fnet = ma = 65 × 1 – 65 N
The man will continue to be stationary with respect to the conveyor belt until the net force on the man is less than or equal to the frictional force fs, exerted by the belt, i. e.,
F’net = fs
ma’ = μmg
∴ a’ =0.2 × 10 = 2 m/s2
Therefore, the maximum acceleration of the belt up to which the man can stand stationary is 2 m/s2.

Question 26.
A stone of mass m tied to the end of a string revolves in a vertical circle of radius R. The net forces at the lowest and highest points of the circle directed vertically downwards are: [Choose the correct alternative]

 Lowest Point Highest Point (a) mg – T1 mg + T2 (b) mg + T1 mg – T1 (c) mg + T1 – (mυ12)/R mg – T2 + (mυ 12) / R (d) mg – T1 – (mυ12 )/ R mg + T2 + (mυ12 ) / R

T1 and υ 1 denote the tension and speed at the lowest point. T2 and υ2denote corresponding values at the highest point.
Solution:
(a) The free body diagram of the stone at the lowest point is shown in the following figure.

According to Newton’s second law. of motion, the net force acting on the stone at this point is equal to the centripetal force, i.e.,
Fnet = mg – T1 ……………….. (i)
where, υ1 = Velocity at the lowest point
The free body diagram of the stone at the highest point is shown in the following figure.

Using Newton’s second law of motion, we have:
Fnet = mg + T2 ……………… (ii)
where, υ 2 = Velocity at the highest point
It is clear from equations (i) and (ii) that the net force acting at the lowest and the highest points are (mg – T1 ) and (mg + T2) respectively.

Question 27.
A helicopter of mass 1000 kg rises with a vertical acceleration of 15 m s-2. The crew and the passengers weigh 300 kg. Give the magnitude and direction of the
(a) force on the floor by the crew and passengers,
(b) action of the rotor of the helicopter on the surrounding air,
(c) force oh the helicopter due to the surrounding air.
Solution:
Mass of the helicopter, mh = 1000 kg
Mass of the crew and passengers, mp = 300 kg
Total mass of the system, m = 1300 kg
Acceleration of the helicopter, a = 15 m/s2

(a) Using Newton’s second law of motion, the reaction force R, on the system by the floor can be calculated as:
R – mpg = mpa
or R = mp(g + a)
= 300 (10 + 15) = 300 × 25
= 7500 N
Since the helicopter is accelerating vertically upward, the reaction force will also be directed upward. Therefore, as per Newton’s third law of motion, the force on the floor by the crew and passengers is 7500 N, directed downward.

(b) Using Newton’s second law of motion, the reaction force R’, experienced by the helicopter can be calculated as:
R’ – mg = ma
or R’ = m(g + a)
= 1300 (10 + 15) = 1300 × 25
= 32500 N
The reaction force experienced by the helicopter from the surrounding air is acting upward. Hence, as per Newton’s third law of motion, the action of the rotor on the surrounding air will be 32500 N, directed downward.

(c) The force on the helicopter due to the surrounding air is 32500 N, directed upward.

Question 28.
A stream of water flowing horizontally with a speed of 15 ms-1 gushes out of a tube of cross-sectional area 10-2 m2, and hits a vertical wall nearby. What is the force exerted on the wall by the impact of water, assuming it does not rebound?
Solution:
Speed of the water stream, υ = 15 m/s
Cross-sectional area of the tube, A = 10-2 m2
Volume of water coming out from the pipe per second,
V = Aυ = 15 × 10-2 m3/s
Density of water, ρ = 103 kg/m3
Mass of water flowing out through the pipe per second = ρ × V =150 kg/s The water strikes the wall and does not rebound. Therefore, the force , exerted by the water on the wall is given by Newton’s second law of motion as:
F = Rate of change of momentum = $$\frac{\Delta P}{\Delta t}=\frac{m v}{t}$$
= 150 × 15 = 2250 N

Question 29.
Ten one-rupee coins are put on top of each other on a table. Each coin has a mass m. Give the magnitude and direction of
(a) the force on the 7th coin (counted from the bottom) due to all the coins on its top,
(b) the force on the 7th coin by the eighth coin,
(c) the reaction of the 6th coin on the 7th coin.
Solution:
(a) Force on the seventh coin is exerted by the weight of the three coins on its top.
Weight of one coin = mg
Weight of three coins = 3 mg
Hence, the force exerted on the 7th coin by the three coins on its top is 3 mg. This force acts vertically downward.

(b) Force on the seventh coin by the eighth coin is because of the weight of the eighth coin and the other two coins (ninth and tenth) on its top.
Weight of the eighth coin = mg
Weight of the ninth coin = mg
Weight of the tenth coin = mg
Total weight of these three coins = 3 mg
Hence, the force exerted on the 7th coin by the eighth coin is 3 mg. This force acts vertically downward.

(c) The 6th coin experiences a downward force because of the weight of the four coins (7th, 8th, 9th and 10th) on its top.
Therefore, the total downward force experienced by the 6th coin is 4 mg.
As per Newton’s third law of motion, the 6th coin will produce an equal reaction force on the 7th coin, but in the opposite direction. Hence, the reaction force of the 6th coin on the 7th coin is of magnitude 4 mg. This force acts in the upward direction.

Question 30.
An aircraft executes a horizontal loop at a speed of 720 km/h with its wings banked at 15°. What is the radius of the loop?
Solution:
Speed of the aircraft, υ = 720 km/h = 720 × $$\frac{5}{18}$$ = 200 m/s
Acceleration due to gravity, g = 10 m/s2
Angle of banking, θ = 15°
For radius r, of the loop, we have the relation:
tan0 =$$\frac{v^{2}}{r g}$$
r = $$\frac{v^{2}}{g \tan \theta}=\frac{200 \times 200}{10 \times \tan 15^{\circ}}=\frac{4000}{0.268}$$
= 14925.37 m = 14.92 km

Question 31.
A train runs along an unbanked circular track of radius 30 m at a speed of 54 km/h. The mass of the train is 106 kg. What provides the centripetal force required for this purpose – The engine or the rails? What is the angle of banking required to prevent wearing out of the rail?
Solution:
Radius of the circular track, r = 30 m
Speed of the train, υ = 54 km/h = 15 m/s
Mass of the train, m = 106 kg
The centripetal force is provided by the lateral thrust of the rail on the wheel. As per Newton’s third law of motion, the wheel exerts an equal and opposite force on the rail. This reaction force is responsible for the wear and rear of the rail.
The angle of banking 0, is related to the radius (r) and speed (υ) by the relation:
tanθ = $$\frac{v^{2}}{r g}=\frac{(15)^{2}}{30 \times 10}=\frac{225}{300}$$
θ = tan-1 (0.75) = 36.87°
Therefore, the angle of banking is about 36.87°.

Question 32.
A block of mass 25 kg is raised by a 50 kg man in two different ways as shown in figure below. What is the action on the floor by the man in the two cases? If the floor yields to a normal force of 700 N, which mode should the man adopt to lift the block without the floor yielding?

Solution:
Mass of the block, m = 25 kg
Mass of the man, M = 50 kg
Acceleration due to gravity, g = 10 m/s2
Force applied on the block, F =25 × 10 = 250 N
Weight of the man, W = 50 × 10 = 500 N

Case (a): When the man lifts the block directly
In this case, the man applies a force in the upward direction. This increases his apparent weight.
.’. Action on the floor by the man = 250 + 500 = 750 N

Case (b): When the man lifts the block using a pulley
In this case, the man applies a force in the downward direction. This decreases his apparent weight.
Action on the floor by the man = 500 – 250 = 250 N
If the floor can yield to a normal force of 700 N, then the man should adopt the second method to easily lift the block by applying lesser force.

Question 33.
A monkey of mass 40 kg climbs on a rope (see figure), which can stand a maximum tension of 600 N. In which of the following cases will the rope break: the monkey
(a) climbs up with an acceleration of 6 m s-2
(b) climbs down with an acceleration of 4 m s-2
(c) climbs up with a uniform speed of 5 m s-1
(d) falls down the rope nearly freely under gravity?
(Ignore the mass of the rope).

Solution:
Case (a)
Mass of the monkey, m = 40 kg
Acceleration due to gravity, g = 10 m/s
Maximum tension that the rope can bear, Tmax = 600 N
Acceleration of the monkey, a = 6 m/s2 upward
Using Newton’s second law of motion, we can write the equation of motion as:
T – mg = ma
∴ T = m(g +a)
= 40(10 + 6)
= 640 N
Since T > Tmax, the rope will break in this case.

Case (b)
Acceleration of the monkey, a = 4 m/s2 downward
Using Newton’s second law of motion, we can write the equation of motion as:
mg – T = ma
∴ T = m(g – a)
= 40(10 – 4)
= 240 N
Since T < Tmax, the rope will not break in this case.

Case (c)
The monkey is climbing with a uniform speed of 5 m/s. Therefore, its acceleration is zero, i.e.,a = 0.
Using Newton’s second law of motion, we can write the equation of motion as:
T – mg = ma
T- mg = 0
∴ T = mg
= 40 × 10
= 400 N
Since T < Tmax, the rope will not break in this case.

Case (d)
When the monkey falls freely under gravity, its will acceleration become equal to the acceleration due to gravity, i.e.,a = g
Using Newton’s second law of motion, we can write the equation of motion as:
mg – T = mg
∴ T = m(g – g) = 0
Since T < Tmax, the rope will not break in this case.

Question 34.
Two bodies A and B of masses 5 kg and 10 kg in contact with each other rest on a table against a rigid wall (see figure). The coefficient of friction between the bodies and the table is 0.15. A force of 200 N is applied horizontally to A. What are (a) the reaction of the partition (b) the action-reaction forces between A and B ? What happens when the wall is removed? Does the answer to (b) change, when the bodies are in motion? Ignore the difference between (μs and μk

Solution:
Mass of body A, mA = 5 kg
Mass of body B, mB =10 kg ,
Applied force, F = 200 N
Coefficient of friction, μs = 0.15

(a) The force of friction is given by the relation:
fs = μ(mA + mB)g
= 0.15(5 + 10) × 10
= 1.5 × 15 = 22.5 N leftward
Net force acting on the partition = 200 – 22.5 = 177.5 N rightward
As per Newton’s third law of motion, the reaction force of the partition will be in the direction opposite to the net applied force.
Hence, the reaction of the partition will be 177.5 N, in the leftward direction.

(b) Force of friction on mass A:
fA = μ mAg
= 0.15 × 5 × 10 = 7.5 N leftward
Net force exerted by mass A on mass B = 200 – 7.5 = 192.5 N rightward
As per Newton’s third law of motion, an equal amount of reaction force will be exerted by mass B on mass A, i. e., 192.5 N acting leftward.
When the wall is removed, the two bodies will move in the direction of the applied force.
Net force acting on the moving system = 177.5 N
The equation of motion for the system of acceleration a, can be written as: Net force = (mA + mB)a
Net force
∴ a = $$\frac{\text { Net force }}{m_{A}+m_{B}}$$
= $$\frac{177.5}{5+10}=\frac{177.5}{15}$$ = 11.83 m/s2
Net force causing mass A to move:
FA =mAa = 5 × 11.83 = 59.15N
Net force exerted by mass A on mass B = 192.5 – 59.15 = 133.35 N
This force will act in the direction of motion. As per Newton’s third law of motion, an equal amount of force will be exerted by mass B on mass A, i, e., 133.35 N, acting opposite to the direction of motion.

Question 35.
A block of mass 15 kg is placed on a long trolley. The coefficient of static friction between the block and the trolley is 0.18. The trolley accelerates from rest with 0.5 ms-2 for 20 s and then moves with uniform velocity. Discuss the motion of the block as viewed by (a) a stationary observer on the ground, (b) an observer moving with the trolley.
Solution:
Mass of the block, m = 15 kg
Coefficient of static friction, μ = 0.18
Acceleiation of the trolley, a = 0.5 m/s2

(a) As per Newton’s second law of motion, the force (F) on the block caused by the motion of the trolley is given by the relation:
F = ma = 15 × 0.5 = 7.5 N
This force is acted in the direction of motion of the trolley.
Force of static friction between the block and the trolley:
f = μ mg = 0.18 × 15 × 10 = 27 N
The force of static friction between the block and the trolley is greater than the applied external force. Hence, for an observer on the ground, the block will appear to be at rest.
When the trolley moves with uniform velocity there will be no applied external force. Only the force of friction will act on the block in this situation.

(b) An observer, moving with the trolley, has some acceleration. This is the case of non-inertial frame of reference. The frictional force, acting on the trolley backward, is opposed by a pseudo force of the same magnitude. However, this force acts in the opposite direction. Thus, the trolley will appear to be at rest for the observer moving with the trolley.

Question 36.
The rear side of a truck is open and a box of 40 kg mass is placed 5 m away from the open end as shown in figure below. The coefficient of friction between the box and the surface below it is 0.15. On a straight road, the truck starts from rest and accelerates with 2 m s-2. At what distance from the starting point does the box fall off the truck? (Ignore the size of the box).

