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#### Page No 11:

#### Question 1:

If two bodies are in thermal equilibrium in one frame, will they be in thermal equilibrium in all frames?

#### Answer:

If two bodies are in thermal equilibrium in one frame, they will be in thermal equilibrium in all the frames. In case there is any change in temperature of one body due to change in frame, the same change will be acquired by the other body.

#### Page No 11:

#### Question 2:

Does the temperature of a body depend on the frame from which it is observed?

#### Answer:

No, the temperature of a body is not dependent on the frame from which it is observed. This is because atoms /molecules of matter move or vibrate in all possible directions. Increase in velocity at a particular direction of the container/ matter does not increase or decrease the overall velocity of the molecules/atoms because of the random collisions the entities suffer. So, there is no net rise in temperature of the system.

#### Page No 11:

#### Question 3:

It is said that mercury is used in defining the temperature scale because it expands uniformly with temperature. If the temperature scale is not yet defined, is it logical to say that a substance expands uniformly with temperature?

#### Answer:

It is not illogical to say that mercury expands uniformly before temperature scale was defined. It's uniform expansion can be studied by comparing the expansion of mercury with expansion of other substances (like alcohol water etc).

#### Page No 11:

#### Question 4:

In defining the ideal gas temperature scale, it is assumed that the pressure of the gas at constant volume is proportional to the temperature* T*. How can we verify whether this is true or not? Do we have to apply the kinetic theory of gases? Do we have to depend on experimental result that the pressure is proportional to temperature?

#### Answer:

The ideal gas thermometer is based on the ideal gas equation,* PV=nRT, *where *P* is pressure of the gas at constant volume *V* with *n* number of moles at temperature *T*. Therefore, *P* = constant$\times $*T*. According to this relation, if the volume of the gas used is constant, the pressure will be directly proportional to the temperature of the gas. We need not use kinetic theory of gases or any experimental results.

#### Page No 11:

#### Question 5:

Can the bulb of a thermometer be made of an adiabatic wall?

#### Answer:

The bulb of a thermometer plays an important role in measuring the temperature of the surrounding body. It is put in contact with the body whose temperature is to be measured. The bulb attains the temperature of the body, which allows calibration of temperature. If the bulb is made of an adiabatic wall, then no heat will be transferred through the wall and the bulb cannot attain thermal equilibrium with the surrounding body. Therefore, the bulb cannot be made of an adiabatic wall.

#### Page No 11:

#### Question 6:

Why do marine animals live deep inside a lake when the surface of the lake freezes?

#### Answer:

Water possesses an anomalous behavour. The volume of a given amount of water decreases as it is cooled from room temperature, until its temperature reaches 4 °C. Below 4 °C, the volume increases, and therefore the density decreases.

When the temperature of the surface of lake falls in winter, the water at the surface becomes denser and sinks. As, the temperature reaches below 4^{ o}C , the density of the water at surface becomes less. Thus, it remains at surface and freezes. As, the ice is a bad conductor of heat, it traps the heat present in the lake's water beneath itself. Hence, no further cooling of water takes place once the top layer of the lake is completely covered by ice. Thus the life of the marine animals inside the lake is possible.

#### Page No 11:

#### Question 7:

The length of a brass rod is found to be less on a hot summer day than on a cold winter day as measured by the same aluminium scale. Can we conclude that brass shrinks on heating?

#### Answer:

On a hot summer day, metals tend to expand due to the heat. Different metals have different expansion coefficients. The coefficient of linear expansion of aluminium is more than that of brass. Therefore, it'll expand more than brass, leading to an apparent decrease in length of the brass rod, as measured by the aluminium scale. So, we cannot conclude that brass shrinks on heating. Instead, aluminium expands more than brass on heating.

#### Page No 11:

#### Question 8:

If mercury and glass had equal coefficients of volume expansion, could we make a mercury thermometer in a glass tube?

#### Answer:

Yes, we can make a mercury thermometer in a glass tube. Mercury and glass have equal coefficients of volume expansion. So, when temperature changes, the increase in the volume of the glass tube as which is equal to the real increase in volume minus the increase in the volume of the container, would be zero. Hence, it will give correct reading at every temperature.

#### Page No 11:

#### Question 9:

The density of water at 4°C is supposed to be 1000 kg m^{–3}. Is it same at sea level and at high altitude?

#### Answer:

At sea level, the pressure is around 1 atmosphere and at high altitude, the density of air reduces.

Pressure of liquid,

$P=h\rho g,\phantom{\rule{0ex}{0ex}}\mathrm{where}\rho =\mathrm{density}\mathrm{of}\mathrm{fluid}$

The above equation shows that pressure depends on density. Therefore at 4^{o}^{}C, the density of water will be less at high altitude, compared to the density at sea level.

#### Page No 11:

#### Question 10:

A tightly closed metal lid of a glass bottle can be opened more easily if it is put in hot water for some time. Explain.

#### Answer:

When a bottle with a tightly-closed metal lid is put in hot water for sometime, its lid can be opened easily because metals have greater coefficient of expansion than glass. Therefore, when the metal lid comes in contact with hot water, it'll expand more than the glass container. As a result, it will be easier to open the bottle.

#### Page No 11:

#### Question 11:

If an automobile engine is overheated, it is cooled by pouring water on it. It is advised that the water should be poured slowly with the engine running. Explain the reason.

#### Answer:

In a hot engine the hot parts are expanded because of heat, if cold water is poured suddenly then there will be uneven thermal contraction in the parts. This will result in a stress to develop between the various parts of the engine and may let the engine to crack down.

#### Page No 11:

#### Question 12:

Is it possible for two bodies to be in thermal equilibrium if they are not in contact?

#### Answer:

Two bodies are said to be in thermal equilibrium if they are at the same temperature. Consider two bodies A and B that are not in contact with each other but in contact with a heat reservoir. Since both the bodies will attain the temperature of the reservoir, they will be at the same temperature and, hence, in thermal equilibrium. Therefore, it is possible to have two bodies in thermal equilibrium even though they are not in contact.

#### Page No 11:

#### Question 13:

A spherical shell is heated. The volume changes according to the equation V_{θ} = V_{0} (1 + γ^{θ}). Does the volume refer to the volume enclosed by the shell or the volume of the material making up the shell?

#### Answer:

When a spherical shell is heated, its volume changes according to the equation, *${V}_{\theta}={V}_{0}\left(1+\gamma \u2206\theta \right)$*. The volume referred to here is the volume of the material used to make up the shell, as its volume expands with the rise of temperature with coefficient of expansion of volume, *$\gamma $.*

#### Page No 11:

#### Question 1:

A system *X* is neither in thermal equilibrium with *Y* nor with *Z*. The systems *Y* and *Z*

(a) must be in thermal equilibrium

(b) cannot be in thermal equilibrium

(c) may be in thermal equilibrium

#### Answer:

(c) may be in thermal equilibrium

The given data in the question is insufficient to specify the relation between the physical conditions of systems *Y* and Z. As system *X* is not in thermal equilibrium with *Y* and *Z*, systems *Y* and *Z* may be at the same temperature or they may or may not be in thermal equilibrium with each other. So, the only possible option is (c).

#### Page No 11:

#### Question 2:

Which of the curves in the figure (23-Q1) represents the relation between Celsius and Fahrenheit temperatures?

Faigure

#### Answer:

(a)

Celsius and Fahrenheit temperatures are related in the following way:

$C=\frac{5}{9}F-\frac{160}{9}$

Here, *F* = temperature in Fahrenheit

*C *= temperature in Celsius

If this equation is plotted on the graph, then the curve will be represented by curve 'a' lying in the fourth quadrant with slope 5/9.

So, the correct option is (a).

#### Page No 11:

#### Question 3:

Which of the following pairs may give equal numerical values of the temperature of a body?

(a) Fahrenheit and Kelvin

(b) Celsius and Kelvin

(c) Kelvin and Platinum

#### Answer:

(a) Fahrenheit and Kelvin

Let *θ *be the temperature in Fahrenheit and Kelvin scales.

We know that the relation between the temperature in Fahrenheit and Kelvin scales is given by

$\frac{{T}_{\mathrm{F}}-32}{180}=\frac{{T}_{\mathrm{K}}-273.15}{100}$

*T*_{F} = *T*_{K} = *θ*

Therefore,

$\frac{\theta -32}{180}=\frac{\theta -273.15}{100}\phantom{\rule{0ex}{0ex}}5\theta -160=9\theta -2458.5\phantom{\rule{0ex}{0ex}}4\theta =2298.35\phantom{\rule{0ex}{0ex}}\theta =574.59{}^{\mathrm{o}}$

If we consider the same for Celsius and Kelvin scales

$\frac{{T}_{\mathrm{C}}-0}{100}=\frac{{T}_{\mathrm{K}}-273.15}{100}$

Let the temperature be *t*

$\frac{t-0}{100}=\frac{t-273.15}{100}\phantom{\rule{0ex}{0ex}}t=t-273.15$

Thus, *t* does not exist.

The Kelvin scale uses mercury as thermometric substance, whereas the platinum scale uses platinum as thermometric substance. The scale depends on the properties of the thermometric substance used to define the scale. The platinum and Kelvin scales do not agree with each other. Therefore, there is no such temperature that has same numerical value in the platinum and Kelvin scale.

#### Page No 11:

#### Question 4:

For a constant-volume gas thermometer, one should fill the gas at

(a) low temperature and low pressure

(b) low temperature and high pressure

(c) high temperature and low pressure

(d) high temperature and high pressure

#### Answer:

(c) high temperature and low pressure.

A constant-volume gas thermometer should be filled with an ideal gas in which particles don't interact with each other and are free to move anywhere, so that the thermometer functions properly. An ideal gas is only a theoretical possibility. Therefore, the gas that is filled in the thermometer should be at high temperature and low pressure, as under these conditions, a gas behaves as an ideal gas.

#### Page No 11:

#### Question 5:

Consider the following statements.

(A) The coefficient of linear expansion has dimension *K*^{–1}.

(B) The coefficient of volume expansion has dimension *K*^{–1}.

*A*and

*B*are correct.

(b)

*A*is correct but

*B*is wrong.

(c)

*B*is correct but

*A*is wrong.

(d)

*A*and

*B*are wrong.

#### Answer:

(a) A and B are correct.

The coefficient of linear expansion,

$\alpha =\frac{1}{L}\frac{\u25b3L}{\u25b3T}\phantom{\rule{0ex}{0ex}}$

$=\frac{\left[L\right]}{\left[LT\right]}={\mathrm{K}}^{-1}$

Here, *L* = initial length

$\u25b3$*L* = change in length

$\u25b3$*T* = change in temperature

On the other hand, the coefficient of volume expansion,

$\gamma =\frac{1}{V}\frac{\u25b3V}{\u25b3T}=\frac{\left[{L}^{3}\right]}{\left[{L}^{3}T\right]}={\mathrm{K}}^{-1}$

Here, *V* = initial volume

$\u25b3$*V* = change in volume

$\u25b3$*T* = change in temperature

K = kelvin, the S.I. unit of temperature

#### Page No 12:

#### Question 6:

A metal sheet with a circular hole is heated. The hole

(a) gets larger

(b) gets smaller

(c) retains its size

(d) is deformed

#### Answer:

(a) gets larger

When a metal sheet is heated, it starts expanding and its surface area will start increasing, which will lead to an increase in the radius of the hole. Hence, the circular hole will become larger.

#### Page No 12:

#### Question 7:

Two identical rectangular strips, one of copper and the other of steel, are riveted together to form a bimetallic strip (α_{copper} > α_{steel}). On heating, this strip will

(a) remain straight

(b) bend with copper on convex side

(c) bend with steel on convex side

(d) get twisted

#### Answer:

(b) bend with copper on convex side

We are provided with two metal strips of copper and steel. On heating, both of them will expand. Expansion coefficient of copper is more than that of steel. So, the copper metal strip will expand more, causing the bimetallic strip to bend with copper at the convex side, as it'll have more surface area compared to the steel sheet, which will be on the concave side.

#### Page No 12:

#### Question 8:

If the temperature of a uniform rod is slightly increased by ∆*t*, its moment of inertia *I *about a perpendicular bisector increases by

(a) zero

(b) α*I*∆*t*

(c) 2α*I*∆*t*

(d) 3α*I*∆*t**.*

#### Answer:

(c) 2α*I∆t*

The change in moment of inertia of uniform rod with change in temperature is given by,

$I\text{'}=I(1+2\alpha \Delta t)$

Here, *I* = initial moment of inertia

*I'* = new moment of inertia due to change in temperature

$\alpha $= expansion coefficient

$\u2206$*t* = change in temperature

So, $I\text{'}-I=2\alpha I\Delta t$

#### Page No 12:

#### Question 9:

If the temperature of a uniform rod is slightly increased by ∆*t*, its moment of inertia *I* about a line parallel to itself will increase by

(a) zero

(b) α*I∆t*

(c) 2α*I∆t*

(d) 3α*I∆t**.*

#### Answer:

(c) 2α*I∆t*

The moment of inertia of a solid body of any shape changes with temperature as

$I\text{'}=I\left(1+2\alpha \Delta t\right)$

Here, *I* = initial moment of inertia

*I'* = new moment of inertia due to change in temperature

$\alpha $ = expansion coefficient

$\Delta $*t *= change in temperature

So, $I\text{'}-I=2\alpha I\Delta t$

#### Page No 12:

#### Question 10:

The temperature of water at the surface of a deep lake is 2°C. The temperature expected at the bottom is

(a) 0 °C

(b) 2 °C

(c) 4 °C

(d) 6 °C

#### Answer:

(c) 4^{ }^{o}C

The density of water is maximum at 4^{ }^{o}C, and the water at the bottom of the lake is most dense, compared to the layers of water above. Therefore, the temperature expected at the bottom is 4^{o}C.

#### Page No 12:

#### Question 11:

An aluminium sphere is dipped into water at 10°C. If the temperature is increased, the force of buoyancy

(a) will increase

(b) will decrease

(c) will remain constant

(d) may increase or decrease depending on the radius of the sphere

#### Answer:

(b) will decrease

When an aluminium sphere is dipped in water and the temperature of water is increased, the aluminium will start expanding leading to increase in its volume. This will lead to increase in the surface area of the shell and it'll exert less pressure on the water such that the volume of the sphere submerged in water will decrease and it'll start float easily on water. Now, the volume of water displaced will be less compared to what was displaced initially. Therefore, the force of buoyancy will decrease, as it is directly proportional to the volume of water displaced.

#### Page No 12:

#### Question 1:

A spinning wheel is brought in contact with an identical wheel spinning at identical speed. The wheels slow down under the action of friction. Which of the following energies of the first wheel decreases?

(a) Kinetic

(b) Total

(c) Mechanical

(d) Internal

#### Answer:

(a) Kinetic

(c) Mechanical

The kinetic energy of a body depends on its speed. Since when a spinning wheel is slowed down, its speed decreases leading to reduction in its kinetic energy. The mechanical energy of a body is defined as the sum of its potential and kinetic energies. Since the kinetic energy of the wheel has been decreased, it'll lead to decrease in its mechanical energy. When the wheel slows down due to friction, its mechanical energy gets converted into heat energy, leading to increase in internal energy, which increases with increase in temperature.

#### Page No 12:

#### Question 2:

A spinning wheel *A* is brought in contact with another wheel *B,* initially at rest. Because of the friction at contact, the second wheel also starts spinning. Which of the following energies of the wheel* B* increases?

(a) Kinetic

(b) Total

(c) Mechanical

(d) Internal

#### Answer:

(a) Kinetic

(b) Total

(c) Mechanical

(d) Internal

When the wheel *B* starts spinning because of the friction at contact, it will gain kinetic energy and, hence, mechanical energy (kinetic + potential energies). Also, internal energy will increase, which increases with rise in temperature. Along with it, the generation of heat energy due to friction will lead to increase in the net sum of all the energies, i.e. total energy.

#### Page No 12:

#### Question 3:

A body *A* is placed on a railway platform and an identical body *B* in a moving train. Which of the following energies of *B* are greater than those of *A, *as seen from the ground?

(a) Kinetic

(b) Total

(c) Mechanical

(d) Internal

#### Answer:

(a) Kinetic

(b) Total

(c) Mechanical

As body *A* is at rest on the ground, it possesses only potential energy, whereas body B, being placed inside a moving train, possesses kinetic energy due to its motion along with the train. Therefore, body B will have greater kinetic, mechanical (energy possessed by the body by virtue of its position and motion = kinetic energy+potential energy) energy and, hence, total (sum of all the energies) energy. No information is given about the temperature of the body so we can not say wheather body B^{'} s internal energy will be or will not be greater than that of body A.

#### Page No 12:

#### Question 4:

In which of the following pairs of temperature scales, the size of a degree is identical?

(a) Mercury scale and ideal gas scale

(b) Celsius scale and mercury scale

(c) Celsius scale and ideal gas scale

(d) Ideal gas scale and absolute scale

#### Answer:

(c) Celsius scale and ideal gas scale

(d) Ideal gas scale and absolute scale

Celsius scale and ideal gas scale measure temperature in kelvin (K) and the ideal gas scale is sometimes also called the absolute scale. A mercury scale gives reading in degrees and its size of degree, which depends on length of mercury column, doesn't match any of the above-mentioned scales.

#### Page No 12:

#### Question 5:

A solid object is placed in water contained in an adiabatic container for some time. The temperature of water falls during this period and there is no appreciable change in the shape of the object. The temperature of the solid object

(a) must have increased

(b) must have decreased

(c) may have increased

(d) may have remained constant

#### Answer:

(a) must have increased.

The whole system (water + solid object) is enclosed in an adiabatic container from which no heat can escape. After some time, the temperature of water falls, which implies that the heat from the water has been transferred to the object, leading to increase in its temperature.

#### Page No 12:

#### Question 6:

As the temperature is increased, the time period of a pendulum

(a) increases proportionately with temperature

(b) increases

(c) decreases

(d) remains constant

#### Answer:

(b) increases

In general, the time period of a pendulum,*t,* is given by

$t=\frac{1}{2\pi}\sqrt{\frac{l}{g}}$.

When the temperature (*T*) is increased, the length of the pendulum (*l*) is given by,

$l={l}_{0}(1+\alpha T)$,

where *l*_{0} = length at 0 ^{o}C

$\alpha $= linear coefficient of expansion.

Therefore, the time period of a pendulum will be

$t=\frac{1}{2\pi}\sqrt{\frac{{l}_{0}(1+\alpha T)}{g}}$

Hence, time period of a pendulum will increase with increase in temperature.

#### Page No 12:

#### Question 1:

The steam point and the ice point of a mercury thermometer are marked as 80° and 20°. What will be the temperature on a centigrade mercury scale when this thermometer reads 32°?

#### Answer:

Given:

Ice point of a mercury thermometer, *T _{0}* = 20° C

Steam point of a mercury thermometer,

*T*= 80° C

_{100}Temperature on thermometer that is to be calculated in centigrade scale,

*T*= 32° C

_{1}Temperature on a centigrade mercury scale,

*T,*is given as:

$T\mathit{=}\frac{{T}_{\mathit{1}}\mathit{-}{T}_{\mathit{0}}}{{T}_{\mathit{100}}\mathit{-}{T}_{\mathit{0}}}\times 100\phantom{\rule{0ex}{0ex}}\Rightarrow T=\frac{32-20}{80-20}\times 100\phantom{\rule{0ex}{0ex}}\Rightarrow T=\frac{12}{60}\times 100\phantom{\rule{0ex}{0ex}}\Rightarrow T=\frac{120}{6}\phantom{\rule{0ex}{0ex}}\Rightarrow T=20\xb0\mathrm{C}$

Therefore, the temperature on a centigrade mercury scale will be 20

^{o}

^{ }C.

#### Page No 12:

#### Question 2:

A constant-volume thermometer registers a pressure of 1.500 × 10^{4} Pa at the triple point of water and a pressure of 2.050 × 10^{4 }Pa at the normal boiling point. What is the temperature at the normal boiling point?

#### Answer:

Given:

Pressure registered by a constant-volume thermometer at the triple point, *P*_{tr} = 1.500 × 10^{4 } Pa

Pressure registered by the thermometer at the normal boiling point, *P* = 2.050 × 10^{4} Pa

We know that for a constant-volume gas thermometer, temperature (*T*) at the normal boiling point is given as:

$T=\frac{P}{{P}_{tr}}\times 273.16\mathrm{K}\phantom{\rule{0ex}{0ex}}\Rightarrow T=\frac{2.050\times {10}^{4}}{1.500\times {10}^{4}}\times 273.16\mathrm{K}\phantom{\rule{0ex}{0ex}}\Rightarrow T=373.31\mathrm{K}$

Therefore, the temperature at the normal point (*T*) is 373.31 K.