Solution:
Mass of the box, m = 40 kg
Coefficient of friction, μ = 0.15
Initial velocity, u = 0
Acceleration, a = 2 m/s2
Distance of the box from the end of the truck, s’ = 5 m
As per Newton’s second law of motion, the force on the box caused by the accelerated motion of the truck is given by:
F = ma – 40 × 2 = 80 N
As per Newton’s third law of motion, a reaction force of 80 N is acting on the box in the backward direction. The backward motion of the box is opposed by the force of friction /, acting between the box and the floor of the truck. This force is given by:
f = μmg = 0.15 × 40 × 10 = 60 N
∴ Net force acting on the block:
Fnet = 80 – 60 = 20 N backward
The backward acceleration produced in the box is given by:
aback = $$\frac{F_{\text {net }}}{m}=\frac{20}{40}$$ = 0.5m/s2
Using the second equation of motion, time t can be calculated as :
s’ =ut + $$\frac{1}{2}$$abackt2
5 = 0 + $$\frac{1}{2}$$ × 0.5 × t2
∴ t = $$\sqrt{20}$$ s
Hence, the box will fall from the truck after $$\sqrt{20}$$ s from start.
The distance s, travelled by the truck in $$\sqrt{20}$$ s is given by the relation :
s = ut + $$\frac{1}{2}$$ at2
= 0 + $$\frac{1}{2}$$ × 2 × ($$\sqrt{20}$$ )2 = 20 m

Question 37.
A disc revolves with a speed of 33$$\frac{1}{3}$$ rev/min, and has a radius of 15 cm. Two coins are placed at 4 cm and 14 cm away from the centre of the record. If the coefficient of friction between the coins and the record is 0.15, which of the coins will revolve with the record?
Solution:
Mass of each coin = m
Radius of the disc, r = 15 cm = 0.15 m
Frequency of revolution, v = 33 $$\frac{1}{3}$$ rev/min =  rev/s
Coefficient of friction, μ = 0.15
In the given situation, the coin having a force of friction greater than or equal to the centripetal force provided by the rotation of the disc will revolve with the disc. If this is not the case, then the coin will slip from the disc.

Coin placed at 4 cm:
Radius of revolution, r’ = 4 cm = 0.04 m
Angular frequency, ω = 2πv = 2 × $$\frac{22}{7}$$ × $$\frac{5}{9}$$ = 3.49 s-1
Frictional force, f = μ mg = 0.15 × m × 10 = 1.5m N
Centripetal force on the coin:
Fcent = mr’ω2
= m × 0.04 × (3.49)2
= 0.49 m N
Since f > Fcent, the coin will revolve along with the record.

Coin placed at 14 cm:
Radius, r” = 14 cm = 0.14 m
Angular frequency, ω = 3.49 s-1
Frictional force, f’ = 1.5 m N
Centripetal force is given as:
Fcent = mr”ω2
= m × 0.14 × (3.49)2 = 1.7m N
Since f < cent, the coin will slip from the surface of the record.

Question 38.
You may have seen in a circus a motorcyclist driving in vertical loops inside a ‘death-well’ (a hollow spherical chamber with holes, so the spectators can watch from outside). Explain clearly why the motorcyclist does not drop down when he is at the uppermost point, with no support from below. What is the minimum speed required at the uppermost position to perform a vertical loop if the radius of the chamber is 25 m?
Solution:
In a death-well, a motorcyclist does not fall at the top point of a vertical loop because both the force of normal reaction and the weight of the motorcyclist act downward and are balanced by the centripetal force. This situation is shown in the following figure.

The net force acting on the motorcyclist is the sum of the normal force (FN) and the force due to gravity (Fg = mg).
The equation of motion for the centripetal acceleration ac, can be written as :
Fnet – mac
FN + Fg = mac
FN + mg = $$\frac{m v^{2}}{r}$$
Normal reaction is provided by the speed of the motorcyclist. At the minimum speed (υmin),
FN = 0
mg = $$\frac{m v_{\min }^{2}}{r}$$
∴ υmin = $$\frac{r}{\sqrt{r g}}=\sqrt{25 \times 10}$$ = 15.8 m/s

Question 39.
A 70 kg man stands in contact against the inner wall of a hollow cylindrical drum of radius 3 m rotating about its vertical axis with 200 rev/min. The coefficient of friction between the wall and his clothing is 0.15. What is the minimum rotational speed of the cylinder to enable the man to remain stuck to the wall (without falling) when the floor is suddenly removed?
Solution:
Mass of the man, m = 70 kg .
Radius of the drum, r = 3 m
Coefficient of friction, μ = 0.15
Frequency of rotation, v = 200 rev/mm = $$\frac{200}{60}=\frac{10}{3}$$ rev/s
The necessary centripetal force required for the rotation of the man is provided by the normal force (FN).
When the floor revolves, the man sticks to the wall of the drum. Hence, the weight of the man (mg) acting downward is balanced by the frictional force (f = μFN) acting upward.
Hence, the man will not fall until:
mg< f
mg< μFN = μmrω
g < μ rω2
ω = $$\sqrt{\frac{g}{\mu r}}$$
The minimum angular speed is given as:
ωmin = $$\sqrt{\frac{g}{\mu r}}$$
= $$\sqrt{\frac{10}{0.15 \times 3}}$$ = 4.71 rad s-1

Question 40.
A thin circular loop of radius R rotates about its vertical diameter with an angular frequency ω. Show that a small bead on the wire loop remains at its lowermost point for ω ≤ $$\sqrt{g / R}$$. What is the angle made by the radius vector joining the centre to the bead with the vertical downward direction for ω = $$\sqrt{2 g / R}$$?  Neglect friction.
Solution:
Let the radius vector joining the bead with the centre makes an angle 0, with the vertical downward direction.

OP = R = Radius of the circle
N = Normal reaction
The respective vertical and horizontal equations of forces can be written as:
mg = N cosθ ……………. (i)
mlω2= N sinθ ………….. (ii)
In Δ OPQ, we have:
sinθ = $$\frac{l}{R}$$
l = R sinθ ………….. (iii)
Substituting equation (iii) in equation (ii), we get:
m (R sinθ) ω2 = N sinθ
mR ω2 = N ……………. (iv)
Substituting equation (iv) in equation (i), we get:
mg = mRω2cosθ
cosθ = $$\frac{g}{R \omega^{2}}$$ …………….. (v)
Since cosθ ≤ 1, the bead will remain at its lowermost point for $$\frac{g}{R \omega^{2}}$$ ≤ 1,
i.e for ω ≤ $$\sqrt{\frac{g}{R}}$$
For ω = $$\sqrt{\frac{2 g}{R}}$$ or ω2 = $$\frac{2 g}{R}$$ …………….. (vi)
On equating equations (v) and (vi), we get:
$$\frac{2 g}{R}=\frac{g}{R \cos \theta}$$
cosθ = $$\frac{1}{2}$$
∴ θ = cos-1 (0.5) = 60°

## PSEB 11th Class Physics Solutions Chapter 4 Motion in a Plane

Punjab State Board PSEB 11th Class Physics Book Solutions Chapter 4 Motion in a Plane Textbook Exercise Questions and Answers.

## PSEB Solutions for Class 11 Physics Chapter 4 Motion in a Plane

PSEB 11th Class Physics Guide Motion in a Plane Textbook Questions and Answers

Question 1.
State, for each of the following physical quantities, if it is a scalar or a vector:
volume, mass, speed, acceleration, density, number of moles, velocity, angular frequency, displacement, angular velocity.
Scalar quantities: Volume, mass, speed, density, number of moles, angular frequency.
Vector quantities: Acceleration, velocity, displacement, angular velocity.

Question 2.
Pick out the two scalar quantities in the following list:
force, angular momentum, work, current, linear momentum, electric field, average velocity, magnetic moment, relative velocity.
Work and current are scalar quantities.

Question 3.
Pick out the only vector quantity in the following list:
Temperature, pressure, impulse, time, power, total path length, energy, gravitational potential, coefficient of friction, charge.
Impulse is only a vector quantity in the given quantities.

Question 4.
State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful:
(b) adding a scalar to a vector of the same dimensions,
(c) multiplying any vector by any scalar,
(d) multiplying any two scalars,
(f) adding a component of a vector to the same vector.
(a) Not meaningful
Explanation: Adding any two scalars is not meaningful because only the scalars of same dimensions can be added.

(b) Not meaningful
Explanation: The addition of a vector quantity with a scalar quantity is not meaningful.

(c) Meaningful
Explanation: A scalar can be multiplied with a vector. For example, force is multiplied with time to give impulse.

(d) Meaningful
Explanation: A scalar, irrespective of the physical quantity it represents, can be multiplied with another scalar having the same or different dimensions.

(e) Not meaningful
Explanation: Adding any two vectors is not meaningful because only vectors of same dimensions can be added.

(f) Meaningful
Explanation: A component of a vector can be added to the same vector as they both have the same dimensions.

Question 5.
Read each statement below carefully and state with reasons, if it is true or false:
(a) The magnitude of a vector is always a scalar,
(b) each component of a vector is always a scalar,
(c) the total path length is always equal to the magnitude of the displacement vector of a particle,
(d) the average speed of a particle (defined as total path length divided by the time taken to cover the path) is either greater or equal to the magnitude of average velocity of the particle over the same interval of time,
(e) Three Vectors not lying in a plane can never add up to give a null vector.
(a) True
Explanation: The magnitude of a vector is a number. Hence, it is a scalar.

(b) False
Explanation: Each component of a vector is also a vector.

(c) False
Explanation: Total path length is a scalar quantity, whereas displacement is a vector quantity. Hence, the total path length is always greater than the magnitude of displacement. It becomes equal to the magnitude of displacement only when a particle is moving in a straight line.

(d) True
Explanation: It is because of the fact that the total path length is always greater than or equal to the magnitude of displacement of a particle.

(e) True
Explanation: Three vectors, which do not he in a plane, cannot be represented by the sides of a triangle taken in the same order.

Question 6.
Establish the following vector inequalities geometrically or otherwise:
(a) $$|\vec{a}+\vec{b}| \leq|\vec{a}|+|\vec{b}|$$
(b) $$|\vec{a}+\vec{b}| \geq|| \vec{a}|-| \vec{b}||$$
(c) $$|\vec{a}-\vec{b}| \leq|\vec{a}|+|\vec{b}|$$
(d) $$|\vec{a}-\vec{b}| \geq|| \vec{a}|-| \vec{b}||$$
When does the equality sign above apply?
Solution:
(a) Let two vectors $$\vec{a}$$ and $$\vec{b}$$ be represented by the adjacent sides of a parallelogram OMNP, as shown in the given figure.

$$|\overrightarrow{O M}|=|\vec{a}|$$ ……………. (i)
$$|\overrightarrow{M N}|=|\overrightarrow{O P}|=|\vec{b}|$$ ……………. (ii)
$$|\overrightarrow{O N}|=|\vec{a}+\vec{b}|$$ ……………. (iii)
In a triangle, each side is smaller than the sum of the other two sides. Therefore, in ∆ OMN, we have:
ON < (OM + MN)
$$|\vec{a}+\vec{b}|<|\vec{a}|+|\vec{b}|$$ ………….. (iv)
If the two vectors $$\vec{a}$$ and $$\vec{b}$$ act along a straight line in the same direction, then we can write: $$|\vec{a}+\vec{b}|=|\vec{a}|+|\vec{b}|$$ …………… (v)
Combining equations (iv) and (v), we get: $$|\vec{a}+\vec{b}| \leq|\vec{a}|+|\vec{b}|$$ (b) Let two vectors $$\vec{a}$$ and $$\vec{b}$$ be represented by the adjacent sides of a parallelogram OMNP, as shown in the given figure.

Here, we have:
$$|\overrightarrow{O M}|=|\vec{a}|$$ …………….. (i)
$$|\overrightarrow{M N}|=|\overrightarrow{O P}|=|\vec{b}|$$ …………….. (ii)
$$|\overrightarrow{O N}|=|\vec{a}+\vec{b}|$$ …………….. (iii)
In a triangle, each side is smaller than the sum of the-other two sides. Therefore, in ∆ OMN, we have: ON + MN > OM
ON + OM > MN
$$|\overrightarrow{O N}|>|\overrightarrow{O M}-\overrightarrow{O P}|$$ (∵ OP = MN)
$$|\vec{a}+\vec{b}|>\| \vec{a}|-| \vec{b}||$$ ……………….. (iv)

If the two vectors $$\vec{a}$$ and $$\vec{b}$$ act along a straight line in the same direction, then we can write:
$$|\vec{a}+\vec{b}|=\|\vec{a}|-| \vec{b}\|$$ ……………. (v)
Combining equations (iv) and (v), we get:
$$|\vec{a}+\vec{b}| \geq \| \vec{a}|-| \vec{b}||$$

(c) Let two vectors $$\vec{a}$$ and $$\vec{b}$$ be represented by the adjacent sides of a parallelogram PORS, as shown in the given figure.