#### Page No 12:

#### Question 3:

A gas thermometer measures the temperature from the variation of pressure of a sample of gas. If the pressure measured at the melting point of lead is 2.20 times the pressure measured at the triple point of water, find the melting point of lead.

#### Answer:

Given:

In a gas thermometer, the pressure measured at the melting point of lead, *P* = 2.20 × Pressure at triple point(*P*_{tr})

So the melting point of lead,(*T*) is given as:

$T=\frac{P}{{P}_{tr}}\times 273.16\mathrm{K}\phantom{\rule{0ex}{0ex}}\Rightarrow T=\frac{2.20\times {P}_{tr}}{{P}_{tr}}\times 273.16\mathrm{K}\phantom{\rule{0ex}{0ex}}\Rightarrow T=2.20\times 273.16K\phantom{\rule{0ex}{0ex}}\Rightarrow T=600.952\mathrm{K}\phantom{\rule{0ex}{0ex}}\Rightarrow T\simeq 601\mathrm{K}$

Therefore, the melting point of lead is 601 K.

#### Page No 12:

#### Question 4:

The pressure measured by a constant volume gas thermometer is 40 kPa at the triple point of water. What will be the pressure measured at the boiling point of water (100°C)?

#### Answer:

Given:

Pressure measured by a constant volume gas thermometer at the triple point of water, *P _{tr}* = 40 kPa = 40 × 10

^{3}Pa

Boiling point of water,

*T*= 100°C = 373.16 K

Let the pressure measured at the boiling point of water be

*P*.

For a constant volume gas thermometer, temperature-pressure relation is given below:

$T=\frac{P}{{P}_{tr}}\times 273.16\mathrm{K}\phantom{\rule{0ex}{0ex}}\Rightarrow P=\frac{T\times {P}_{tr}}{273.16}\phantom{\rule{0ex}{0ex}}\Rightarrow P=\frac{373.16\times 40\times {10}^{3}}{273.16}\phantom{\rule{0ex}{0ex}}\Rightarrow P=54643\mathrm{Pa}\phantom{\rule{0ex}{0ex}}\Rightarrow P=54.6\times {10}^{3}\mathrm{Pa}\phantom{\rule{0ex}{0ex}}\Rightarrow P\simeq 55\mathrm{kPa}$

Therefore, the pressure measured at the boiling point of water is 55 kPa.

#### Page No 12:

#### Question 5:

The pressure of the gas in a constant volume gas thermometer is 70 kPa at the ice point. Find the pressure at the steam point.

#### Answer:

Given:

Temperature of ice point, *T _{1}* = 273.15 K

Temperature of steam point,

*T*= 373.15 K

_{2}Pressure of the gas in a constant volume thermometer at the ice point,

*P*

_{1}_{}= 70 kPa,

Let

*P*be the pressure at the triple point and

_{tr}_{ }*P*be the pressure at the steam point.

_{2}_{ }The temperature-pressure relations for ice point and steam point are given below:

For ice point,

${T}_{1}=\frac{{P}_{1}}{{P}_{tr}}\times 273.16\mathrm{K}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

$\Rightarrow 273.15=\frac{70}{{P}_{tr}}\times {10}^{3}\times 273.16\phantom{\rule{0ex}{0ex}}\Rightarrow {P}_{tr}=\frac{70\times 273.16\times {10}^{3}}{273.15}\mathrm{Pa}\phantom{\rule{0ex}{0ex}}$

For steam point,

${T}_{2}=\frac{{P}_{2}\times 273.16}{{P}_{tr}}\mathrm{K}\phantom{\rule{0ex}{0ex}}$

On substituting the value of

*P*

_{tr},we get:

$373.15=\frac{{P}_{2}\times 273.15\times 273.16}{70\times 273.16\times {10}^{3}}\phantom{\rule{0ex}{0ex}}\Rightarrow {P}_{2}=\frac{373.15\times 70\times {10}^{3}}{273.15}\phantom{\rule{0ex}{0ex}}\Rightarrow {P}_{2}=95.626\times {10}^{3}\mathrm{Pa}\phantom{\rule{0ex}{0ex}}\Rightarrow {P}_{2}\simeq 96\mathrm{kPa}$

Therefore, the pressure at steam point is 96 kPa.

#### Page No 12:

#### Question 6:

The pressures of the gas in a constant volume gas thermometer are 80 cm, 90 cm and 100 cm of mercury at the ice point, the steam point and in a heated wax bath, respectively. Find the temperature of the wax bath.

#### Answer:

Given:

In a constant volume gas thermometer,

Pressure of the gas at the ice point, *P*_{0} = 80 cm of Hg

Pressure of the gas at the steam point, *P*_{100} = 90 cm of Hg

Pressure of the gas in a heated wax bath, *P* = 100 cm of Hg

The temperature of the wax bath $\left(T\right)$ is given by:

$T=\frac{P-{P}_{0}}{{P}_{100}-{P}_{0}}\times 100\xb0\mathrm{C}\phantom{\rule{0ex}{0ex}}\Rightarrow T=\frac{100-80}{90-80}\times 100\phantom{\rule{0ex}{0ex}}\Rightarrow T=\frac{20}{10}\times 100\phantom{\rule{0ex}{0ex}}\Rightarrow T=200\xb0\mathrm{C}$

Therefore, the temperature of the wax bath is 200^{o} C.

#### Page No 12:

#### Question 7:

In a Callender's compensated constant pressure air thermometer, the volume of the bulb is 1800 cc. When the bulb is kept immersed in a vessel, 200 cc of mercury has to be poured out. Calculate the temperature of the vessel.

#### Answer:

Given:

Volume of the bulb in a Callender's compensated constant pressure air thermometer, (*V*) =

1800 cc

Volume of mercury that has to be poured out, *V'* = 200 cc

Temperature of ice bath, *T*_{o} = 273.15 K

So the temperature of the vessel(*T'*) is given by:

$T\text{'}=\frac{V}{V-V\text{'}}\times {T}_{0}\phantom{\rule{0ex}{0ex}}\Rightarrow T\text{'}=\frac{1800}{1600}\times 273.15\mathrm{K}\phantom{\rule{0ex}{0ex}}\Rightarrow T\text{'}=307.293\phantom{\rule{0ex}{0ex}}\Rightarrow T\text{'}\simeq 307\mathrm{K}$

Therefore, the temperature of the vessel is 307 K.

#### Page No 12:

#### Question 8:

A platinum resistance thermometer reads 0° when its resistance is 80 Ω and 100° when its resistance is 90 Ω.

Find the temperature at the platinum scale at which the resistance is 86 Ω.

#### Answer:

Given:

Resistance at 0^{o}C, *R*_{0} = 80 $\mathrm{\Omega}$

Resistance at 100^{o}C, *R*_{100} = 90 $\mathrm{\Omega}$

Let *t *be the temperature at which the resistance (*R _{t}*) is 86 $\mathrm{\Omega}$.

$t=\frac{{R}_{t}-{R}_{0}}{{R}_{100}-{R}_{0}}\times 100\phantom{\rule{0ex}{0ex}}\Rightarrow t=\frac{86-80}{90-80}\times 100\phantom{\rule{0ex}{0ex}}\Rightarrow t=\frac{6}{10}\times 100\phantom{\rule{0ex}{0ex}}\Rightarrow t=60\xb0$

Therefore, the resistance

*is 86 $\mathrm{\Omega}$ at 60*

^{o}C.

#### Page No 13:

#### Question 9:

A resistance thermometer reads R = 20.0 Ω, 27.5 Ω, and 50.0 Ω at the ice point (0°C), the steam point (100°C) and the zinc point (420°C), respectively. Assuming that the resistance varies with temperature as R_{θ} = R_{0} (1 + αθ + βθ^{2}), find the values of R_{0}, α and β. Here θ represents the temperature on the Celsius scale.

#### Answer:

Given:

Reading on resistance thermometer at ice point, *R*_{0} = 20 Ω

Reading on resistance thermometer at steam point, *R*_{100} = 27.5 Ω

Reading on resistance thermometer at zinc point, *R*_{420} = 50 Ω

The variation of resistance with temperature in Celsius scale,θ, is given as:

${R}_{100}={R}_{0}\left(1+\alpha \theta +\beta {\theta}^{2}\right)\phantom{\rule{0ex}{0ex}}\Rightarrow {R}_{100}={R}_{0}+{R}_{0}\alpha \theta +{R}_{0}\beta {\theta}^{2}\phantom{\rule{0ex}{0ex}}\Rightarrow {R}_{100}={R}_{0}+{R}_{0}\alpha \theta +{R}_{0}\beta {\theta}^{2}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{\left({R}_{100}-{R}_{0}\right)}{{R}_{0}}=\alpha \theta +\beta {\theta}^{2}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{27.5-20}{20}=\alpha \theta +\beta {\theta}^{2}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{7.5}{20}=\alpha \times 100+\beta \times 10000...\left(i\right)\phantom{\rule{0ex}{0ex}}\mathrm{Also},{R}_{420}={R}_{0}\left(1+\alpha \theta +\beta {\theta}^{2}\right)\phantom{\rule{0ex}{0ex}}\Rightarrow {R}_{420}={R}_{0}+{R}_{0}\alpha \theta +{R}_{0}\beta {\theta}^{2}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{{R}_{420}-{R}_{0}}{{R}_{0}}=\alpha \theta +\beta {\theta}^{2}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{50-20}{20}=420\alpha +176400\beta \phantom{\rule{0ex}{0ex}}\Rightarrow \frac{3}{2}=420\alpha +176400\beta ...\left(ii\right)$

Solving (*i*) and (*ii*), we get:

*α* = 3.8 ×10^{–3}°C^{-1}

*β* = –5.6 ×10^{–7}°C^{-1}

Therefore, resistance *R*_{0 }is 20 Ω and the value of *α* is 3.8 ×10^{–3}°C^{-1 }and that of *β* is –5.6 ×10^{–7}°C^{-1}.

#### Page No 13:

#### Question 10:

A concrete slab has a length of 10 m on a winter night when the temperature is 0°C. Find the length of the slab on a summer day when the temperature is 35°C. The coefficient of linear expansion of concrete is 1.0 × 10^{–5} °C^{–1}.

#### Answer:

Given:

Length of the slab when the temperature is 0°C, *L*_{0} = 10 m

Temperature on the summer day, *t = *35 °C

Let *L*_{1} be the length of the slab on a summer day when the temperature is 35°C.

The coefficient of linear expansion of concrete, *α* = 1 ×10^{–5} °C^{}^{-1}

${L}_{1}={L}_{0}\left(1+\alpha t\right)$

$\Rightarrow {L}_{1}$= 10 (1 + 10^{–5} × 35)

$\Rightarrow {L}_{1}$ = 10 + 35 × 10^{–4}

$\Rightarrow {L}_{1}$ = 10.0035 m

So, the length of the slab on summer day when the temperature is 35^{o}C is 10.0035 m.

#### Page No 13:

#### Question 11:

A metre scale made of steel is calibrated at 20°C to give correct reading. Find the distance between the 50 cm mark and the 51 cm mark if the scale is used at 10°C. Coefficient of linear expansion of steel is 1.1 × 10^{–5} °C^{–1}.

#### Answer:

Given:

Temperature at which the steel metre scale is calibrated, *t*_{1} = 20^{o}C

Temperature at which the scale is used, *t*_{2} = 10^{o}C

So, the change in temperature, $\Delta $*t* = (20^{o}$-$10^{o}) C

The distance to be measured by the metre scale, *L*_{o} = (51$-$50) = 1 cm = 0.01 m

Coefficient of linear expansion of steel, ${\alpha}_{steel}$ = 1.1 × 10^{–5} °C^{–1}

Let the new length measured by the scale due to expansion of steel be* L*_{2}, Change in length is given by,

$\u2206L={L}_{1}{\alpha}_{steel}\Delta t\phantom{\rule{0ex}{0ex}}\Rightarrow \u2206L=1\times 1.1\times {10}^{\u20135}\times 10\phantom{\rule{0ex}{0ex}}\Rightarrow \u2206L=0.00011\mathrm{cm}\phantom{\rule{0ex}{0ex}}$

As the temperature is decreasing, therefore length will decrease by $\u2206L$.

Therefore, the new length measured by the scale due to expansion of steel (*L*_{2}) will be,

*L*_{2} = 1 cm $-$ 0.00011 cm = 0.99989 cm

#### Page No 13:

#### Question 12:

A railway track (made of iron) is laid in winter when the average temperature is 18°C. The track consists of sections of 12.0 m placed one after the other. How much gap should be left between two such sections, so that there is no compression during summer when the maximum temperature rises to 48°C? Coefficient of linear expansion of iron = 11 × 10^{–6} °C^{–1}.

#### Answer:

Given:

Length of the iron sections when there's no effect of temperature on them, *L*_{o} = 12.0 m

Temperature at which the iron track is laid in winter, *t _{w}* = 18

^{ o}C

Maximum temperature during summers,

*t*= 48

_{s}^{ o}C

Coefficient of linear expansion of iron, $\alpha $ = 11 × 10

^{–6}°C

^{–1}

Let the new lengths attained by each section due to expansion of iron in winter and summer be

*L*

_{w}_{ }and

*L*

_{s}_{,}_{ }respectively, which can be calculated as follows:

${L}_{w}={L}_{0}\left(1+\alpha {t}_{w}\right)\phantom{\rule{0ex}{0ex}}\Rightarrow {L}_{w}=12\left(1+11\times {10}^{-6}\times 18\right)\phantom{\rule{0ex}{0ex}}\Rightarrow {L}_{w}=12.00237\mathrm{m}\phantom{\rule{0ex}{0ex}}{L}_{s}={L}_{0}\left(1+\alpha {t}_{s}\right)\phantom{\rule{0ex}{0ex}}\Rightarrow {L}_{s}=12\left(1+11\times {10}^{-6}\times 48\right)\phantom{\rule{0ex}{0ex}}\Rightarrow {L}_{s}=12.006336\mathrm{m}\phantom{\rule{0ex}{0ex}}\therefore \u2206L={L}_{s}-{L}_{w}\phantom{\rule{0ex}{0ex}}\Rightarrow \Delta L=12.006336-12.002376\phantom{\rule{0ex}{0ex}}\Rightarrow \Delta L=0.00396\mathrm{m}\phantom{\rule{0ex}{0ex}}\Rightarrow \Delta L\approx 0.4\mathrm{cm}$

Therefore, the gap ($\Delta $

*L)*that should be left between two iron sections, so that there is no compression during summer, is 0.4 cm.

#### Page No 13:

#### Question 13:

A circular hole of diameter 2.00 cm is made in an aluminium plate at 0°C. What will be the diameter at 100°C? α for aluminium = 2.3 × 10^{–5} °C^{–1}.

#### Answer:

Given:

Diameter of a circular hole in an aluminium plate at 0°C, *d*_{1} = 2 cm = 2 × 10^{–2} m

Initial temperature, *t*_{1} = 0 °C

Final temperature, *t*_{2} = 100 °C

So, the change in temperature, ($\Delta $*t*) = 100°C - 0°C = 100°C

The linear expansion coefficient of aluminium, *α _{al}*

_{}= 2.3 × 10

^{–5}°C

^{–1}

Let the diameter of the circular hole in the plate at 100

^{o}C be

*d*

_{2}

_{ }, which can be written as:

${d}_{2}={d}_{1}\left(1+\alpha \Delta t\right)$

$\Rightarrow {d}_{2}$= 2 × 10

^{–2 }(1 + 2.3 × 10

^{–5 }× 10

^{2})

$\Rightarrow {d}_{2}$= 2 × 10

^{–2}(1 + 2.3 × 10

^{–3})

$\Rightarrow {d}_{2}$= 2 × 10

^{–2}+ 2.3 × 2 × 10

^{–5}

$\Rightarrow {d}_{2}$ = 0.02 + 0.000046

$\Rightarrow {d}_{2}$= 0.020046 m

$\Rightarrow {d}_{2}$≈ 2.0046 cm

Therefore, the diameter of the circular hole in the aluminium plate at 100

^{o}C is 2.0046 cm.

#### Page No 13:

#### Question 14:

Two metre scales, one of steel and the other of aluminium, agree at 20°C. Calculate the ratio aluminium-centimetre/steel-centimetre at (a) 0°C, (b) 40°C and (c) 100°C. α for steel = 1.1 × 10^{–5} °C^{–1} and for aluminium = 2.3 × 10^{–5}°C^{–1}.

#### Answer:

Given:

At 20°C, length of the metre scale made up of steel, *L*_{st}= length of the metre scale made up of aluminium, *L _{al}*

Coefficient of linear expansion for aluminium,

*α*

_{al}= 2.3 × 10

^{–5 }°C

^{-1}

Coefficient of linear expansion for steel,

*α*

_{st}= 1.1 × 10

^{–5}

^{ }°C

^{-1}

Let the length of the aluminium scale at 0°C, 40°C and 100°C be

*L*

_{0al},

_{ }

*L*

_{4}

_{0al}

_{ }and

*L*

_{10}

_{0al}

*.*

And let the length of the steel scale at 0°C, 40°C and 100°C be

*L*

_{0st},

_{ }

*L*

_{4}

_{0st }and

*L*

_{10}

_{0st}

*.*

(a) So,

*L*

_{0st}(1 –

*α*

_{st}× 20) =

*L*

_{0al}(1 –

*α*

_{al}× 20)

$\frac{{L}_{0st}}{{L}_{0al}}=\frac{\left(1-{\alpha}_{al}\times 20\right)}{\left(1-{\alpha}_{st}\times 20\right)}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{{L}_{0st}}{{L}_{0al}}=\frac{1-2.3\times {{10}^{-}}^{5}\times 20}{1-1.1\times {10}^{-5}\times 20}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{{L}_{0st}}{{L}_{0al}}=\frac{0.99954}{0.99978}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{{L}_{0st}}{{L}_{0al}}=0.999759$

(b)

$\frac{{L}_{40al}}{{L}_{40st}}=\frac{{L}_{0al}\left(1+\alpha {}_{al}\times 40\right)}{{L}_{0st}\left(1+{\alpha}_{st}\times 40\right)}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{{L}_{40al}}{{L}_{40st}}=\frac{{L}_{0al}}{{L}_{0st}}\times \frac{\left(1+2.3\times {10}^{-5}\times 40\right)}{\left(1+1.1\times {10}^{-5}\times 40\right)}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{{L}_{40al}}{{L}_{40st}}=\frac{0.99977\times 1.00092}{1.00044}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{{L}_{40al}}{{L}_{40st}}=1.0002496\phantom{\rule{0ex}{0ex}}$

(c)

$\frac{{L}_{100al}}{{L}_{100st}}=\frac{{L}_{0al}\left(1+\alpha {}_{al}\times 100\right)}{{L}_{0st}\left(1+{\alpha}_{st}\times 100\right)}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{{L}_{100al}}{{L}_{100st}}=\frac{0.99977\times 1.0023}{1.0023}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{{L}_{100al}}{{L}_{100st}}=1.00096\phantom{\rule{0ex}{0ex}}$

#### Page No 13:

#### Question 15:

A metre scale is made up of steel and measures correct length at 16°C. What will be the percentage error if this scale is used (a) on a summer day when the temperature is 46°C and (b) on a winter day when the temperature is 6°C? Coefficient of linear expansion of steel = 11 × 10^{–6} °C^{–1}.

#### Answer:

(a) Let the correct length measured by a metre scale made up of steel 16 °C be *L.*

Initial temperature, *t*_{1} = 16 °C

Temperature on a hot summer day,* **t*_{2} = 46 °C

So, change in temperature, Δθ = *t*_{2}_{ }$-$_{ }*t*_{1}_{ }_{ }= 30 °C

Coefficient of linear expansion of steel, $\alpha $ = 1.1 × 10^{–5 }°C^{-1}

Therefore, change in length,

Δ*L* = *L* αΔθ = *L *× 1.1 × 10^{–5} × 30

$\%\mathrm{of}\mathrm{error}=\left(\frac{\u2206L}{L}\times 100\right)\%\phantom{\rule{0ex}{0ex}}=\left(\frac{L\alpha \Delta \theta}{L}\times 100\right)\%\phantom{\rule{0ex}{0ex}}=\left[1.1\times {10}^{-5}\times 30\times 100\right]\%\phantom{\rule{0ex}{0ex}}=3.3\times {10}^{-2}\%\phantom{\rule{0ex}{0ex}}$

(b) Temperature on a winter day, *t*_{2} = 6 °C

So, change in temperature, Δθ = *t*_{1}_{ }$-$ *t*_{2}_{ }= 10 °C

Δ*L** *=* L*_{2 }$-$ *L*_{1 }= *L* αΔθ = *L *× 1.1 × 10^{–5} × 10

$\%\mathrm{of}\mathrm{error}=\left(\frac{\u2206L}{L}\times 100\right)\%\phantom{\rule{0ex}{0ex}}=\left(\frac{L\alpha \Delta \theta}{L}\times 100\right)\%\phantom{\rule{0ex}{0ex}}=1.1\times {10}^{-5}\times 10\times 100\%\phantom{\rule{0ex}{0ex}}=1.1\times {10}^{-2}\phantom{\rule{0ex}{0ex}}$

#### Page No 13:

#### Question 16:

A metre scale made of steel reads accurately at 20°C. In a sensitive experiment, distances accurate up to 0.055 mm in 1 m are required. Find the range of temperature in which the experiment can be performed with this metre scale. Coefficient of linear expansion of steel = 11 × 10^{–6} °C^{–1}.