Here we have:
$$|\overrightarrow{O R}|=|\overrightarrow{P S}|=|-\vec{b}|$$ ……………… (i)
$$|\overrightarrow{O P}|=|\vec{a}|$$ …………….. (ii)
$$|\overrightarrow{O S}|=|\vec{a}-\vec{b}|$$ …………….. (iii)
In a triangle, each side is smaller than the sum of the other two sides. Therefore, in ∆ OPS, we have:
OS < OP + PS
$$|\vec{a}-\vec{b}|<|\vec{a}|+|-\vec{b}|$$
$$|\vec{a}-\vec{b}|<|\vec{a}|+|\vec{b}|$$ …………… (iv)
If the two vectors act in a straight line but in opposite directions, then we can write:
$$|\vec{a}-\vec{b}|=|\vec{a}|+|\vec{b}|$$ …………….. (v)
Combining equations (iv) and (v), we get
$$|\vec{a}-\vec{b}| \leq|\vec{a}|+|\vec{b}|$$

(d) Let two vectors $$\vec{a}$$ and $$\vec{b}$$ be represented by the adjacent sides of a parallelogram PORS, as shown in the given figure.

The following relations can be written for the given parallelogram. OS + PS > OP …………… (i)
OS > OP – PS ……………. (ii)
$$|\vec{a}-\vec{b}|>|\vec{a}|-|\vec{b}|$$ ……………….. (iii)
The quantity on the LHS is always positive and that on the RHS can be positive or negative. To make both quantities positive, we take modulus on both sides as:
$$\|\vec{a}-\vec{b}\|>\|\vec{a}|-| \vec{b}\|$$
$$|\vec{a}-\vec{b}|>|| \vec{a}|-| \vec{b}||$$ ………………. (iv)
If the two vectors act in a straight line but in the opposite directions, then we can write:
$$|\vec{a}-\vec{b}|=\| \vec{a}|-| \vec{b}||$$ …………….(v)
Combining equations (iv) and (v), we get
$$|\vec{a}-\vec{b}| \geq|| \vec{a}|-| \vec{b}||$$

Question 7.
Given $$\vec{a}+\vec{b}+\vec{c}+\vec{d}$$ = 0, which of the following statements are correct:
(a) $$\vec{a}, \vec{b}, \vec{c}$$ and $$\overrightarrow{\boldsymbol{d}}$$ must each be a null vector,
(b) The magnitude of $$(\vec{a}+\vec{c})$$ equals the magnitude of $$(\vec{b}+\vec{d})$$,
(c) The magnitude of a can never be greater than the sum of the magnitudes of $$\vec{b}, \vec{c}$$ and $$\vec{d}$$,
(d) $$\vec{b}+\vec{c}$$ must lie in the plane of $$\overrightarrow{\boldsymbol{a}}$$ and $$\overrightarrow{\boldsymbol{b}}$$ if $$$\overrightarrow{\boldsymbol{a}}$$$ and $$\overrightarrow{\boldsymbol{d}}$$ are not collinear, and in the line of $$\overrightarrow{\boldsymbol{a}}$$ and $$\overrightarrow{\boldsymbol{d}}$$, if they are collinear?
Solution:
(a) Incorrect
In order to make $$\vec{a}+\vec{b}+\vec{c}+\vec{d}$$ = 0, it is not necessary to have all the four given vectors to be null vectors. There are many other combinations which can give the sum zero.

(b) Correct
$$\vec{a}+\vec{b}+\vec{c}+\vec{d}$$ = 0
$$\vec{a}+\vec{c}=-(\vec{b}+\vec{d})$$
Taking modulus on both the sides, we get:
$$|\vec{a}+\vec{c}|=|-(\vec{b}+\vec{d})|=|\vec{b}+\vec{d}|$$
Hence, the magnitude of ($$\vec{a}+\vec{c}$$) is the same as the magnitude of ($$\vec{b}+\vec{d}$$).

(c) Correct $$\vec{a}+\vec{b}+\vec{c}+\vec{d}$$ = 0
$$\vec{a}=(\vec{b}+\vec{c}+\vec{d})$$

Taking modulus on both sides, we get
$$|\vec{a}|=|\vec{b}+\vec{c}+\vec{d}|$$
$$|\vec{a}| \leq|\vec{a}|+|\vec{b}|+|\vec{c}|$$ ………………. (i)

Equation (i) shows that the magnitude of a is equal to or less than the sum of the magnitudes of $$\vec{b}$$, $$\vec{c}$$ and $$\vec{d}$$.
Hence, the magnitude of vector a can never be greater than the sum of the magnitudes of $$\vec{b}$$, $$\vec{c}$$ and $$\vec{d}$$.

(d) Correct For $$\vec{a}+\vec{b}+\vec{c}+\vec{d}$$ = 0
$$\vec{a}+(\vec{b}+\vec{c})+\vec{d}$$ = 0
The resultant sum of the three vectors $$\vec{a},(\vec{b}+\vec{c})$$ and $$\vec{d}$$ can be zero
only if ($$\vec{b}+\vec{c}$$) lie in a plane containing a and d, assuming that these
three vectors are represented by the three sides of a triangle.

If a and d are collinear/ then it implies that the vector ($$\vec{b}+\vec{c}$$ ) is in the line of $$\vec{a}$$ and $$\vec{d}$$. This implication holds only then the vector sum of all the vectors will be zero.

Question 8.
Three girls skating on a circular ice ground of radius 200m start from a point P on the edge of the ground and reach a point Q diametrically opposite to P following different paths as shown in figure below. What is the magnitude of the displacement vector for each? For which girl is this equal to the actual length of the path skate?

Solution:
Displacement is given by the minimum distance between the initial and final positions of a particle. In the given case, all the girls start from point P and reach point Q. The magnitudes of their displacements will be equal to the diameter of the ground.
Radius of the ground = 200 m
Diameter of the ground = 2 × 200 = 400 m
Hence, the magnitude of the displacement for each girl is 400 m. This is equal to the actual length of the path skated by girl B.

Question 9.
A cyclist starts from the centre O of a circular park of radius 1 km, reaches the edge P of the park, then cycles along the circumference, and returns to the centre along QO as shown in figure below. If the round trip takes 10 min, what is the (a) net displacement, (b) average velocity, and (c) average speed of the cyclist?

Solution:
(a) Displacement is given by the minimum distance between the initial and final positions of a body. In the given case, the cyclist comes to the starting point after cycling for 10 minutes. Hence, his net displacement is zero.

(b) Average velocity is given by the relation;

Since the net displacement of the cyclist is zero, his average velocity will also be zero.

(c) Average speed of the cyclist is given by the relation

Question 10.
On an open ground, a motorist follows a track that turns to his left by an angle of 60° after every 500 m. Starting from a given turn, specify the displacement of the motorist at the third, sixth and eighth turn. Compare the magnitude of the displacement with the total path length covered by the motorist in each case.
Solution:
The path followed by the motorist is a regular hexagon with side 500 m, as shown in the given figure

Let the motorist start from point P.
The motorist takes the third turn at S.
∴ Magnitude of displacement PS = PV + VS = 500 + 500 = 1000 m
Total path length = PQ + QR +RS = 500 + 500 + 500 = 1500 m
The motorist takes the sixth turn at point P, which is the starting point.
∴ Magnitude of displacement = 0
Total path length = PQ + QR + RS + ST +TU + UP
= 500 + 500 + 500 + 500 + 500 + 500 = 3000 m
The motorist takes the eight turn at point R
∴ Magnitude of displacement = PR

= 866.03 m
If it is inclined at an angle β from the direction of PQ, then

or β = 30°
Therefore, the magnitude of displacement is 866.03 m at an angle of 30° with PR.
Total path length = Circumference of the hexagon + PQ + QR
= 6 × 500 + 500 + 500 = 4000 m
The magnitude of displacement and the total path length corresponding to the required turns is shown in the given

 Turn Magnitude of dispalcement (m) Total path length (m) Third 1000 1500 Sixth 0 3000 Eighth 866.03; 30° 4000

Comparison of the magnitude of displacement with the total path length in each case:

Question 11.
A passenger arriving in a new town wishes to go from the station to a hotel located 10 km away on a straight road from the station. A dishonest cabman takes him along a circuitous path 23 km long and reaches the hotel in 28 min. What is (a) the average speed of the taxi, (b) the magnitude of average velocity? Are the two equal?
Solution:
Total distance travelled = 23 km
Total time taken = 28 min = $$\frac{28}{60}$$ h

(b) Distance between the hotel and the station =10 km = Displacement of the taxi
∴ Average velocity = $$\frac{\frac{10}{28}}{\frac{60}{60}}$$ = 21.43 km/ h

Therefore, the two physical quantities (average speed and average velocity) are not equal.

Question 12.
Rain is falling vertically with a speed of 30 m s-1. A woman rides a bicycle with a speed of 10 ms-1 in the north to south direction. What is the direction in which she should hold her umbrella?
Solution:
The described situation is shown in the given figure.

Here, υc = Velocity of the cyclist
υr = Velocity of falling rain
In order to protect herself from the rain, the woman must hold her umbrella in the direction of the relative velocity (v) of the rain with respect to the woman.
υ = υr + (-υc)
= 30 + (-10) = 20 m/s
tanθ = $$\frac{v_{c}}{v_{r}}=\frac{10}{30}$$
θ = tan-1 ($$\frac{1}{3}$$)
= tan-1 (0.333) ≈ 18°
Hence, the woman must hold the umbrella toward the south, at an angle of nearly 18° with the vertical.

Question 13.
A man can swim with a speed of 4.0 km/h in still water. How long does he take to cross a river 1.0 km wide if the river flows steadily at 3.0 km/h and he makes his strokes normal to the river current? How far down the river does he go when he reaches the other hank?
Solution:

Speed of the man, υm = 4 km/h
Width of the river = 1 km

= $$\frac{1}{4}$$ h = $$\frac{1}{4}$$ × 60 = 15 min
Speed of the river, υr = 3 km/h
Distance covered with flow of the river = υr × t
= 3 × $$\frac{1}{4}$$ = $$\frac{3}{4}$$ km
= $$\frac{3}{4}$$ × 1000 = 750 m

Question 14.
In a harbour, wind is blowing at the speed of 72 km/h and the flag on the mast of a boat anchored in the harbour flutters along the N-E direction. If the boat starts moving at a speed of 51 km/h to the north, what is the direction of the flag on the mast of the boat?
Solution:
Velocity of the boat, υb = 51 km/h
Velocity of the wind, υw = 72 km/h
The flag is fluttering in the north-east direction. It shows that the wind is blowing toward the north-east direction. When the ship begins sailing toward the north, the flag will move along the direction of the relative velocity (υwb) of the wind with respect to the boat.

∴ β = tan-1 (1.0038) = 45.18°
Angle with respect to the east direction = 45.18° – 45° = 0.18°
Hence, the flag will flutter almost due east.

Question 15.
The ceiling of a long hall is 25 m high. What is the maximum horizontal distance that a ball thrown with a speed of 40 m s 1 can go without hitting the ceiling of the hall?
Solution:
Speed of the ball, u = 40 m/s
Maximum height, h = 25 m
In projectile motion, the maximum height reached by a body projected at an angle 0, is given by the relation:

Question 16.
A cricketer can throw a ball to a maximum horizontal distance of 100 m. How much high above the ground can the cricketer throw the same ball?
Solution:
Maximum horizontal distance, R = 100 m
The cricketer will only be able to throw the ball to the maximum horizontal distance when the angle of projection is 45°, i.e., θ = 45°.
The horizontal range for a projection velocity v, is given by the relation:
R = $$\frac{u^{2} \sin 2 \theta}{g}$$
100 = $$\frac{u^{2}}{g}$$ sin90°
$$\frac{u^{2}}{g}$$ = 100 ……………… (i)
The ball will achieve the maximum height when it is thrown vertically upward. For such motion, the final velocity v is zero at the maximum height H.
Acceleration, a = -g
Using the third equation of motion:
υ2 – u2 = -2gH
H = $$\frac{1}{2}$$ × $$\frac{u^{2}}{g}$$ = $$\frac{1}{2}$$ × 100 = 50 m

Question 17.
A stone tied to the end of a string 80 cm long is whirled in a horizontal circle with a constant speed. If the stone makes 14 revolutions in 25 s, what is the magnitude and direction of acceleration of the stone?
Solution:
Length of the string, l = 80 cm = 0.8 m
Number of revolutions = 14
Time taken = 25 s

= 9.91 m/s2
The direction of centripetal acceleration is always directed along the string, toward the centre, at all points.

Question 18.
An aircraft executes a horizontal loop of radius 1.00 km with a steady speed of 900 km/h. Compare its centripetal acceleration with the acceleration due to gravity.
Solution:
Radius of the loop, r = 1 km = 1000 m
Speed of the aircraft,υ = 900 km/h = 900 × $$\frac{5}{18}$$ = 250 m/s
Centripetal acceleration, ac = $$\frac{v^{2}}{r}$$
= $$\frac{(250)^{2}}{1000}$$ = 62.5 m/s2
Acceleration due to gravity, g = 9.8 m/s2
$$\frac{a_{c}}{g}=\frac{62.5}{9.8}$$ = 6.38
ac = 6.38 g

Question 19.
Read each statement below carefully and state, with reasons, if it is true or false:
(a) The net acceleration of a particle in circular motion is always along the radius of the circle towards the centre.
(b) The velocity vector of a particle at a point is always along the tangent to the path of the particle at that point.
(c) The acceleration vector of a particle in uniform circular motion averaged over one cycle is a null vector.
(a) False
Reason: The net acceleration of a particle in circular motion is not always directed along the radius of the circle towards the centre. It happens only in the case of uniform circular motion.

(b) True
Reason: At a point on a circular path, a particle appears to move tangentially to the circular path. Hence, the velocity vector of the particle is always along the tangent at a point.