#### Answer:

Given:

Temperature at which a metre scale gives an accurate reading, *T*_{1} = 20 °C

The value of variation admissible, *ΔL* = 0.055 mm = 0.055 × 10^{–3} m, in the length, *L*_{0} = 1 m

Coefficient of linear expansion of steel, *α* = 11 × 10^{–6} °C^{–1}

Let the range of temperature in which the experiment can be performed be *T*_{2}.

We know: *ΔL = **L*_{0}* αΔT*

$\Rightarrow 0.055\times {10}^{-3}=1\times 11\times {10}^{-6}\times \left({T}_{1}\pm {T}_{2}\right)\phantom{\rule{0ex}{0ex}}\Rightarrow 5\times {10}^{-3}=\left(20\pm {T}_{2}\right)\times {10}^{-3}\phantom{\rule{0ex}{0ex}}\Rightarrow 20\pm {T}_{2}=5\phantom{\rule{0ex}{0ex}}\Rightarrow \text{E}\mathrm{ither}{T}_{2}=20+5=25\xb0\mathrm{C}\phantom{\rule{0ex}{0ex}}\mathrm{or}{T}_{2}=20-5=15\xb0\mathrm{C}$

Hence, the experiment can be performed in the temperature range of 15 °C to 25 °C .

#### Page No 13:

#### Question 17:

The density of water at 0°C is 0.998 g cm^{–3 }and at 4°C is 1.000 g cm^{–1}. Calculate the average coefficient of volume expansion of water in the temperature range of 0 to 4°C.

#### Answer:

Given:

Density of water at 0°C, ( *f*_{0})= 0.998 g cm^{-3}

Density of water at 4°C, (*f*_{4}) = 1.000 g cm-3

Change in temperature, (Δ*t*) = 4^{o}C

Let the average coefficient of volume expansion of water in the temperature range of 0 to 4°C be *γ*.

$\mathrm{We}\mathrm{know}:{f}_{4}={f}_{0}\left(1+\gamma \u2206t\right)\phantom{\rule{0ex}{0ex}}\Rightarrow {f}_{0}=\frac{{f}_{4}}{1+\gamma \u2206t}\phantom{\rule{0ex}{0ex}}\Rightarrow 0.998=\frac{1}{1+\gamma .4}\phantom{\rule{0ex}{0ex}}\Rightarrow 1+4\gamma =\frac{1}{0.998}\phantom{\rule{0ex}{0ex}}\Rightarrow 4\gamma =\left(\frac{1}{0.998}\right)-1\phantom{\rule{0ex}{0ex}}\Rightarrow \gamma =0.0005=5\times {10}^{-4}{}^{\mathrm{o}}{\mathrm{C}}^{-1}$

As the density decreases,

$\gamma =-5\times {10}^{-4}{}^{\mathrm{o}}{\mathrm{C}}^{-1}$

Therefore,the average coefficient of volume expansion of water in the temperature range of 0 to 4°C will be $\gamma =-5\times {10}^{-4}$^{o}C^{-1}.

#### Page No 13:

#### Question 18:

Find the ratio of the lengths of an iron rod and an aluminium rod for which the difference in the lengths is independent of temperature. Coefficients of linear expansion of iron and aluminium are 12 × 10^{–6} °C^{–1} and 23 × 10^{–6} °C^{–1} respectively.

#### Answer:

Let the original length of iron rod be *L** _{Fe}* and

*L*

^{'}^{}^{}*be its length when temperature is increased by Δ*

_{Fe }*T*.

Let the original length of aluminium rod be

*L*

*and*

_{Al}*L*

^{'}^{}^{}*be its length when temperature is increased by Δ*

_{Al }*T*.

Coefficient of linear expansion of iron, ${\alpha}_{Fe}$ = 12 × 10

^{–6}°C

^{$-$1}

Coefficient of linear expansion of aluminium,

*α*= 23 × 10

_{Al}^{–6}°C

^{}

^{$-$1 }

Since the difference in length is independent of temperature, the difference is always constant.

$L{\text{'}}_{Fe}={L}_{Fe}\left(1+{\alpha}_{Fe}\times \u2206T\right)\phantom{\rule{0ex}{0ex}}\mathrm{and}L{\text{'}}_{Al}={L}_{Al}\left(1+{\alpha}_{Al}\times \u2206T\right)\phantom{\rule{0ex}{0ex}}\Rightarrow \begin{array}{cc}L{\text{'}}_{Fe}-L{\text{'}}_{Al}={L}_{Fe}-{L}_{Al}+{L}_{Fe}\times {\alpha}_{Fe}\u2206T-{L}_{Al}\times {\alpha}_{Al}\times \u2206T& -\left(1\right)\end{array}\phantom{\rule{0ex}{0ex}}\mathrm{Given}:\phantom{\rule{0ex}{0ex}}L{\text{'}}_{Fe}-L{\text{'}}_{Al}={L}_{Fe}-{L}_{Al}\phantom{\rule{0ex}{0ex}}\mathrm{Hence},{L}_{Fe}{\alpha}_{Fe}={L}_{Al}{\alpha}_{Al}[\mathrm{using}(1\left)\right]\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{{L}_{Fe}}{{L}_{Al}}=\frac{23}{12}$

The ratio of the lengths of the iron to the aluminium rod is 23:12.

#### Page No 13:

#### Question 19:

A pendulum clock shows correct time at 20°C at a place where *g* = 9.800 m s^{–2}. The pendulum consists of a light steel rod connected to a heavy ball. It is taken to a different place where* g* = 9.788 m s^{–1}. At what temperature will the clock show correct time? Coefficient of linear expansion of steel = 12 × 10^{–6} °C^{–1}.

#### Answer:

Given:

The temperature at which the pendulum shows the correct time, *T*_{1} = 20 °C

Coefficient of linear expansion of steel, $\alpha $ = 12 × 10^{–6} °C^{–1}

Let* **T*_{2} be the temperature at which the value of *g* is 9.788 ms^{–2 }and $\Delta $*T* be the change in temperature.

^{}So, the time periods of pendulum at different values of *g* will be *t*_{1} and *t*_{2} , such that

${t}_{1}=2\mathrm{\pi}\sqrt{\frac{{l}_{\mathit{1}}}{{g}_{1}}}\phantom{\rule{0ex}{0ex}}{t}_{2}=2\mathrm{\pi}\sqrt{\frac{{l}_{2}}{{g}_{2}}}\phantom{\rule{0ex}{0ex}}=\frac{2\mathrm{\pi}\sqrt{{l}_{1}\left(1+\alpha \Delta T\right)}}{\sqrt{{g}_{2}}}\left(\because {l}_{2}={l}_{1}\left(1+\alpha \u2206T\right)\right)\phantom{\rule{0ex}{0ex}}\mathrm{Given},{t}_{1}={t}_{2}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{2\pi \sqrt{{l}_{1}}}{\sqrt{{g}_{1}}}=\frac{2\pi \sqrt{{l}_{1}\left(1+\alpha \u2206T\right)}}{\sqrt{{g}_{2}}}\phantom{\rule{0ex}{0ex}}\Rightarrow \sqrt{\left(\frac{{l}_{1}}{{g}_{1}}\right)}=\frac{\sqrt{{l}_{1}\left(1+\alpha \u2206T\right)}}{\sqrt{{g}_{2}}}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{1}{9.8}=\frac{1+12\times {10}^{-6}\times \u2206T}{9.788}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{9.788}{9.8}=1+12\times {10}^{-6}\times \u2206T\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{9.788}{9.8}-1=12\times {10}^{-6}\times \u2206T\phantom{\rule{0ex}{0ex}}\Rightarrow \u2206T=\frac{-0.00122}{12\times {10}^{-6}}\phantom{\rule{0ex}{0ex}}\Rightarrow {T}_{2}-20=-102.4\phantom{\rule{0ex}{0ex}}\Rightarrow {T}_{2}=-102.4+20\phantom{\rule{0ex}{0ex}}=-82.4\phantom{\rule{0ex}{0ex}}\Rightarrow {T}_{2}\approx -82\xb0\mathrm{C}$

Therefore, for a pendulum clock to give correct time, the temperature at which the value of *g* is 9.788 ms^{–2} should be $-$ 82^{ }^{o}C.

#### Page No 13:

#### Question 20:

An aluminium plate fixed in a horizontal position has a hole of diameter 2.000 cm. A steel sphere of diameter 2.005 cm rests on this hole. All the lengths refer to a temperature of 10 °C. The temperature of the entire system is slowly increased. At what temperature will the ball fall down? Coefficient of linear expansion of aluminium is 23 × 10^{–6} °C^{–1}^{ }and that of steel is 11 × 10^{–6} °C^{–1}.

#### Answer:

Given:

Diameter of the steel sphere at temperature (*T*_{1}_{ }= 10 °C) , *d*_{st} = 2.005 cm

Diameter of the aluminium sphere, *d*_{Al} = 2.000 cm

Coefficient of linear expansion of steel, *α*_{st} = 11 × 10^{$-6$} °C^{$-1$}

Coefficient of linear expansion of aluminium, *α*_{Al} = 23 × 10^{$-$6} °C^{$-$1}^{}

Let the temperature at which the ball will fall be *T*_{2}_{ , }so that change in temperature be Δ*T*.