(c) True
Reason: In uniform circular motion (UCM), the direction of the acceleration vector points toward the centre of the circle. However, it constantly changes with time. The average of these vectors over one cycle is a null vector.

Question 20.
The position of a particle is given by
r̂ = 3.0 t î – 2.0 t2ĵ + 4.0k̂ m
where t is in seconds and the coefficients have the proper units for r to be in metres.
(a) Find the υ and a of the particle?
(b) What is the magnitude and direction of velocity of the particle at t = 3.0 s?
Solution:

= – tan-1 (2.667)
= -69.45°
The negative sign indicates that the direction of velocity is below the x-axis.

Question 21.
A particle starts from the origin at t 0 s with a velocity of 10.0 ĵ m/s and moves in the x – y plane with a constant acceleration of (8.0î + 2.0ĵ) ms-2.
(a) At what time is the x – coordinate of the particle 16 m? What is the y-coordinate of the particle at that time?
(b) What is the speed of the particle at the time?
Solution:

Question 22.
î and ĵ are unit vectors along x- and y-axis respectively. What is the magnitude and direction of the vectors î + ĵ and î – ĵ? What are the components of a vector $$\overrightarrow{\boldsymbol{A}}$$ = 2î + 3ĵ along the directions of î + ĵ and î – ĵ? [You may use graphical method]
Solution:
Consider a vector $$\vec{P}$$, given as:
$$\vec{P}$$ = î + ĵ
Pxî +Py ĵ = î + ĵ
On comparing the components on both sides, we get:
Px = Py = 1
$$|\vec{P}|=\sqrt{P_{x}^{2}+P_{y}^{2}}=\sqrt{1^{2}+1^{2}}=\sqrt{2}$$ …………… (i)
Hence, the magnitude of the vector î + ĵ is √2.
Let 0 be the angle made by the vector , with the x-axis, as shwon in the following figure.

Hence, the vector î + ĵ makes an angle of 45° with the x-axis.
Let $$\vec{Q}$$ = î – ĵ

Question 23.
For any arbitrary motion in space, which of the following relations are true:

(The ‘average’ stands for average of the quantity over the time interval t1 to t2)
Solution:
(b) and (e)
(a) It is given that the motion of the particle is arbitrary. Therefore, the average velocity of the particle cannot be given by this equation.
(b) The arbitrary motion of the particle can be represented by this equation.
(c) The motion of the particle is arbitrary. The acceleration of the particle may also be non-uniform. Hence, this equation cannot represent the motion of the particle in space.
(d) The motion of the particle is arbitrary; acceleration of the particle may also be non-uniform. Hence, this equation cannot represent the motion of particle in space.
(e) The arbitrary motion of the particle can be represented by this equation.

Question 24.
Read each statement below carefully and state, with reasons and examples, if it is true or false:
A scalar quantity is one that
(a) is conserved in a process
(b) can never take negative values
(c) must be dimensionless
(d) does not vary from one point to another in space
(e) has the same value for observers with different orientations of axes.
(a) False
Reason: Despite being a scalar quantity, energy is not conserved in inelastic collisions.

(b) False
Reason: Despite being a scalar quantity, temperature can take negative values.

(c) False
Reason: Total path length is a scalar quantity. Yet it has the dimension of length.

(d) False
Reason: A scalar quantity such as gravitational potential can vary from one point to another in space.

(e) True
Reason: The value of a scalar does not vary for observers with different orientations of axes.

Question 25.
An aircraft is flying at a height of 3400 m above the ground. If the angle subtended at a ground observation point by the aircraft positions 10.0 s apart is 30°, what is the speed of the aircraft?
Solution:
The positions of the observer and the aircraft are shown in the given figure.

Height of the aircraft from ground, OR = 3400 m
Angle subtended between the positions, ∠POQ =30°
Time = 10 s
In Δ PRO:
tan15° = $$\frac{P R}{O R}$$
PR = OR tan 15°
= 3400 × tan15°
Δ PRO is similar to Δ RQO.
PR =RQ
Motion in a Plane 81
PQ = PR + RQ
= 2PR = 2 × 3400 tanl5°
= 6800 × 0.268 = 1822.4 m
speed of the aircraft = $$\frac{1822.4}{10}$$ = 182.24 m/s

Question 26.
A vector has magnitude and direction. Does it have a location in space? Can it vary with time? Will two equal vectors a and b at different locations in space necessarily have identical physical ‘ effects? Give examples in support of your answer.
No; Yes; No
Generally speaking, a vector has no definite locations in space. This is because a vector remains invariant when displaced in such a way that its magnitude and direction remain the same. However, a position vector has a definite location in space.

A vector can vary with time. For example, the displacement vector of a particle moving with a certain velocity varies with time.

Two equal vectors located at different locations in space need not produce the same physical effect. For example, two equal forces acting on an object at different points can cause the body to rotate, but their combination cannot produce an equal turning effect.

Question 27.
A vector has both magnitude and direction. Does it mean that anything that has magnitude and direction is necessarily a vector? The rotation of a body can be specified by the direction of the axis of rotation, and the angle of rotation about the axis. Does that make any rotation a vector?
No; No
A physical quantity having both magnitude and direction need not be considered a vector. For example, despite having magnitude and direction, current is a scalar quantity. The essential requirement for a physical quantity to be considered a vector is that it should follow the law of vector addition.

Generally speaking, the rotation of a body about an axis is not a vector quantity as it does not follow the law of vector addition. However, a rotation by a certain small angle follows the law of vector addition and is therefore considered a vector.

Question 28.
Can you associate vectors with (a) the length of a wire bent into a loop, (b) a plane area, (c) a sphere? Explain.
No; Yes; No
One cannot associate a vector with the length of a wire bent into a loop. One can associate an area vector with a plane area. The direction of this vector is normal, inward or outward to the plane area.
One cannot associate a vector with the volume of a sphere. However, an area vector can be associated with the area of a sphere.

Question 29.
A bullet fired at an angle of 30° with the horizontal hits the ground 3.0 km away. By adjusting its angle of projection, can one hope to hit a target 5.0 km away? Assume the muzzle speed to the fixed, and neglect air resistance.
Solution:
No
Range, R = 3 km
Angle of projection, θ = 30°
Acceleration due to gravity, g = 9.8 m/s2
Horizontal range for the projection velocity u0, is given by the relation :
R = $$\frac{u_{0}^{2} \sin 2 \theta}{g}$$
3 = $$\frac{u_{0}^{2}}{g}$$ sin 60°
$$\frac{u_{0}^{2}}{g}$$ = 2√3 ……………… (i)
The maximum range (Rmax) is achieved by the bullet when it is fired at an angle of 45° with the horizontal, that is,
Rmax = $$\frac{u_{0}^{2}}{g}$$ = ………………. (ii)
On comparing equations (i) and (ii), we get:
Rmax = 2√3 × 1.732 = 3.46 km
Hence, the bullet will not hit a target 5 km away.

Question 30.
A fighter plane flying horizontally at an altitude of 1.5 km with speed 720 km/h passes directly overhead an dnti-aircraft gun. At what angle from the vertical should the gun be fired for the shell with muzzle speed 600 ms-1 to hit the plane? At what minimum altitude should the pilot fly the plane to avoid being hit? (Take g = 10ms-2)
Solution:
Height of the fighter plane = 1.5 km = 1500 m
Speed of the fighter plane, υ = 720 km/h = 200 m/s
Let θ be the angle with the vertical so that the shell hits the plane. The
situation is shown in the given figure.

Muzzle velocity of the gun, u = 600 m/s
Time taken by the shell to hit the plane = t
Horizontal distance travelled by the shell = uxt
Distance travelled by the plane = υt
The shell hits the plane. Hence, these two distances must be equal.
uxt = υt
usinθ = υ
sinθ = $$\frac{v}{u}=\frac{200}{600}=\frac{1}{3}$$ 0.33
θ = sin-1 (0.33) = 19.5°
In order to avoid being hit by the shell, the pilot must fly the plane at
an altitude (H) higher than the maximum height achieved by the shell.

Question 31.
A cyclist is riding with a speed of 27 km/h. As he approaches a circular turn on the road of radius 80 m, he applies brakes and reduces his speed at the constant rate of 0.50 m/s every second. What is the magnitude and direction of the net acceleration of the cyclist on the circular turn?
Solution:
Speed of the cyclist, υ = 27 km/h = 7.5 m/s
Radius of the circular turn , r = 80m
Centripetal acceleration is given as:
a = $$\frac{v^{2}}{r}$$
= $$\frac{(7.5)^{2}}{80}$$ = 0.7 m/s2
The situation is shown in the given figure:

Suppose the cyclist begins cycling from point P and moves toward point Q. At point Q, he applies the brakes and decelerates the speed of the bicycle by 0.5 m/s2.
This acceleration is along the tangent at Q and opposite to the direction of motion of the cyclist.
Since the angle between ac and aT is 90°, the resultant acceleration a is given by:
a = $$\sqrt{a_{c}^{2}+a_{T}^{2}}=\sqrt{(0.7)^{2}+(0.5)^{2}}=\sqrt{0.74}$$ = 0.86 m/s2
and
tan θ = $$\frac{a_{c}}{a_{T}}$$
where θ is the angle of the resultant with the direction of velocity
tanθ = $$\frac{0.7}{0.5}$$ = 1.4
θ = tan-1 (1.4) = 54.46°
Hence, the net acceleration of the cyclist is 0.86 rn/s2, 54.60 0 with the direction of velocity.

Question 32.
(a) Show that for a projectile the angle between the velocity and the x-axis as a function of time is
θ(t) = tan ($$\frac{v_{0 y}-g t}{v_{0 x}}$$)

(b) Show that the projection angle θ0 for a projectile launched from the origin is given by
θ0 = tan-1 ($$\frac{\mathbf{4} \boldsymbol{h}_{\boldsymbol{m}}}{\boldsymbol{R}}$$)
where the symbols have their usual meaning.
Solution:
Let y Ox and y 0, respectively be the initial components of the velocity of the projectile along horizontal (x) and vertical (y) directions.
Let y and y , respectively be the horizontal and vertical components of velocity at a point P.

Time taken by the projectile to reach point P = t.
Applying the first equation of motion along the vertical and horizontal directions, we get:

## PSEB 11th Class Physics Solutions Chapter 8 Gravitation

Punjab State Board PSEB 11th Class Physics Book Solutions Chapter 8 Gravitation Textbook Exercise Questions and Answers.

## PSEB Solutions for Class 11 Physics Chapter 8 Gravitation

PSEB 11th Class Physics Guide Gravitation Textbook Questions and Answers

Question 1.
(a) You can shield a charge from electrical forces by putting it inside a hollow conductor. Can you shield a body from the gravitational influence of nearby matter by putting it inside a hollow sphere or by some other means?
(b) An astronaut inside a small spaceship orbiting around the earth cannot detect gravity. If the space station orbiting around the earth has a large size, can he hope to detect gravity?
(c) If you compare the gravitational force on the earth due to the sun to that due to the moon, you would find that the Sun’s pull is greater than the moon’s pull. (You can check this yourself using the data available in the succeeding exercises). However, the tidal effect of the moon’s pull is greater than the tidal effect of sun. Why?
(a) No, Gravitational influence of matter on nearby objects cannot be screened by any means. This is because gravitational force unlike electrical forces is independent of the nature of the material medium. Also, it is independent of the status of other objects.

(b) Yes, If the size of the space station is large enough, then the astronaut will detect the change in Earth’s gravity (g).

(c) Tindal effect depends inversely upon the cube of the distance while gravitational force depends inversely on the square of the distance. Since the distance between the Moon and the Earth is smaller than the distance between the Sun and the Earth, the tidal effect of the Moon’s pull is greater than the tidal effect of the Sun’s pull.

Question 2.
Choose the correct alternative:
(a) Acceleration due to gravity increases/decreases with increasing altitude.
(b) Acceleration due to gravity increases/decreases with increasing depth (assume the earth to be a sphere of uniform density).
(c) Acceleration due to gravity is independent of mass of the earth/mass of the body.
(d) The formula -G Mm(1 /r2 -1/r1) is more/less accurate than the formula mg (r2 – r1) for the difference of potential energy between two points r2 and r1 distance away from the centre of the earth.
Solution:
(a) Decreases,
Explanation : Acceleration due to gravity at altitude h is given by the relation:
gh = $$\left(1-\frac{2 h}{R_{e}}\right) g$$
where, Re = Radius of the Earth
g = Acceleration due to gravity on the surface of the Earth
It is clear from the given relation that acceleration due to gravity decreases with an increase in height.

(b) Decreases,
Explanation : Acceleration due to gravity at depth d is given by the relation:
gd = $$\left(1-\frac{d}{R_{e}}\right) g$$
It is clear from the given relation that acceleration due to gravity decreases with an increase in depth.

(c) Mass of the body,
Explanation : Acceleration due to gravity of body of mass m is given by the relation:
g = $$\frac{G M_{e}}{R^{2}}$$
where, G = Universal gravitational constant
Me = Mass of the Earth
Re = Radius of the Earth
Hence, it can be inferred that acceleration due to gravity is independent of the mass of the body.

(d) More,
Explanation : Gravitational potential energy of two points r2 and r1 distance away from the centre of the Earth is respectively given by:
V(r1) = -GmM
V(r2) = – $$\frac{G m M}{r_{2}}$$
V = V (r2) – V(r1) = -GmM$$\left(\frac{1}{r_{2}}-\frac{1}{r_{1}}\right)$$.
Hence, this formula is more accurate than the formula mg (r2 – r1).

Question 3.
Suppose there existed a planet that went around the Sun twice as fast as the Earth. What would be its orbital size as compared to that of the Earth?
Solution:
Time taken by the Earth to complete one revolution around the Sun,
Te = 1 year
Orbital radius of the Earth in its orbit, Re = 1 AU
Time taken by the planet to complete one revolution around the Sun,
Tp = $$\frac{1}{2} T_{e}=\frac{1}{2}$$ year
Orbital radius of the planet = Rp
From Kepler’s third law of planetary motion, we can write

Hence, the orbital radius of the planet will be 0.63 times smaller than that of the Earth.

Question 4.
I0, one of the satellites of Jupiter, has an orbital period of 1.769 days and the radius of the orbit is 4.22 x 108 m. Show that the mass of Jupiter is about one-thousandth that of the sun.
Solution:
Orbital period of I0, TI0 = 1.769 days = 1.769 x 24 x 60 x 60 s
Orbital radius of II0, RI0 = 4.22 x 108 m
Satellite I0 is revolving around the Jupiter Mass of the latter is given by the relation:
Mj = $$\frac{4 \pi^{2} R_{I_{O}}^{3}}{G T_{I_{O}}^{2}}$$ ……………………….. (i)

where Mj = Mass of Jupiter
G = Universal gravitational constant Orbital period of the Earth,
Te = 365.25 days = 365.25 x 24 x 60 x 60 s Orbital radius of the Earth,
Re =1 AU = 1.496 x 1011 m
Mass of Sun is given as:
Ms = $$\frac{4 \pi^{2} R_{e}^{3}}{G T_{e}^{2}}$$ ………………………. (ii)

= 1045.04
∴ $$\frac{M_{s}}{M_{j}} \sim 1000$$
$$M_{s} \sim 1000 \times M_{j}$$
Hence, it can be inferred that the mass of Jupiter is about one thousandth that of the Sun.

Question 5.
Let us assume that our only consists of 2.5 x 1011 stars each of one solar mass. How long will a star at a distance of 50,000 ly from the galactic center take to complete one revolution? Take the diameter of the Milky Way to be 105 ly.
Solution:
Number of stars in our galaxy, = 2.5 x 1011
Mass of each star = one solar mass =2 x 1030 kg
Mass of our galaxy,M =2.5 x 1011 x 2 x 1030 = 5 x 1041 kg
Diameter of Milky Way, d = 105 ly
Radius of Milky Way,r= 5 x 104 ly
∵ 1 ly=946 x 1015m
∴ r=5 x 104 x 9.46 x 1015
=4.73 x 1020m

Since a star revolves around the galactic centre of the Milky Way, its time period is given by the relation:
T = $$\left(\frac{4 \pi^{2} r^{3}}{G M}\right)^{1 / 2}$$

1 year = 365 x 324 x 60 x 60 s
∵ 1 s = $$\frac{1}{365 \times 24 \times 60 \times 60}$$ years
∴ 1.12 x 1016 s = $$\frac{1.12 \times 10^{16}}{365 \times 24 \times 60 \times 60}$$ = 3.55 x 10 8 years

Question 6.
Choose the correct alternative:
(a) If the zero of potential energy Is at infinity, the total energy of an orbiting satellite is negative of its kinetic/potential energy.
(b) The energy required to launch an orbiting satellite out of Earth’s gravitational influence Is more/less than the energy required to project a stationary object at the same height (as the satellite) out of Earth’s influence.
Solution:
(a) Kinetic energy
Explanation: Total mechanical energy of a satellite is the sum of its kinetic energy (always positive) and potential energy (maybe negative). At infinity, the gravitational potential energy of the satellite is zero. As the Earth-satellite system is a bound system, the total energy of the satellite is negative.

Thus, the total energy of an orbiting satellite at infinity is equal to the negative of its kinetic energy.

(b) Less
Explanation: An orbiting satellite acquires certain amount of energy that enables it to revolve around the Earth. This energy is provided by its orbit. It requires relatively lesser energy to move out of the influence of the Earth’s gravitational field than a stationary object on the Earth’s surface that initially contains no energy.

Question 7.
Does the escape speed of a body from the Earth depend on
(a) the mass of the body,
(b) the location from where it is projected,
(c) the direction of projection,
(d) the height of the location from where the body is launched?
(a) No, we know that the escape velocity of the body is given by
Ve = $$\sqrt{\frac{2 G M}{R}}$$ , where M and R are mass and radius of Earth. Thus clearly, it does not depend on the grass of the body as Ve is independent of it.

(b) Yes, we know that Ve depends upon the gravitational potential at the point from where the body is launched. Since the gravitational potential depends on the latitude and height of the point, therefore the escape velocity depends on the location of the point from where it is projected. It can also be experienced as:
Ve = $$\sqrt{2 g r}$$ . As g has different values at different heights. Therefore, Ve depends upon the height of location.

(c) No, it does not depend on the direction of projection as Ve is independent of the direction of projection.

(d) Yes, it depends on the height of location from where the body is launched as explained in (b).

Question 8.
A comet orbits the Sun in a highly elliptical orbit. Does the comet have a constant
(a) linear speed,
(b) angular speed,
(c) angular momentum,
(d) kinetic energy,
(e) potential energy,
(f) total energy throughout its orbit? Neglect any mass loss of the comet when it comes very close to the Sun.
Solution:
(a) No, according to law of conservation of linear momentum L = mvr constant, therefore the comet moves faster when it is close to the sun and moves slower when it is farther away from the sun. Therefore, the speed of the comet does not remain constant.

(b) No, as the linear speed varies, the angular speed also varies. Therefore, angular speed of the comet does not remain constant.

(c) Yes, as no external torque is acting on the comet, therefore, according to law of conservation of angular momentum, the angular momentum of the comet remain constant.

(d) No, kinetic energy of the comet = $$\frac{1}{2}$$ mν2 As the linear speed of the comet changes as its kinetic energy also changes. Therefore, its KE does not remains constant.

(e) No, potential energy of the comet changes as its kinetic energy changes.

(f) Yes, total energy of a comet remain constant throughout its orbit.

Question 9.
Which of the following symptoms is likely to afflict an astronaut in space
(a) swollen feet,
(b) swollen face,
(d) orientational problem?
(b), (c), and (d)
Explanations:
(a) Legs hold the entire mass of a body in standing position due to gravitational pull. In space, an astronaut feels weightlessness because of the absence of gravity. Therefore, swollen feet of an astronaut do not affect him/her in space.

(b) A swollen face is caused generally because of apparent weightlessness in space sense organs such as eyes, ears, nose, and mouth constitute a person’s face. This symptom can affect an astronaut in space.

(c) Headaches are caused because of mental strain. It can affect the working of an astronaut in space.

(d) Space has different orientations. Therefore, orientational problems can affect an astronaut in space.

Question 10.
In the following two exercises, choose the correct answer from among the given ones:
The gravitational intensity at the center of a hemispherical shell of uniform mass density has the direction indicated by the arrow (see figure) (i) a, (ii) b, (iii) c, (iv) 0.

Solution:
(iii) Gravitational potential (V) is constant at all points in a spherical shell.
Hence, the gravitational potential gradient $$\left(\frac{d V}{d r}\right)$$ is zero everywhere inside the spherical shell. The gravitational potential gradient is equal to the negative of gravitational intensity. Hence, intensity is also zero at all points inside the spherical shell. This indicates that gravitational forces acting at a point in a spherical shell are symmetric.

If the upper half of a spherical shell is cut out (as shown in the given figure), then the net gravitational force acting on a particle located at center O will be in the downward direction.

Since gravitational intensity at a point is defined as the gravitational force per unit mass at that point, it will also act in the downward direction. Thus, the gravitational intensity at center O of the given hemispherical shell has the direction as indicated by arrow c.

Question 11.
For the above problem, the direction of the gravitational intensity at an arbitrary point P is indicated by the arrow
(i) d,
(ii) e,
(iii) f,
(iv) g.
Solution:
(ii) Gravitational potential (V) is constant at all points in a spherical shell.
Hence, the gravitational potential gradient $$\left(\frac{d V}{d r}\right)$$ inside the spherical shell. The gravitational potential gradient is equal to the negative of gravitational intensity. Hence, intensity is also zero at all points inside the spherical shell. Ibis indicates that gravitational forces acting at a point in a spherical shell are symmetric. If the upper half of a spherical shell is cut out (as shown in the given figure), then the net gravitational force acting on a particle at an arbitrary point P will be in the downward direction.

Since gravitational intensity at a point is defined as the gravitational force per unit mass at that point, it will also act in the downward direction. Thus, the gravitational intensity at an arbitrary point P of the hemispherical shell has the direction as indicated by arrow e.

Question 12.
A rocket Is fired from the Earth towards the Sun. At what distance from the Earth’s centre is the gravitational force on the rocket zero? Mass of the Sun = 2 x 1030kg, mass of the Earth = 6 x 1024 kg. Neglect the effect of other planets etc. (orbital radius = 1.5 x 1011 m)
Solution:
Mass of the Sun, Ms = 2 x 1030 kg
Mass of the Earth, Me = 6 x 10 24 kg
Orbital radius, r = 1.5 x 1011 m
Mass of the rocket = m

Let x be the distance from the center of the Earth where the gravitational force acting on satellite P becomes zero.
From Newton’s law of gravitation, we can equate gravitational forces acting on satellite P under the influence of the Sun and the Earth as:

= 2.59 x 108 m

Question 13.
How will you ‘weigh the Sun’, that is estimate its mass? The mean orbital radius of the Earth around the Sun is 1.5 x 108 km.
Solution:
Orbital radius of the Earth around the Sun, r = 1.5 x 108 km = 1.5 x 1011 m
Time taken by the Earth to complete one revolution around the Sun,
T = 1 year = 365.25 days = 365.25 x 24 x 60 x 60 s
Universal gravitational constant, G = 6.67 x 10-11N-m2 kg-2
Thus, mass of the Sun can be calculated using the relation,
M = $$\frac{4 \pi^{2} r^{3}}{G T^{2}}$$
= $$\frac{4 \times(3.14)^{2} \times\left(1.5 \times 10^{11}\right)^{3}}{6.67 \times 10^{-11} \times(365.25 \times 24 \times 60 \times 60)^{2}}$$
= $$\frac{133.24 \times 10^{33}}{6.64 \times 10^{4}}$$ = 2.0 x 1030kg
Hence, the mass of the Sun is 2.0 x 1030 kg.

Question 14.
A Saáurn year is 29.5 times the Earth year. How far is the Saturn from the Sun if the Earth is 1.5 x 108 km away from the Sun?
Solution:
Distance of the Earth from the Sun, re = 1.5x 108 km= 1.5 x 1011 m
Time period of the Earth = Te
Time period of Saturn, Ts = 29.5 Te
Distance of Saturn from the Sun = rs
From Kepler’s third law of planetary motion, we have
T=$$\left(\frac{4 \pi^{2} r^{3}}{G M}\right)^{\frac{1}{2}}$$
For Saturn and Sun, we can write

= 1.43 x 1012 m

Question 15.
A body weigths 63 N on the surface of the Earth. What is the gravitational force on it due to the Earth at a height equal to half the radius of the Earth?
Solution:
Weight of the body, W =63 N
Acceleration due to gravity at height h from the Earth’s surface is given by the relation:
g’ = $$\frac{g}{\left(1+\frac{h}{R_{e}}\right)^{2}}$$
where, g = Acceleration due to gravity on the Earth’s surface
For h= $$\frac{R_{e}}{2}$$
g’ = $$\frac{g}{\left(1+\frac{R_{e}}{2 \times R_{e}}\right)^{2}}=\frac{g}{\left(1+\frac{1}{2}\right)^{2}}=\frac{4}{9} g$$

Weight of a body of mass m at height h is given as
W’ = mg’
= m x $$\frac{4}{9}$$ g = $$\frac{4}{9}$$ x mg
= $$\frac{4}{9}$$ W
= $$\frac{4}{9}$$ x 63 =28 N

Question 16.
Assuming the Earth to be a sphere of uniform mass density, how much would a body weigh halfway down to the centre of the Earth if it weighed 250 N on the surface?
Solution:
Weight of a body of mass m at the Earth’s surface, W = mg = 250 N
Body of mass m is located at depth, d = $$\frac{1}{2}$$ Re
where, Re = Radius of the Earth
Acceleration due to gravity at depth d is given by the relation
g’ = $$\left(1-\frac{d}{R_{e}}\right) g=\left(1-\frac{R_{e}}{2 \times R_{e}}\right) g=\frac{1}{2} g$$
Weight of the body at depth d,
W’ = mg’
= m x $$\frac{1}{2} g=\frac{1}{2}$$ mg = $$\frac{1}{2}$$ W
= $$\frac{1}{2}$$ x 250 = 125 N