*d*'_{st}_{ }= 2.005(1 + *α _{st} *Δ

*T*)

$\Rightarrow d{\text{'}}_{\mathrm{st}}=2.005+2.005\times 11\times {10}^{-6}\times \u2206T\phantom{\rule{0ex}{0ex}}d{\text{'}}_{\mathrm{Al}}=2\left(1+{\alpha}_{\mathrm{Al}}\times \u2206T\right)\phantom{\rule{0ex}{0ex}}\Rightarrow d{\text{'}}_{\mathrm{Al}}=2+2\times 23\times {10}^{-6}\times \u2206T$

The steel ball will fall when both the diameters become equal.

So,

*d*'

_{st}

_{ }

*=*

*d*'

_{Al}

$\Rightarrow 2.005+2.005\times 11\times {10}^{-6}\u2206T=2+2\times 23\times {10}^{-6}\u2206T\phantom{\rule{0ex}{0ex}}\Rightarrow \left(46-22.055\right)\times {10}^{-6}\u2206T=0.005\phantom{\rule{0ex}{0ex}}\Rightarrow \u2206T=\frac{0.005\times {10}^{6}}{23.945}=208.81\phantom{\rule{0ex}{0ex}}\mathrm{Now},\u2206T={T}_{2}-{T}_{1}={T}_{2}-10\xb0\mathrm{C}\phantom{\rule{0ex}{0ex}}\Rightarrow {T}_{2}=\u2206T+{T}_{1}=208.81+10\phantom{\rule{0ex}{0ex}}\Rightarrow {T}_{2}=218.8\cong 219\xb0\mathrm{C}$

Therefore, the temperature at which the ball will fall is 219 °C.

#### Page No 13:

#### Question 21:

A glass window is to be fit in an aluminium frame. The temperature on the working day is 40°C and the glass window measures exactly 20 cm × 30 cm. What should be the size of the aluminium frame so that there is no stress on the glass in winter even if the temperature drops to 0°C? Coefficients of linear expansion for glass and aluminium are 9.0 × 10^{–6} °C^{–1} and 24 ×100^{–6}°C^{–1} , respectively.

#### Answer:

Given:

At 40^{o}C, the length and breadth of the glass window are 20 cm and 30 cm, respectively.

Coefficient of linear expansion of glass, ${\alpha}_{g}$ = 9.0 × 10^{–6} °C^{–1}

Coefficient of linear expansion for aluminium, ${\alpha}_{Al}$ = 24 ×100^{–6} °C^{–1}

The final length of aluminium should be equal to the final length of glass so that there is no stress on the glass in winter, even if the temperature drops to 0 °C.

Change in temperature, $\Delta \theta $ = 40 °C

Let the initial length of aluminium be *l*.

$l\left(1-{\alpha}_{Al}\u2206\theta \right)=20\left(1-{\alpha}_{g}\u2206\theta \right)\phantom{\rule{0ex}{0ex}}\Rightarrow l\left(1-24\times {10}^{-6}\times 40\right)=20\left(1-9\times {10}^{-6}\times 40\right)\phantom{\rule{0ex}{0ex}}\Rightarrow l\left(1-0.00096\right)=20\left(1-0.00036\right)\phantom{\rule{0ex}{0ex}}\Rightarrow l=\frac{20\times 0.99964}{1-0.00096}\phantom{\rule{0ex}{0ex}}=\frac{20\times 0.99964}{0.99904}\phantom{\rule{0ex}{0ex}}\Rightarrow l=20.012\mathrm{cm}$

Let the initial breadth of aluminium be *b.*

$b\left(1-{\alpha}_{Al}\u2206\mathrm{\theta}\right)=30\left(1-{\alpha}_{g}\u2206\mathrm{\theta}\right)\phantom{\rule{0ex}{0ex}}\Rightarrow b=\frac{30\times \left(1-9\times {10}^{-6}\times 40\right)}{\left(1-24\times {10}^{-6}\times 40\right)}\phantom{\rule{0ex}{0ex}}=\frac{30\times 0.99964}{0.99904}\phantom{\rule{0ex}{0ex}}\Rightarrow b=30.018\mathrm{cm}$

Therefore, the size of the aluminium frame should be 20.012 cm × 30.018 cm.

#### Page No 13:

#### Question 22:

The volume of a glass vessel is 1000 cc at 20°C. What volume of mercury should be poured into it at this temperature so that the volume of the remaining space does not change with temperature? Coefficients of cubical expansion of mercury and glass are 1.8 × 10^{–6} °C^{–1} and 9.0 × 10^{–6} °C^{–1} , respectively.

#### Answer:

At *T* = 20°C, the volume of the glass vessel, *V*_{g}_{ }= 1000 cc.

Let the volume of mercury be *V*_{Hg} .

Coefficient of cubical expansion of mercury, *γ*_{Hg} = 1.8 × 10^{–4} /°C

Coefficient of cubical expansion of glass,* **γ*_{g} = 9 × 10^{–6} /°C

Change in temperature,* *Δ*T**, *is same for glass and mercury.

Let the volume of glass and mercury after rise in temperature be *V'*_{g} and *V'*_{Hg} respectively.

Volume of remaining space after change in temperature,(*V'*_{g}* – V'*_{Hg}) = Volume of the remaining space (initial),(*V*_{g}* – V*_{Hg})

We know: *V'*_{g} = *V*_{g} (1 + *γ*_{g}* ΔT*) ...(1)

* V'*_{Hg}* = V*_{Hg} (1 + *γ*_{Hg }*ΔT*) ...(2)

Subtracting (2) from (1), we get:

$V{\text{'}}_{\mathrm{g}}-V{\text{'}}_{\mathrm{Hg}}={V}_{\mathrm{g}}-{V}_{\mathrm{Hg}}+{V}_{\mathrm{g}}{\gamma}_{\mathrm{g}}\u2206T-{V}_{\mathrm{Hg}}{\gamma}_{\mathrm{Hg}}\u2206T\phantom{\rule{0ex}{0ex}}\Rightarrow {V}_{\mathrm{g}}{\gamma}_{\mathrm{g}}\u2206T-{V}_{\mathrm{Hg}}{\gamma}_{\mathrm{Hg}}\u2206T=0\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{{V}_{\mathrm{g}}}{{V}_{\mathrm{Hg}}}=\frac{{\gamma}_{\mathrm{Hg}}}{{\gamma}_{\mathrm{g}}}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{1000}{{V}_{\mathrm{Hg}}}=\frac{1.8\times {10}^{-4}}{9\times {10}^{-6}}\phantom{\rule{0ex}{0ex}}\Rightarrow {V}_{\mathrm{Hg}}=\frac{9\times {10}^{-3}}{1.8\times {10}^{-4}}\phantom{\rule{0ex}{0ex}}\Rightarrow {V}_{\mathrm{Hg}}=50\mathrm{cc}$

Therefore, the volume of mercury that should be poured into the glass vessel is 50 cc.

#### Page No 13:

#### Question 23:

An aluminium can of cylindrical shape contains 500 cm^{3} of water. The area of the inner cross section of the can is 125 cm^{2}. All measurements refer to 10°C.

Find the rise in the water level if the temperature increases to 80°C. The coefficient of linear expansion of aluminium is 23 × 10^{–6} °C^{–1} and the average coefficient of the volume expansion of water is 3.2 × 10^{–4} °C^{–1}.

#### Answer:

Given:

Volume of water contained in the aluminium can, *V*_{0} = 500 cm^{3}

Area of inner cross-section of the can, *A *= 125 cm^{2}

^{}Coefficient of volume expansion of water, *γ* = 3.2 × 10^{–4} °C^{–1}

Coefficient of linear expansion of aluminium, ${\alpha}_{AL}$ = 23 × 10^{–6} °C^{–1}

If $\u2206\theta $ is the change in temperature, then final volume of water$\left(V\right)$ due to expansion,

*V* = V_{0}(1 +* γΔθ*)

= 500 [1 + 3.2 × 10^{–4} × (80 – 10)]

= 500 [1 + 3.2 × 10^{–4 }× 70]

= 511.2 cm^{3}

The aluminium vessel expands in its length only.

So, area of expansion of the base can be neglected.

Increase in volume of water = 11.2 cm^{3}^{ }

Consider a cylinder of volume 11.2 cm^{3}

∴ Increase in height of the water $=\frac{11.2}{125}$ = 0.0896

= 0.089 cm

#### Page No 13:

#### Question 24:

A glass vessel measures exactly 10 cm × 10 cm × 10 cm at 0°C. It is filled completely with mercury at this temperature. When the temperature is raised to 10°C, 1.6 cm^{3} of mercury overflows. Calculate the coefficient of volume expansion of mercury. Coefficient of linear expansion of glass = 6.5 × 10^{–1 }°C^{–1}.

#### Answer:

Given: At 0^{o}^{}C, volume of glass vessel, *V*_{g} = 10 × 10 × 10 = 1000 cc = volume of mercury, *V*_{Hg}

Let the volume of mercury at 10°C be *V'*_{Hg }and that of glass be *V' _{g}*.

At 10

^{o}C, the additional volume of mercury than glass, due to heating,

*V'*

_{Hg}– V'_{g}_{ }= 1.6 cm

^{3}

So change in temperature, Δ

*T*= 10°C

Coefficient of linear expansion of glass,

*α*= 6.5 × 10

_{g}^{–6 }°C

^{–1}

Therefore, the coefficient of volume expansion of glass,

*γ*

_{g}= 3 × 6.5 × 10

^{–6}°C

^{–1}

Let the coefficient of volume expansion of mercury be

*γ*

_{Hg}

*.*

We know that

*V'*

_{Hg}

*= V*

_{Hg}

*(*

*1 + γ*

_{Hg }Δ

*T*)

*...*(1)

*V'*

_{g}

_{ }=*V*

_{g}(

*1 +*

*γ*

_{g}Δ

*T*)

*...*(2)

Subtracting (2) from (1) we get,

*V'*

_{Hg}

_{ }–*V'*

_{g}

_{ }=*V*

_{H}_{g}

*–*

*V*

_{g}

*+*

*V*

_{Hg}

*γ*

_{Hg}

*Δ*

*T*

*–*

*V*

_{g}

*γ*

_{g}Δ

*T*(as

*V*

_{Hg}

*=*

*V*

_{g})

$\Rightarrow 1.6=1000\times {\gamma}_{\mathrm{Hg}}\times 10-1000\times 6.5\times 3\times {10}^{-6}\times 10\phantom{\rule{0ex}{0ex}}\Rightarrow {\gamma}_{\mathrm{Hg}}=\frac{1.6+19.5\times {10}^{-2}}{10000}\phantom{\rule{0ex}{0ex}}\Rightarrow {\gamma}_{\mathrm{Hg}}=\frac{1.6+0.195}{10000}\phantom{\rule{0ex}{0ex}}\Rightarrow {\gamma}_{Hg}=\frac{1.795}{10000}\phantom{\rule{0ex}{0ex}}\Rightarrow {\gamma}_{Hg}=1.795\times {10}^{-4}\phantom{\rule{0ex}{0ex}}\Rightarrow {\gamma}_{\mathrm{Hg}}\cong 1.8\times {10}^{-4}\xb0{\mathrm{C}}^{-1}$

Therefore, the coefficient of volume expansion of mercury is 1.8× 10

^{–4 }°C

^{–1}.