Question 17.
A rocket is fired vertically with a speed of 5 km s-1 from the Earth’s surface. How far from the Earth does the rocket ‘ go before returning to the earth? Mass of the Earth = 6.0 x 1024 kg; mean radius of the Earth = 6.4 x 106 m;
G = 6.67 x 10 -11 N-m2 kg-2.
Solution:
Velocity of the rocket, ν = 5 km/s = 5 x 103 m/s
Mass of the Earth, Me = 6.0 x 1024 kg
Radius of the Earth, Re = 6.4 x 106 m
Height reached by rocket mass, m = h

At the surface of the Earth,
Total energy of the rocket = Kinetic energy + Potential energy
= $$\frac{1}{2} m v^{2}+\left(\frac{-G M_{e} m}{R_{e}}\right)$$
At highest point h,
ν = 0
and, Potential energy = – $$\frac{G M_{e} m}{R_{e}+h}$$
Total energy of the rocket = 0+ $$\left(-\frac{G M_{e} m}{R_{e}+h}\right)=-\frac{G M_{e} m}{R_{e}+h}$$
From the law of conservation of energy, we have
Total energy of the rocket at the Earth’s surface=Total energy at height h

Height achieved by the rocket with respect to the centre of the Earth = Re +h
= 6.4 x 106 +1.6 x 106 = 8.0 x 106 m

Question 18.
The escape speed of a projectile on the Earth’s surface is 11.2 km s-1. A body is projected out with thrice this speed. What is the speed of the body far away from the Earth? Ignore the presence of the Sun and other planets.
Solution:
Escape velocity of a projectile from the Earth, υesc = 11.2 km/s
Projection velocity of the projectile, vp = 3 vesc
Mass of the projectile = m
Velocity of the projectile far away from the Earth = vf
Total energy of the projectile on the Earth = $$\frac{1}{2} m v_{p}^{2}-\frac{1}{2} m v_{\mathrm{esc}}^{2}$$
The gravitational potential energy of the projectile far away from the Earth is zero.
Total energy of the projectile far away from the Earth = $$\frac{1}{2}$$ mv2f
From the law of conservation of energy, we have

Question 19.
A satellite orbits the Earth at a height of 400 km above the surface. How much energy must be expended to rocket the satellite out Of the Earth’s gravitational influence? Mass of the satellite = 200 kg mass of the Earth = 6.0 x 1024 kg; radius of the Earth=6.4x 106 m;G=6.67 x 10-11 N-m2 kg-2.
Solution:
Mass of the Earth, Me =6.0 x 1024 kg
Mass of the satellite, m = 200 kg
Radius of the Earth, Re = 6.4 x 106 m
Universal gravitational constant, G = 6.67 x 10-11 N-m2kg2
Heightofthesatellite,h =400 km=4 x 105 m=0.4 x 106 m

Total energy of the satellite at height h = $$\frac{1}{2} m v^{2}+\left(\frac{-G M_{e} m}{R_{e}+h}\right)$$
Orbital velocity of the satellite, ν = $$\sqrt{\frac{G M_{e}}{R_{e}+h}}$$
Total energy of height, h = $$\frac{1}{2} m\left(\frac{G M_{e}}{R_{e}+h}\right)-\frac{G M_{e} m}{R_{e}+h}=-\frac{1}{2}\left(\frac{G M_{e} m}{R_{e}+h}\right)$$

The negative sign indicates that the satellite is bound to the Earth. This is called bound energy of the satellite.
Energy required to send the satellite out of its orbit = – (Bound energy)

Question 20.
Two stars each of one solar mass (= 2x 1030 kg) are approaching each other for a head-on collision. When they are a distance 109 km, their speeds are negligible. What is the speed with which they collide? The radius of each star is 104 km. Assume the stars to remain undistorted until they collide. (Use the known value of G).
Solution:
Mass of each star, M = 2 x 1030 kg
Radius of each star, R = 104 km = 107 m
Distance between the stars, r = 109 km = 1012 m
For negligible speeds, ν = 0 total energy of two stars separated at distance r
= $$\frac{-G M M}{r}+\frac{1}{2} m v^{2}$$
= $$\frac{-G M M}{r}+0$$ = 0 ………………………… (i)

Now, consider the case when the stars are about to collide:
The velocity of the stars = ν
Distance between the centers of the stars = 2R
Total kinetic energy of both stars = $$\frac{1}{2} M v^{2}+\frac{1}{2} M v^{2}$$ = Mv2
Total potential energy of both stars = $$\frac{-G M M}{2 R}$$
Total energy of the two stars = Mv2 – $$\frac{-G M M}{2 R}$$ ………………………. (ii)
Using the law of conservation of energy, we can write

Question 21.
Two heavy spheres each of mass 100 kg and radius 0.10 m are placed 1.0 m apart on a horizontal table. What is the gravitational force and potential at the midpoint of the line joining the centers of the spheres? Is an object placed at that point in equilibrium? If so, is the equilibrium stable or unstable?
Solution:
The situation is represented in the following figure :

Mass of each sphere, M = 100 kg
Separation between the spheres, r = 1 m
X is the mid-point between the spheres. Gravitational force at point X will be zero. This is because gravitational force exerted by each sphere will act in opposite directions. Gravitational potential at point X

Any object placed at point X will be in equilibrium state, but the equilibrium is unstable. This is because any change in the position of the object will change the effective force in that direction.

Question 22.
As you have learnt in the text, a geostationary satellite orbits the Earth at a height of nearly 36,000 km from the surface of the Earth. What is the potential due to Earth’s gravity at the site of this satellite? (Take the potential energy at infinity to be zero). Mass of the Earth = 6.0 x 1024 kg, radius = 6400 km.
Solution:
Mass of the Earth, M = 6.0 x 1024 kg
Radius of the Earth, R = 6400 km = 6.4 x 106 m
Height of a geostationary satellite from the surface of the Earth,
h = 36000 km = 3.6 x 107m
Gravitational potential energy due to Earth’s gravity at height h,

Question 23.
A star 2.5 times the mass of the Sun and collapsed to a size of 12 km rotates with a speed of 1.2 revs. per second. (Extremely compact stars of this kind are known as neutron stars. Certain stellar objects called pulsars belong to this category). Will an object placed on its equator remain stuck to its surface due to gravity? (Mass of the Sun = 2 x 1030 kg).
Solution:
Yes, A body gets stuck to the surface of a star if the inward gravitational force is greater than the outward centrifugal force caused by the rotation of the star.
Gravitational force, fg = $$\frac{G M m}{R^{2}}$$
where,M =Mass of the star=2.5 x 2 x 1030 = 5 x 1030 kg
m = Mass of the body

∴ fg = $$\frac{6.67 \times 10^{-11} \times 5 \times 10^{30} \times m}{\left(1.2 \times 10^{4}\right)^{2}}$$
= 2.31 x 1012 mN
Centrifugal force, fc = mrω2

where, ω = Angular speed = 2 πv
ν = Angular frequency = 1.2 rev s-1
f c=mR(2πv)2
= m x (1.2 x 104) x 4 x(3.14)2 x (1.2)2 = 1.7 x 105 mN
Since fg > fc, the body will remain stuck to the surface of the star.

Question 24.
A spaceship is stationed on Mars. How much energy must be expended on the spaceship to launch it out of the solar system? Mass of the spaceship = 1000 kg; mass of the Sun = 2 x 1030 kg; mass of mars = 6.4 x 1023 kg; radius of mars = 3395 km; radius of the orbit of mars = 2.28 x 108 km;
G = 6.67 x 10-11 N-m2kg-2.
Solution:
Given, mass of the Sun, M = 2 x 1030 kg

Mass of the mars, m = 6.4 x 1023 kg
Mass of spaceship, Δm = 1000 kg
Radius of orbit of the mars, r0 = 2.28 x 1011
Radius of the mars, r = 3.395 x 106 m
If v is the orbital velocity of mars, then

Since the velocity of the spaceship is the same as that of the mars, Kinetic energy of the spaceship,

Total potential energy of the spaceship,
U = Potential energy of the spaceship due to its being in the gravitational
field of the mars + potential energy of the spaceship due to its being in the gravitational field of the Sun.

Total energy of the spaceship E=K +U = 2.925 x 1011 J – 5.977 x 1011J
= – 3.025 x 1011 J = -3.1 x 1011 J
Negative energy denotes that the spaceship is bound to the solar system. Thus, the energy needed by the spaceship to escape from the solar system = 3.1 x 1011 J.

Question 25.
A rocket is fired ‘vertically’ from file surface of mars with a speed of 2 kms-1. If 20% of its initial energy is lost due to martian atmospheric resistance, how far will the rocket go from the surface of Mars before returning to it?
Mass of mars = 6.4x 1023 kg; radius of mars = 3395 km;
G = 6.67 x 10-11 N-m2 kg2.
Solution:
Initial velocity of the rocket, ν = 2 km/s = 2 x 103 m/s
Mass of Mars, M = 6.4 x 1023 kg .
Radius of Mars, R = 3395 km = 3.395 x 106 m
Universal gravitational constant, G = 6.67 x 10-11 N-m2 kg-2
Mass of the rocket = m
Initial kinetic energy of the rocket = $$\frac{1}{2} m v^{2}$$
Initial potential energy of the rocket = $$\frac{-G M m}{R}$$
Total initial energy = $$\frac{1}{2} m v^{2}-\frac{G M m}{R}$$

If 20 % of initial kinetic energy is lost due to martian atmospheric resistance, then only 80 % of its kinetic energy helps in reaching a height.
Total initial energy available = $$\frac{80}{100} \times \frac{1}{2} m v^{2}-\frac{G M m}{R}=0.4 m v^{2}-\frac{G M m}{R}$$
Maximum height reached by the rocket = h
At this height, the velocity and hence, the kinetic energy of the rocket will become zero.
Total energy of the rocket at height h = $$-\frac{G M m}{(R+h)}$$
Applying the law of conservation of energy for the rocket, we can write

= 495 x 103 m = 495 km

## PSEB 11th Class Biology Solutions Chapter 1 The Living World

Punjab State Board PSEB 11th Class Biology Book Solutions Chapter 1 The Living World Textbook Exercise Questions and Answers.

## PSEB Solutions for Class 11 Biology Chapter 1 The Living World

PSEB 11th Class Biology Guide The Living World Textbook Questions and Answers

Question 1.
Why are living organisms classified?
Living organisms are classified because:

• there are millions of organisms on the earth, which need a proper system of classification for identification.
• a number of new organisms are discovered each year. They require a particular system to be identified and to find out their correct position in a group.

Question 2.
Why are the classification systems changing every now and then?
Evolution is the major factor responsible for the change in classification systems. Since, evolution still continues, so many different species of plants and animals are added in the already existed biodiversity. These newly discovered plant and animal specimens are then identified, classified and named according to the already existing classification systems. Due to evolution, animal and plant species keep on changing, so necessary changes in the already existed classification systems are necessary to place every newly discovered plant and animal in their respective ranks.

Question 3.
What different criteria would you choose to classify people that you meet often?
The different scientific criteria to classify people that we meet often would be :
(i) Nomenclature: It is the science of providing distinct and proper names to the organisms. It is the determination of correct name as per established universal practices and rules.

(ii) Classification: It is the arrangement of organisms into categories based on systematic planning. In classification various categories used are class, order, family, genus and species.

(iii) Identification: It is the determination of correct name and place of an organism. Identification is used to tell that a particular species is similar to other organism of known identity. This includes assigning an organism to a particular taxonomic group.
The same criteria can be applied to the people we meet daily. We can identify them will their names classify them according to their living areas, profession, etc.

Question 4.
What do we learn from identification of individuals and populations?
Identification of individuals and population categorized it into a species. Each species has unique characteristic features. On the basis of these features, it can be distinguished from other closely related species, e. g.,

Question 5.
Given below is the scientific name of Mango. Identify the correctly written name.
(i) Mangifera Indica
(ii) Mangifera indica.
(ii) Mangifera indica (the name of species can never begins with a capital letter).

Question 6.
Define a taxon. Give some examples of taxa at different hierarchical levels.
Taxon is a grouping of organisms of any level in hierarchical classification, which is based on some common characteristics, e.g., insects represent a class of phylum – Arthropoda. All the insects possess common characters of three pairs of jointed legs. The term ‘taxon’ was introduced by ICBN in 1956. Examples of taxa are kingdom, phylum or division, class, order, family, genus and species. These taxa form taxonomic hierarchy, e.g., taxa for human :
Phylum – Chordata
Class – Mammalia
Order – Primata
Family – Hominidae
Genus – Homo
Species – sapiens

Question 7.
Can you identify the correct sequence of taxonomical categories?
(a) Species → Order → Phylum → Kingdom
(b) Genus → Species → Order → Kingdom
(c) Species → Genus → Order → Phylum
The correct sequence of taxonomical categories is as follows :
Species → Genus → Order → Phylum

Question 8.
Try to collect all the currently accepted meanings for the word ‘species’. Discuss with your teacher the meanings of species in case of higher plants and animals on one hand, and bacteria on the other hand.
A group of individual organisms with fundamental similarities is called species. It can be distinguished from other closely related species on the basis of distinct morphological differences.
In case of higher plants and animals, one genus may have one or more than one species, e.g.,Panthera leo (lion) and Panthera tigris (tiger).
In this example, Panthera is genus, which includes leo (lion) and tigris (tiger) as species.
In case of bacteria, different categories are present on the basis of shape. These are spherical, coccus, rod-shaped, comma and spiral-shaped. Thus, meaning of species in case of higher organisms and bacteria are different.

Question 9.
Define and understand the following terms:
(i) Phylum,
(ii) Class,
(iii) Family,
(iv) Order,
(v) Genus.
(i) Phylum: Phylum comes next to Kingdom in the taxonomical hierarchy. All broad characteristics of an animal or plant are defined in a phylum. For example all chordates have a notochord and gill at some stage of life cycle. Similarly all arthropods have jointed legs made of chitin.

(ii) Class: The category class includes related orders. It is higher than order and lower than phylum. For example, class – Mammalia has order – Carnivora, Primata, etc.

(iii) Family: It is the category higher than genus and lower than order, which has one or more related genera having some common features. For example, Felidae, Canidae, etc.

(iv) Order: Order further zeroes down on characteristics and includes related genus. For example humans and monkeys belong to the order primates. Both humans and monkeys can use their hands to manipulate objects and can walk on their hind legs.

(v) Genus: It comprises a group of related species which has more characters in common in comparison to species of other genera. We can say that genera are aggregates of closely related species. For example, j potato, tomato and brinjal are three different species but all belong to the genus Solatium. Lion (Panthera leo), leopard (P. pardus) and tiger (P. tigris) with several common features, are all species of the genus Panthera. This genus differs from another genus Felis which includes cats.

Question 10.
How is a key helpful in the identification and classification of an
organism?
Key is a device (scheme) of diagnostic alternate (contrasting) characters, which provide an easy means for the identification of unknown organism. The keys are taxonomic literature based on the contrasting characters generally a pair called couplet. Each statement in the key is called a lead. Separate taxonomic keys are required for each taxonomic category such as family, genus and species for identification purposes. Being analytical ‘ in nature, two types of keys are commonly used-indented key and bracketed key.

(i) Indented key provides sequence of choice between two or more statements of characters of species. The user has to make correct choise for identification.

(ii) Bracketed key (1) are used for contrasting characters like indented key but they are not repeated by intervening sub-dividing character and each character is given a number in brackets.

Question 11.
Illustrate the taxonomical hierarchy with suitable examples of a plant and an animal.

Taxonomical hierarchy is the system of arrangement of taxonomic categories in a descending order depending upon their relative dimensions. It was introduced by Linnaeus (1751) and is therefore, also called Linnaeus hierarchy. Each category, referred to as a unit of classification commonly called as taxon (pi. taxa), e.g., taxonomic categories and hierarchy can be illustrated by a group of organisms, i.e., insects. The common features of insects is ‘three pair of jointed legs’. It means insects are recognisable objects which can be classified, so given a rank or category.
Category further denotes a rank. Each rank or taxon, represents a unit of classification taxonomic studies of all plants and animals led to the development of common categories such as kingdom, phylum or division (for plants), class, order, family, genus and species. All organisms, including those in the plant and animal, kingdoms have ‘species’ as the lowest category.

To place an organism in various categories is to have the knowledge of characters of an individual or group of organism. This helps to identify similarities and dissimilarities among the individual of the same kind of organisms as well as of other kinds of organism. Some organisms with their taxonomical categories are given in following table:

## PSEB 11th Class Biology Solutions Chapter 21 Neural Control and Coordination

Punjab State Board PSEB 11th Class Biology Book Solutions Chapter 21 Neural Control and Coordination Textbook Exercise Questions and Answers.

## PSEB Solutions for Class 11 Biology Chapter 21 Neural Control and Coordination

PSEB 11th Class Biology Guide Neural Control and Coordination Textbook Questions and Answers

Question 1.
Briefly describe the structure of the following:
(a) Brain
(b) Eye
(c) Ear
(a) Brain
The human brain has the following parts :
(i) Cerebrum
A deep cleft called longitudinal fissure divides the brain/cerebrum into two halves-cerebral hemispheres.
The two cerebral hemispheres are joined together by bundles of densely packed nerve fibers, called corpus callosum.
The outer surface of cerebrum, the cerebral cortex, is called grey matter, due to its greyish appearance; the cell bodies of the neurons are concentrated in this region; it contains motor areas, sensory areas and association areas.
Inner to the cortex is the white matter, that consists of myelinated nerve fibers in the form of nerve fibre tracts.

(ii) Thalamus
Thalamus is the major coordinating centre for sensory and motor signals.

(iii) Hypothalamus
It has centers to control body temperature, hunger, thirst, etc.
It contains several groups of neurosecretory cells, which secrete hormones.

(iv) Limbic System
The inner parts of the cerebral hemispheres and a group of deep structures called amygdala, hippocampus, etc. form a complex structure, called limbic system. Along with the hypothalamus, it is involved in the regulation of sexual behavior, expression of emotions, motivation, etc.

(v) Midbrain
Midbrain is located between the hypothalamus of the forebrain and the pons of the hindbrain. The dorsal portion of the midbrain consists of four small lobes, called corpora quadrigemina. A canal, called cerebral aqueduct passes through the midbrain.

(vi) Hindbrain
It consists of pons, cerebellum and medulla oblongata. The medulla contains centres which control vital functions like respiration, cardiovascular reflexes and gastric secretions. The medulla continues down as the spinal cord.

(b) Eye

• Each eye ball consists of three concentric layers, the outermost sclera, middle choroid and innermost retina.
• The sclera in the front (l/6th) forms the transparent cornea.
• The middle choroid is highly vascular and pigmented, that prevents internally reflected light within the eye; just behind the junction between cornea and sclera, the choroid becomes thicker forming the ciliary body.
• The iris extends from the ciliary body in front of the lens; it controls the dilation or constriction of pupil.
• The lens is suspended from the ciliary body, by suspensory ligaments.
• The anterior chamber of eye is filled with an aqueous clear fluid, aqueous humor and the posterior chamber has a gelatinous material, vitreous humor.
• The retina is composed of three layers of cells; the photoreceptor layer contains rods and cones, the intermediate layer has bipolar neurons, which synapse with retinal ganglion cells and their axons bundle to form optic nerve.
• The photoreceptor cells (rods and cones) contain the light sensitive proteins, called photopigments.
• The point in the retina where the optic nerve leaves the eye and the retinal blood vessels enter the eye is called a blind spot; there are no photoreceptor cells in this region.
• Lateral to blindspot, there is a yellowish pigmented spot, called macula lutea with a central pit called fovea.
• The fovea is the region where only cones are densely packed and it is the point where acuity (resolution) vision is the greatest.

(c) Ear
The ear performs two sensory functions, namely
(a) hearing and

(b) maintenance of body balance.

• Ear consists of three parts: external ear, middle ear and internal ear.
• The external ear consists of the pinna, and external auditory meatus.
• The tympanic membrane separates the middle ear from the external ear.
• The middle ear (tympanic cavity) is an air- filled chamber, which is connected to pharynx by Eustachian tube.
• The middle ear lodges three small bones, the ear ossicles namely, the malleus, incus and stapes.
• The middle ear communicates with the internal ear through the oval window and round window.
• The inner ear is a fluid-filled chamber and called labyrinth; it has two parts, an outer bony labyrinth, inside
• which a membranous labyrinth is floating in the perilymph; the membranous labyrinth is filled with a fluid, called endolymph.
• The labyrinth is divided into two parts, the cochlea and vestibular apparatus.
• Cochlea is the coiled portion of the labyrinth and its membranes, Reissner’s membrane and basilar membrane divide the perilymph-filled bony labyrinth into an upper scala vestibule, middle scala media and a lower scala tympani; scala media is filled with endolymph.
• At the base of the cochlea, scala vestibuli ends at the oval window, while the scala tympani terminates at the round window, that opens to the middle ear.
• Organ of Corti is the structural unit of hearing; it consists of hair cells which are the auditory receptors and is located on the basilar membrane.
• A thin elastic tectorial membrane lies over the row of hair cells.
• The vestibular apparatus is composed of three semicircular canals and an otolith organ or vestibule.
• The otolith organ has two parts namely the utricle and saccule.
• The utricle and saccule also contain a projecting ridge, called macula.
• The crista ampullar and macula are the specific receptors of the vestibular apparatus, for maintaining body balance.

Question 2.
Compare the following:
(a) Central Neural System (CNS) and Peripheral Neural System (PNS)
(b) Resting potential and action potential
(c) Choroid and retina
(a) Comparison between Central Neural System (CNS) and Peripheral Neural System (PNS): The CNS includes the brain and the spinal cord and is the site of information processing and control. The PNS comprises of all the nerves of the body associated with the CNS (brain and spinal cord). The nerve fibers of the PNS are of two types :
(i) Afferent fibers, (ii) Efferent fibers

(b) Comparison between Resting Potential and Action Potential:
The electrical potential difference across the resting plasma membrane A is called the resting potential. The electrical potential difference across the plasma membrane at the site A is called the action potential, which is in fact termed as a nerve impulse.

(c) Comparison between Choroid and Retina: The middle layer of eyeball which contains many blood vessels and looks bluish in colour is known as choroid. The choroid layer is thin over the posterior two-thirds of the eyeball, but it becomes thick in the anterior part to form the ciliary body. The ciliary body itself continues forward to form a pigmented and opaque structure called the iris.

Retina is the inner layer of eye ball and it contains three layers of cells from inside to outside, i. e., ganglion cells, bipolar cells and photoreceptor cells. There are two types of photoreceptor cells, namely, rods and cones. These cells contain the light-sensitive proteins called the photopigments.

Question 3.
Explain the following processes:
(a) Polarisation of the membrane of a nerve fibre
(b) Depolarisation of the membrane of a nerve fibre
(c) Conduction of a nerve impulse along a nerve fibre
(d) Transmission \of a nerve impulse across a chemical synapse
(a) Polarisation of the Membrane of a Nerve Fibre: During resting condition, the concentration of K+ ions is more inside the axoplasm while the concentration of Na+ ions is more outside the axoplasm. As a result, the potassium ions move faster from inside to outside as compared to sodium ions. Therefore, the membrane becomes positively charged outside and negatively charged inside. This is known as polarisation of membrane or polarised nerve.

(b) Depolarisation of the Membrane of a Nerve Fibre: When an electrical stimulus is given to a nerve fibre, an action potential is generated. The membrane becomes permeable to sodium ions than to potassium ions. This results into positive charge inside and negative charge outside the nerve fibre. Hence, the membrane is said to be depolarised.

(c) Conduction of a Nerve Impulse Along a Nerve Fibre: There are two types of nerve fibers-myelinated and non-myelinated. In myelinated nerve fibre, the action potential is conducted from node to node in jumping manner. This is because the myelinated nerve fibre is coated with myelin sheath.

The myelin sheath is impermeable to ions. As a result, the ionic exchange and depolarization of nerve fiber is not possible along the whole length of nerve fiber. It takes place only at some point, known as nodes of Ranvier, whereas in non-myelinated nerve fiber, the ionic exchange and depolarization of nerve fiber takes place along the whole length of the nerve fiber. Because of this ionic exchange, the depolarised area becomes repolarised and the next polarised area becomes depolarised.

(d) Transmission of a Nerve Impulse Across a Chemical Synapse:
Synapse is a small gap that occurs between the last portion of the axon of one neuron and the dendrite of next neuron. When an impulse reaches at the endplate of axon, vesicles consisting of chemical substances or neurotransmitters, such as acetylcholine, fuse with the plasma membrane.

This chemical moves across the cleft and attaches to chemo-receptors present on the membrane of the dendrite of next neuron. This binding of chemical with chemo-receptors leads to the depolarization of membrane and generates a nerve impulse across nerve fibre. The chemical, acetylcholine, is inactivated by enzyme acetylcholinesterase. The enzyme is present in the postsynaptic membrane of the dendrite. It hydrolyses acetylcholine and this allows the membrane to repolarise.

Question 4.
Draw labeled diagrams of the following:
(a) Neuron
(b) Brain
(c) Eye
(d) Ear
(a) Neuron

(b) Brain

(c) Eye

(d) Ear

Question 5.
Write short notes on the following:
(a) Neural coordination
(b) Forebrain
(c) Midbrain
(d) Hindbrain
(e) Retina
(f) Ear ossicles
(g) Cochlea
(h) Organ of Corti
(i) Synapse
(a) Neural Coordination: The organized network of point-to-point connections for quick coordination provided by neural system is called neural coordination. The mechanism of neural coordination involves transmission of nerve impulses, impulse conduction across a synapse, and the physiology of reflex action.

(b) Forebrain: The forebrain consists of :
1. Olfactory lobes: The anterior part of the brain is formed by a pair of short club-shaped structures, the olfactory lobes. These are concerned with the sense of smell.

2. Cerebrum: It is the largest and most complex of all the parts of the human brain. A deep cleft divides the cerebrum longitudinally into two halves, which are termed as the left and right cerebral hemispheres connected by a large bundle of myelinated fibres the corpus callosum. The outer cover of cerebral hemisphere is called cerebral cortex. The cerebral cortex is referred to as the grey matter due to its greyish appearance (as neuron cell bodies are concentrated here).

The cerebral cortex is greatly folded. The upward folds, gyri, alternate with the downward grooves or sulci. Beneath the grey matter, there are millions of medullated nerve fibers, which constitute the inner part of the cerebral hemisphere. The large concentration of medullated nerve fibers gives this tissue an opaque white appearance. Hence, it is called the white matter.