#### Page No 13:

#### Question 25:

The densities of wood and benzene at 0°C are 880 kg m^{3} and 900 kg m^{–3} , respectively. The coefficients of volume expansion are 1.2 × 10^{–3 }°C^{–1} for wood and 1.5 × 10^{–3 }°C^{–1} for benzene. At what temperature will a piece of wood just sink in benzene?

#### Answer:

Given:

Density of wood at 0 °C, *f*_{w}_{ }= 880 kgm^{-3}

Density of benzene at 0 °C, *f _{b}* = 900 kgm

^{-}

^{3}

Coefficient of volume expansion for wood, γ

_{w}= Coefficient of volume expansion for benzene,

*γ*= 1.5 × 10

_{b}^{–3 }°C

^{–1}

So, initial temperature,

*T*

_{1}= 0 °C

Let

*T*

_{2}

_{ }be the temperature at which the piece of wood will just sink in benzene and $\Delta $

*T =*

*T*

_{2}$-$

*T*

_{1}.

The piece of wood begins to sink when its weight is equal to the weight of the benzene displaced.

Mass = volume $\times $density

$\mathrm{Therefore},Vf{\text{'}}_{w}g=Vf{\text{'}}_{b}g\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{{f}_{w}}{1+{\gamma}_{w}\Delta T}=\frac{{f}_{b}}{1+{\gamma}_{b}\Delta T}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{880}{1+1.2\times {10}^{-3}\Delta T}=\frac{900}{1+1.5\times {10}^{-3}\Delta T}\phantom{\rule{0ex}{0ex}}\Rightarrow 880+880\times 1.5\times {10}^{-3}\Delta T=900+900\times 1.2\times {10}^{-3}\Delta T\phantom{\rule{0ex}{0ex}}\Rightarrow \left(1320-1080\right)\times {10}^{-3}\Delta T=20\phantom{\rule{0ex}{0ex}}\Rightarrow \Delta T=83.3\xb0\mathrm{C}\phantom{\rule{0ex}{0ex}}\Rightarrow {T}_{2}-{T}_{1}\cong {83}^{o}\phantom{\rule{0ex}{0ex}}\Rightarrow {T}_{2}-0\xb0\cong {83}^{o}\phantom{\rule{0ex}{0ex}}\Rightarrow {T}_{2}\cong 83\xb0\mathrm{C}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

Therefore, the piece of wood will just sink in benzene at 83

^{o}C.

#### Page No 13:

#### Question 26:

A steel rod of length 1 m rests on a smooth horizontal base. If it is heated from 0°C to 100°C, what is the longitudinal strain developed?

#### Answer:

The steel rod is resting on a horizontal base at 0 °C. When the temperature is increased to 100 °C, it will lead to an increase in the length of the steel due to expansion on heating. Since, there is no opposition in expansion of length, no longitudinal strain will be developed.

#### Page No 13:

#### Question 27:

A steel rod is clamped at its two ends and rests on a fixed horizontal base. The rod is unstrained at 20°C.

Find the longitudinal strain developed in the rod if the temperature rises to 50°C. Coefficient of linear expansion of steel = 1.2 × 10^{–5} °C^{–1}.

#### Answer:

Given:

Temperature at which rod is resting on a fixed horizontal base without any strain, *T*_{1}=20 °C. Then the rod is heated to temperature, *T*_{2} = 50 °C

So change in temperature, $\Delta T={T}_{2}-{T}_{1}=30{}^{o}\mathrm{C}$

Coefficient of linear expansion of steel, *α* = 1.2 × 10^{–5} °C^{$-$1 }

Let* L* be the length of the rod without heating and *L' *be the length of the rod on heating.

Let longitudinal strain developed in the rod be *S*.

We know that

$L\text{'}=L(1+\alpha \Delta T)\phantom{\rule{0ex}{0ex}}\Rightarrow \Delta L=L\alpha \Delta T\phantom{\rule{0ex}{0ex}}\mathrm{Strain},S=\frac{\Delta L}{L}\phantom{\rule{0ex}{0ex}}=\frac{L\alpha \Delta T}{L}\phantom{\rule{0ex}{0ex}}=\alpha \Delta T\phantom{\rule{0ex}{0ex}}\Rightarrow S=1.2\times {10}^{-5}\times \left(50-20\right)\phantom{\rule{0ex}{0ex}}=1.2\times {10}^{-5}\times 30\phantom{\rule{0ex}{0ex}}=36\times {10}^{-5}\phantom{\rule{0ex}{0ex}}S=3.6\times {10}^{-4}$

The strain of 3.6 × 10^{$-$4 }will be opposite to the direction of expansion.

#### Page No 13:

#### Question 28:

A steel wire of cross-sectional area 0.5 mm^{2}^{ }is held between two fixed supports. If the wire is just taut at 20°C, determine the tension when the temperature falls to 0°C. Coefficient of linear expansion of steel is 1.2 × 10^{–5} °C^{–1} and its Young's modulus is 2.0 × 10^{–11} Nm^{–2}.

#### Answer:

Given:

Cross-sectional area of the steel wire, *A* = 0.5 mm^{2} = 0.5 × 10^{–6} m^{2}

The wire is taut at a temperature, *T*_{1}_{ }= 20 °C,

After this, the temperature is reduced to *T*_{2} = 0 °C

So, the change in temperature, *Δθ* = *T*_{1}$-$*T*_{2}_{ }= 20 °C

Coefficient of linear expansion of steel, *α* = 1.2 ×10^{–5} °C^{-1}

Young's modulus, *γ* = 2 ×10^{11} Nm^{$-2$}

Let *L* be the initial length of the steel wire and *L*' be the length of the steel wire when temperature is reduced to 0°C.

Decrease in length due to compression, *ΔL = L'$-$L= LαΔθ* ...(1)

Let the tension applied be *F*.

^{$\gamma =\frac{\mathrm{stress}}{\mathrm{strain}}=\raisebox{1ex}{$\left(\frac{F}{A}\right)$}\!\left/ \!\raisebox{-1ex}{$\left({\displaystyle \frac{\Delta L}{L}}\right)$}\right.\phantom{\rule{0ex}{0ex}}\Rightarrow \gamma =\frac{F}{A}\times \frac{L}{\Delta L}\phantom{\rule{0ex}{0ex}}\Rightarrow \Delta L=\frac{FL}{AY}...\left(2\right)$}

Change in length due to tension produced is given by (1) and (2).

So, on equating (1) and (2), we get:

$L\alpha \Delta \theta =\frac{FL}{AY}\phantom{\rule{0ex}{0ex}}\Rightarrow F=\alpha \Delta \theta AY\phantom{\rule{0ex}{0ex}}=1.2\times {10}^{-5}\times \left(20-0\right)\times 0.5\times {10}^{-6}\times 2\times {10}^{11}\phantom{\rule{0ex}{0ex}}=1.2\times 20\phantom{\rule{0ex}{0ex}}\Rightarrow F=24\mathrm{N}$

Therefore, the tension produced when the temperature falls to 0°C is 24 N.

#### Page No 13:

#### Question 29:

A steel rod is rigidly clamped at its two ends. The rod is under zero tension at 20°C. If the temperature rises to 100°C, what force will the rod exert on one of the clamps? Area of cross-section of the rod is 2.00 mm^{2}. Coefficient of linear expansion of steel is 12.0 × 10^{–6} °C^{–1} and Young's modulus of steel is 2.00 × 10^{11} Nm^{–2}.

#### Answer:

Given:

Temperature of the rod at zero tension, *T*_{1} = 20 °C

Final temperature, *T*_{2} = 100 °C

Change in temperature, $\Delta \theta $ = 80 °C

Cross-sectional area of the rod, *A* = 2 mm^{2} = 2 × 10^{$-$6} m^{2}

Coefficient of linear expansion of steel, *α*_{ }= 12 ×10^{–6 }°C^{}^{$-$1}

Young's modulus of steel, *Y*_{ }= 2 × 10^{11 }Nm^{$-2$}

Let *L* be the length of the steel rod at 20 °C and *L'* be the length of steel rod at 100 °C.

Change of length of the rod, $\u2206L$ = *L'$-$* *L*

If *F* be the force exerted by the rod on one of the clamps due to rise in temperature, then

$Y=\frac{\mathrm{stress}}{\mathrm{strain}}=\frac{F/A}{{\displaystyle \raisebox{1ex}{$\Delta L$}\!\left/ \!\raisebox{-1ex}{$L$}\right.}}\phantom{\rule{0ex}{0ex}}\Rightarrow F=\frac{Y\times \Delta L}{L}\times A\phantom{\rule{0ex}{0ex}}\Delta L=L\alpha \Delta \theta \phantom{\rule{0ex}{0ex}}\Rightarrow F=\frac{YL\alpha \Delta \theta A}{L}\phantom{\rule{0ex}{0ex}}\Rightarrow F=YA\alpha \Delta \theta \phantom{\rule{0ex}{0ex}}=2\times {10}^{11}\times 2\times {10}^{-6}\times 12\times {10}^{-6}\times 80\phantom{\rule{0ex}{0ex}}=48\times 80\times {10}^{-1}\phantom{\rule{0ex}{0ex}}\mathrm{So},F=384\mathrm{N}$

Therefore, the rod will exert a force of 384 N on one of the clamps.

#### Page No 14:

#### Question 30:

Two steel rods and an aluminium rod of equal length *l*_{0} and equal cross-section are joined rigidly at their ends, as shown in the figure below. All the rods are in a state of zero tension at 0°C. Find the length of the system when the temperature is raised to θ. Coefficient of linear expansion of aluminium and steel are α* _{a}* and α

_{s}

_{,}respectively. Young's modulus of aluminium is Y

_{a}and of steel is Y

_{s}.