3. Lobes: A very deep and a longitudinal fissure, separates the two cerebral hemispheres. Each cerebral hemisphere of the cerebrum is divided into four lobes, i.e., frontal, parietal, temporal, and occipital lobes.

In each cerebral hemisphere, there are three types of functional areas:
(i) Sensory areas receive impulses from the receptors and motor areas transmit impulses to the effectors.
(ii) Association areas are large regions that are neither clearly sensory nor motor injunction. They interpret the input, store the input and initiate a response in light of similar past experiences. Thus, these areas are responsible for complex functions like memory, learning, reasoning, and other intersensory associations.

(iii) Diencephalon is the posteroventral part of forebrain. Its main parts are as follows :
Epithalamus is a thin membrane of non-neural tissue. It is the posterior segment of the diencephalon. The cerebrum wraps around a structure called thalamus, which is a major coordinating center for sensory and motor signaling. The hypothalamus, that lies at the base of thalamus contains a number of centers, which control body temperature, urge for eating and drinking. It also contains several groups of neurosecretory cells, which secrete hormones called hypothalamic hormones.

(c) Midbrain: The midbrain is located between the thalamus and hypothalamus of the forebrain and pons of the hindbrain. A canal called the cerebral aqueduct passes through, the midbrain.
The dorsal portion of the midbrain mainly consists of two pairs (i.e., four) of rounded swellings (lobes) called corpora qua trigeminal.

(d) Hindbrain: The hindbrain consists of :

(i) Pons: It consists of fiber tracts that interconnect different regions of the brain.

(ii) Cerebellum: It is the second-largest part of the human brain (means little cerebrum). It has very convoluted surface in order to provide the additional space for many more neurons.

(iii) Medulla: It (oblongata) is connected to the spinal cord and contains centers, which control respiration, cardiovascular reflexes, and gastric secretions.

(e) Retina: The inner layer of eyeball is the retina and it contains three layers of cells from inside to outside—ganglion cells, bipolar cells and photoreceptor cells. There are two types of photoreceptor cells namely, rods and cones. These cells contain the light-sensitive proteins called the photopigments.

(f) Ear Ossicles: The middle ear contains three ossicles called malleus, incus and stapes which are attached to one another in a chain-like fashion. The malleus is attached to the tympanic membrane and the stapes is attached to the oval window or the cochlea. The ear ossicles increase the efficiency of transmission of sound waves to the inner ear.

(g) Cochlear: The membranous labyrinth of inner ear is filled with a fluid called endolymph. The coiled portion of the labyrinth is called cochlea. The membranes constituting cochlea, the Meissner’s and basilar, divide the surrounding perilymph-filled bony labyrinth into an upper scale vestibule and a lower scala tympani. The space within cochlea called scala media is filled with endolymph. At the base of the cochlea, the scala vestibule ends at the oval window, while the scala tympani terminates at the round window which opens to the middle ear.

(h) Organ of Corti: The organ of Corti is a structure located on the basilar membrane of inner ear, which contains hair cells that act as auditory receptors. The hair cells are present in rows on the internal side of the organ of Corti. The basal end of the hair cell is in close contact with the afferent nerve fibers. A large number of processes called stereocilia are projected from the apical part of each hair cell. Above the rows of the hair cells is a thin elastic membrane called tectorial membrane.

(i) Synapse: It is a junction between two neurons, where one neuron expands and comes in near contact with another neuron. A synapse is formed by the membranes of a pre-synaptic neuron, and a post-synaptic neuron, which may or may not be separated by a gap called synaptic cleft.

There are two types of synapses-an electrical synapse and a chemical synapse. In electrical synapse, membranes of pre and post-synaptic neurons are is very close proximity field. In chemical synapse, these membranes are separated by a fluid-filled space called synaptic cleft.

Question 6.
Give a brief account of:
(a) Mechanism of synaptic transmission
(b) Mechanism of vision
(c) Mechanism of hearing
(a) Mechanism of Synaptic Transmission: Synapse is a junction between two neurons. It is present between the axon terminal of one neuron and the dendrite of next neuron separated by a cleft.

There are two ways of synaptic transmission :
1. Chemical transmission: When a nerve impulse reaches the end plate of axon, it releases a neurotransmitter (acetylcholine) across the synaptic cleft. This chemical is synthesized in cell body of the neuron and is transported to the axon terminal. The acetylcholine diffuses across the cleft and binds to the receptors present on the membrane of next neuron. This causes depolarization of membrane and initiates an action potential.

2. Electrical transmission: In this type of transmission, an electric current is formed in the neuron. This electric current generates an action potential and leads to transmission of nerve impulses across the nerve fiber. This represents a faster method of nerve conduction than the chemical method of transmission.

(b) Mechanism of Vision: Retina is the innermost layer of eye. It contains three layers of cells-inner ganglion cells, middle bipolar cells and outermost photoreceptor cells. A photoreceptor cell is composed of a protein called opsin and an aldehyde of vitamin A called retinal. When light rays are focused on the retina through cornea, it leads to the dissociation of retinal from opsin protein.

This changes the structure of opsin. As the structure of opsin changes, the permeability of membrane changes, generating a potential difference in the cells. This generates an action potential in the ganglionic cells and is transmitted to the visual cortex of the brain via optic nerves. In the cortex region of brain, the impulses are analyzed and image is formed on the retina.

(c) Mechanism of Hearing: The pinna of the external region collects the sound waves and directs it towards ear drum or external auditory canal. These waves strike the tympanic membrane and vibrations are created. Then, these vibrations are transmitted to the oval window, fenestra ovalis, through three ear ossicles, named as malleus, incus, and stapes. These ear ossicles act as lever and transmit the sound waves to internal ear.

These vibrations from fenestra ovalis are transmitted into cochlear fluid. This generates sound waves in the lymph. The formation of waves generates a ripple in the basilar membrane. This movement bends the sensory hair cells present on the organ of corti against tectorial membrane. As a result of this, sound waves are converted into nerve impulses. These impulses are then carried to auditory cortex of brain via auditory nerves. In cerebral cortex of brain, the impulses are analysed and sound is recognised.

Question 7.
(a) How do you perceive the colour of an object?
(b) Which part of our body helps us in maintaining the body balance?
(c) How does the eye regulate the amount of light that falls on the retina?
(a) The daylight (photopic) vision and colour vision are functions of cones. In the human eye, there are three types of cones which possess their own characteristic photopigments that respond to red, green and blue lights. The sensations of different colours are produced by various combinations of these cones and their photopigments. When these cones are stimulated equally a sensation of white light is produced.

(b) The crista and macula are the specific receptors of the vestibular apparatus of inner ear which are responsible for the maintenance of balance of the body and posture.

(c) The diameter of the pupil is regulated by the muscle fiber of iris. Photoreceptors, rods, and cones regulate the amount of light that falls on the retina.

Question 8.
Explain the following:
(a) Role of Na+ in the generation of action potential.
(b) Mechanism of generation of light-induced impulse in the retina.
(c) Mechanism through which a sound produces a nerve impulse in the inner ear.
(a) Role of Na+ in the Generation of Action Potential: When a stimulus is applied to a nerve, the membrane of the nerve becomes freely permeable to Na + . This leads to a rapid influx of Na+ followed by the reversal of the polarity at that site, i. e., the outer surface of the membrane becomes negatively charged and the inner side becomes positively charged. The electrical potential difference across plasma membrane at the membrane is called the action potential, which is in fact termed as a nerve impulse. Thus, this shows that Na+ ions play an important role in the conduction of nerve impulses.

(b) Mechanism of Generation of Light-induced Impulse in the Retina: Light induces dissociation of the retina from opsin resulting in changes in the structure of the opsin. This causes membrane permeability changes. As a result, potential differences are generated in the photoreceptor cells. This produces a signal that generates action potentials in the ganglion cells through the bipolar cells.

These action potentials (impulses) are transmitted by the optic nerves to the visual corted area of the brain, where the nerve impulses are analysed and the image formed on the retina is recognised.

(c) Mechanism through which a Sound Produces a Nerve Impulse in the Inner Ear: In the inner ear, the vibrations are passed through the oval window on to the fluid of the cochlea, where they generate waves in the lymph.
The waves in the lymphs induce a ripple in the basilar membrane.

These movements of the basilar membrane bend \ the hair cells, pressing them against the tectorial membrane. As a result, nerve impulses are generated in the associated afferent neurons. These impulses are transmitted by the afferent fibres via auditory nerves to the auditory cortex of the brain, where the impulses are analyzed and the sound is recognized.

Question 9.
Differentiate between:
(a) Myelinated and non-myelinated axons
(b) Dendrites and axons
(c) Rods and cones
(d) Thalamus and hypothalamus
(e) Cerebrum and cerebellum
(a) Differences between Myelinated and Non-myelinated Axons

 Myelinated Axon Non-myelinated Axon 1. The myelinated nerve fibers are enveloped with Schwann cells, which form a myelin sheath around the axon. 1. Unmyelinatcd nerve fibers are enclosed by a Schwann cell that does not form a myelin sheath around the axon. 2. Myelinated nerve fibres are found in spinal and cranial nerves. 2. They are commonly found in autonomous and the somatic neural systems.

(b) Differences between Dendrite and Axon

 Dendrite Axon 1. These are short fibres which branch repeatedly and project out of the cell body also contain Nissl’s granules The axon is a long branched fibre, which terminates as a bulb-like structure called. synaptic knob. It possess synaptic vesicles containing chemicals called neurotransmitters. 2. These fibres transmit impulses towards the cell body. The axons transmit nerve impulses away from the cell body to a synapse.

(c) Differences between Rods and Cones

 Rod Cone 1. The twilight vision is the function of rods. The daylight vision and colour vision are functions of cones. 2. The rods contain a purplish-red protein called the rhodopsin or visual purple, which contains a derivative of Vitamin-A In the human eye, there are three types of cones which possess their own characteristic photopigments that respond to red, green and blue lights.

(d) Differences between Thalamus and Hypothalamus

 Thalamus Hypothalamus 1. The cerebrum wraps around a structure called thalamus. It lies at the base of the thalamus. 2. All types of sensory input passes synapses in the thalamus It contains neurosecretory cells that secrete hypothalamus hormones. 3. It controls emotional and memory functions. It regulates, sexual behavior, expression of emotional reactions and motivation.

.(e) Differences between Cerebrum and Cerebellum

 Cerebrum Cerebellum 1. It is the most developed part in brain. It is the second developed part of brain also called as little cerebrum 2. A deep cleft divides cerebrum into two cerebral hemispheres. Externally the whole surface contains gyri and sulci. 3. Its functions are intelligence, learning, memory, speech, etc. It contains centres for coordination and error checking during motor and cognition.

Question 10.
(a) Which part of the ear determines the pitch of a sound?
(b) Which part of the human brain is the most developed?
(c) Which part of our central neural, system acts as a master clock?
(a) Inner ear
(b) Cerebrum
(c) Brain

Question 11.
The region of the vertebrate eye, where the optic nerve passes out of the retina, is called the
(a) fovea
(b) iris
(c) blind spot
(d) optic chiasma
(d) Optic Charisma

Question 12.
Distinguish between:
(a) Afferent neurons and efferent neurons.
(b) Impulse conduction in a myelinated nerve fibre and unmyelinated nerve fibre.
(c) Aqueous humour and vitreous humour.
(d) Blind spot and yellow spot.
(e) Cranial nerves and spinal nerves.
(a) Differences between Afferent neurons and Efferent neurons

 Afferent Neurons Efferent Neurons The afferent nerve fibres transmit impulses from tissues/organs to the CNS. The efferent fibres transmit regulatory impulses from the CNS to the concerned peripheral tissues/organs.

(b) Differences between Myelinated and Non-myelinated Axons

 Myelinated Axon Non-myelinated Axon 1. The myelinated nerve fibers are enveloped with Schwann cells, which form a myelin sheath around the axon. 1. Unmyelinatcd nerve fibers are enclosed by a Schwann cell that does not form a myelin sheath around the axon. 2. Myelinated nerve fibres are found in spinal and cranial nerves. 2. They are commonly found in autonomous and the somatic neural systems.

(c) Differences between Aqueous humour and Vitreous humour

 Aqueous Humour Vitreous Humour 1. It is the space between the cornea and the lens. The space between the lens and the retina is called the vitreous chamber. 2. It contains a thin watery fluid. It is filled with a transparent gel.

(d) Differences between Blindspot and Yellow spot

 Blind Spot Yellow Spot 1. Photoreceptor cells are not present in this region. Yellow spot or macula lutea is located at the posterior pole of the eye lateral to the blind spot. It has a central pit called fovea. 2. The light focuses on that part of the retina is not detected. The fovea of yellow spot is a thinned-out portion of retina where only the cones are densely packed is the point where visual cavity is greatest.

(e) Differences between Cranial nerves and spinal nerves

 Cranial Nerves Spinal Nerves 1. The cranial nerves originate in the brain and terminate mostly in organs head and upper body. The spinal nerves originate in the spinal cord and extend to parts of the body below the head. 2. There are 12 pairs of cranial nerves. There are 31 pairs of spinal nerves. 3. Most of the cranial nerves contain axons and both sensory and motor neurons. All of the spinal nerves contain axons of both sensory and motor neurons.