Steel |

Aluminium |

Steel |

#### Answer:

Given:

Initial length of the system (two identical steel rods + an aluminium rod) at 0 °C = *l*_{0}

Coefficient of linear expansion of steel and aluminium are *α _{s}*

_{ }and

*α*, respectively.

_{Al}Temperature is raised by

*θ.*

_{}So, the change in temperature, $\u2206\theta $ =

*θ*$-$ 0 °C =

*θ*

Young's modulus of steel and aluminium are

*γ*and

_{s}*γ*

_{Al}

_{, }respectively.

If

*l*be the final length of the system at temperature

*θ,*

strain on the system $=\frac{\left(l-{l}_{0}\right)}{{l}_{0}}$ ...(1)

Young's modulus = Stress/ Strain

Therefore, the total strain on the system $=\frac{\text{T}\mathrm{otal}\mathrm{stress}\text{onthe}\mathrm{system}}{\text{T}\mathrm{otal}\mathrm{Young}\text{'}\mathrm{s}\mathrm{modulus}\mathrm{of}\text{the}\mathrm{system}}$

Now, total stress = stress due to the two steel rods + stress due to the aluminium rod

$\text{S}\mathrm{tress}=\frac{F}{A}=\alpha Y\u2206\theta $

Total stress =

*γ*

_{s}α_{s}θ + γ_{s}α_{s}θ + γ_{Al}α_{Al}θ= 2

*γ*

_{s}*α*

_{s}θ*+ γ*...(2)

_{Al}α_{Al}θYoung's modulus of the system,

*Y*=

*γ*2

_{s}+ γ_{s}+ γ_{Al}=*γ*

_{s}*+*

*γ*

_{Al}

_{ }...(3)

Using (1), (2) and (3), we get:

$\mathrm{Strain}\text{onthe}\mathrm{system}=\frac{2{\gamma}_{s}{\alpha}_{s}\theta +{\gamma}_{Al}{\alpha}_{Al}\theta}{2{\gamma}_{s}+{\gamma}_{Al}}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{l-{l}_{0}}{{l}_{0}}=\frac{2{\gamma}_{s}{\alpha}_{s}\theta +{\gamma}_{Al}{\alpha}_{Al}\theta}{2{\gamma}_{s}+{\gamma}_{Al}}\phantom{\rule{0ex}{0ex}}\Rightarrow l={l}_{0}\left[1+\frac{2{\gamma}_{s}{\alpha}_{s}\theta +{\gamma}_{Al}{\alpha}_{Al}\theta}{2{\gamma}_{s}+{\gamma}_{Al}}\right]\phantom{\rule{0ex}{0ex}}$

Therefore, the final length of the system will be ${l}_{0}\left[1+\frac{2{\gamma}_{s}{\alpha}_{s}\theta +{\gamma}_{Al}{\alpha}_{Al}\theta}{2{\gamma}_{s}+{\gamma}_{Al}}\right]$, where

*l*

_{0}is its initial length.

#### Page No 14:

#### Question 31:

A steel ball that is initially at a pressure of 1.0 × 10^{5 }Pa is heated from 20°C to 120°C, keeping its volume constant.

Find the pressure inside the ball. Coefficient of linear expansion of steel = 12 × 10^{–6} °C^{–1} and bulk modulus of steel = 1.6 × 10^{11} Nm^{–2}.

#### Answer:

Given:

Initial pressure on the steel ball = 1.0 × 10^{5 }Pa

The ball is heated from 20 °C to 120 °C.

So, change in temperature, $\Delta \theta $ = 100 °C.

Coefficient of linear expansion of steel, $\alpha $ = 12 × 10^{–6} °C^{–1}

Bulk modulus of steel,* B* = 1.6 × 10^{11} Nm^{–2}

Pressure is given as,

$\Rightarrow P=B\times \gamma \u2206\theta \phantom{\rule{0ex}{0ex}}\Rightarrow P=B\times 3\alpha \u2206\theta \left(\because \gamma =3\alpha \right)\phantom{\rule{0ex}{0ex}}\Rightarrow P=1.6\times {10}^{11}\times 3\times 12\times {10}^{-6}\times \left(120-20\right)\phantom{\rule{0ex}{0ex}}=1.6\times 3\times 12\times {10}^{11}\times {10}^{-6}\times {10}^{2}\phantom{\rule{0ex}{0ex}}=57.6\times {10}^{7}\phantom{\rule{0ex}{0ex}}\Rightarrow P=5.8\times {10}^{8}\mathrm{Pa}$

Therefore, the pressure inside the ball is 5.8 × 10^{8 }Pa.

#### Page No 14:

#### Question 32:

Show that the moment of inertia of a solid body of any shape changes with temperature as *I = **I*_{0} (1 + 2αθ), where *I*_{0} is the moment of inertia at 0°C and α is the coefficient of linear expansion of the solid.

#### Answer:

Given:

Coefficient of linear expansion of solid = *α *

Moment of inertia at 0 °C = *I*_{0}

If temperature changes to *θ* from 0 °C, then change in temperature, $\left(\u2206T\right)$ = $\theta $

Let *I* be the new moment of inertia attained due to rise in temperature.

Let *R*_{0} be the radius of gyration at 0 °C.

We know that on heating, radius of gyration will change as

*R*_{ }= *R*_{0}(1 + *αθ*)

Here, *R* is the radius of gyration after heating.

*I*_{0} = *MR*_{0}^{2} , where *M* = mass of the body

Now, *I* = *MR*^{2} = *MR*_{0}^{2}(1 + *αθ*)^{2}

Expanding binomially and neglecting the higher terms of order (*αθ*) that will be very small, we get

*I* = *MR*_{0}^{2}(1 + 2 *αθ*)

So, *I* = *I*_{0}(1 + 2 *αθ*)

Hence, proved.

#### Page No 14:

#### Question 33:

A torsional pendulum consists of a solid disc connected to a thin wire (α = 2.4 × 10^{–5 }°C^{–1}) at its centre. Find the percentage change in the time period between peak winter (5°C) and peak summer (45°C).

#### Answer:

Given:

Coefficient of linear expansion of the wire, α = 2.4 × 10^{–5 }°C^{–1}

Let *I*_{0}_{ }be the moment of inertia of the torsional pendulum at 0 °C.

If *K* is the torsional constant of the wire, then time period of torsional pendulum $\left(T\right)$:

$T=2\pi \sqrt{\frac{I}{K}}...\left(1\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

Here, *I* = moment of inertia after change in temperature

When the temperature is changed by $\u2206\theta $, moment of inertia $\left(I\right)$,

*I = **I*_{0}(1+2$\alpha $$\u2206\theta $)

On substituting the value of *I *in equation(1), we get:

$T=2\mathrm{\pi}\frac{\sqrt{{I}_{0}\left(1+2\alpha \u2206\theta \right)}}{\sqrt{K}}$

In winter, $\u2206\theta $ = 5 °C

$\therefore $ Time period$\left({T}_{1}\right)$

$=2\mathrm{\pi}\frac{\sqrt{{I}_{0}\left(1+2\alpha \times 5\right)}}{\sqrt{K}}$

In summer, $\u2206\theta $ = 45 °C

Time period $\left({T}_{2}\right)$

$=2\mathrm{\pi}\frac{\sqrt{{\mathrm{I}}_{0}\left(1+2\mathrm{\alpha}\times 45\right)}}{\sqrt{\mathrm{K}}}$

$\mathrm{So},\phantom{\rule{0ex}{0ex}}\frac{{T}_{2}}{{T}_{1}}=\frac{\sqrt{1+90\alpha}}{\sqrt{1+10\alpha}}\phantom{\rule{0ex}{0ex}}=\frac{\sqrt{1+90\times 2.4\times {10}^{-5}}}{\sqrt{1+10\times 2.4\times {10}^{-5}}}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{{T}_{2}}{{T}_{1}}=\frac{\sqrt{1.00216}}{\sqrt{1.00024}}\phantom{\rule{0ex}{0ex}}\%\mathrm{change}=\left(\frac{{T}_{2}}{{T}_{1}}-1\right)\times 100\phantom{\rule{0ex}{0ex}}=0.0959\%\phantom{\rule{0ex}{0ex}}\Rightarrow \%\mathrm{change}\mathrm{in}\mathrm{time}\mathrm{period}\approx 9.6\times {10}^{-2}\%\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

Therefore, the percentage change in time period of a torsional pendulum between peak winters and peak summers is 9.6 × 10^{–2 }% .

#### Page No 14:

#### Question 34:

A circular disc made of iron is rotated about its axis at a constant velocity ω. Calculate the percentage change in the linear speed of a particle of the rim as the disc is slowly heated from 20°C to 50°C, keeping the angular velocity constant. Coefficient of linear expansion of iron = 1.2 × 10^{–5} °C^{–1}.

#### Answer:

Let initial radius of the circular disc at 20^{ }^{ₒ}C = ${r}_{20}$

Let final radius of the circular disc at 50^{ }^{ₒ}C = ${r}_{50}$

Coefficient of linear expansion of iron, $\alpha $ = 1.2 × 10^{–5} °C^{–1}.

Change in temperature, $\u2206T$ = 30 ^{₀}C

Let *R*' and *R* be the radius of the paricle at 50^{ }^{ₒ}C and 20^{ }^{ₒ}C respectively.

If *v* and *v*' be the linear speed of the particle at 50^{ }^{ₒ}C and 20^{ }^{ₒ}C respectively, as the angular velocity remains($\omega $) constant.

Therefore,

$\omega =\frac{v}{R}=\frac{v\text{'}}{R\text{'}}....\left(1\right)$

Now,

*R*' = *R*(1+$\alpha \u2206T$)

*$\Rightarrow $R*' = *R* + *R* $\times $ 1.2 × 10^{–5} °C^{–1}^{ }$\times \u2206T$.

$\Rightarrow $ *R*' = 1.00036*R*

Using equation(1) we have,

$\frac{v}{R}=\frac{v\text{'}}{R\text{'}}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{v}{R}=\frac{v\text{'}}{1.00036R}\phantom{\rule{0ex}{0ex}}\Rightarrow v\text{'}=1.00036v$

Percentage change in linear speed will be,

$=\frac{v\text{'}-v}{v}\times 100\phantom{\rule{0ex}{0ex}}=\frac{1.00036v-v}{v}\times 100\phantom{\rule{0ex}{0ex}}=3.6\times {10}^{-2}\phantom{\rule{0ex}{0ex}}$

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