Thermal Expansion

Q. Charges $Q_1$ and $Q_2$ are at points $A$ and $B$ of a right angle triangle $OAB$ see figure. The resultant electric feild at point $O$ is perpendicular to the hypotenuse, then $Q_1/Q_2$ is proportional to :

1. $\frac{x_1^3}{x_2^3}$
2. $\frac{x_2}{x_1}$
3. $\frac{x_1}{x_2}$
4. $\frac{x_2^2}{x_1^2}$

Q. The equation of an equipotential line in an electric field is $y=2x$, then the electric field strength vector at $(1,2)$ may be :

1. $4\hat{i}+3\hat{j}$
2. $4\hat{i}+8\hat{j}$
3. $8\hat{i}+4\hat{j}$
4. $-8\hat{i}+4\hat{j}$

Q. The coefficient of linear expansion of steel and brass are $11\times {10}^{-6}{/}^{\circ }\text{C}$ and $19\times {10}^{-6}{/}^{\circ }\text{C}$, respectively. If their differences in lengths at all temperatures has to kept constant at $30\text{cm}$, their lengths at $0^{\circ }\text{C}$ should be

1. $71.25\text{cm}$ and $41.25\text{cm}$
2. $82\text{cm}$ and $52\text{cm}$
3. $92\text{cm}$ and $62\text{cm}$
4. $62.25\text{cm}$ and $32.25\text{cm}$

Q.

The moment of inertia of a uniform circular disc of radius $R$ and mass $M$ about an axis passing from the edge of the disc and normal to the disc is

Q. A physical quantity X is related to four measurable quantities a, b, c and d as follows $X=a^2{b}^3{c}^{5/2}{d}^{-2}$
The percentage error in the measurement of a, b, c and d are $1\%,2\%,2\%$ and $4\%$ respectively. What is the percentage error in quantity X?

1. $21\%$
2. $17\%$
3. $23\%$
4. $15\%$

Q. Two wires of same length and radius, are joined end to end and loaded. The Young's modulii of the materials of the two wires are $Y_1$ and $Y_2.$ The combination behaves as a single wire whose Young's modulus is :

1. $Y=\frac{Y_1{Y}_2}{2(Y_1+Y_2)}$
2. $Y=\frac{2Y_1{Y}_2}{3(Y_1+Y_2)}$
3. $Y=\frac{Y_1{Y}_2}{Y_1+Y_2}$
4. $Y=\frac{2Y_1{Y}_2}{Y_1+Y_2}$

Q.

Two different wires having lengths ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$, and respective temperature coefficient of linear expansion ${\mathrm{ɑ}}_1$ and ${\mathrm{ɑ}}_2$, are joined end-to-end. Then the effective temperature coefficient of linear expansion is:

1. $[{\mathrm{ɑ}}_1{\mathrm{L}}_1+{\mathrm{ɑ}}_2{\mathrm{L}}_2]/[{\mathrm{L}}_1+{\mathrm{L}}_2]$

2. $2\sqrt{{\mathrm{ɑ}}_1{\mathrm{ɑ}}_2}$

3. $4\left[\right({\mathrm{ɑ}}_1{\mathrm{ɑ}}_2)/[{\mathrm{ɑ}}_1+{\mathrm{ɑ}}_2\left]\right)][({\mathrm{L}}_2{\mathrm{L}}_1/{({\mathrm{L}}_2+{\mathrm{L}}_1)}^2]$

4. $[{\mathrm{ɑ}}_1+{\mathrm{ɑ}}_2]/2$

Q. Co-efficient of linear expansion of brass and steel rods are ${\alpha }_1$ and ${\alpha }_2$ respectively. Lengths of brass and steel rods are $L_1$ and $L_2$ respectively. If $(L_2-L_1)$ is maintained same at all the temperatures, which one of the following relations holds good?

1. ${\alpha }_1{L}_1={\alpha }_2{L}_2$
2. ${\alpha }_1{L}_2^2={\alpha }_2{L}_1^2$
3. ${\alpha }_1{L}_2={\alpha }_2{L}_1$
4. ${\alpha }_1^2{L}_2={\alpha }_2^2{L}_1$

Q. Two rods of different materials having coefficients of thermal expansion ${\alpha }_1,{\alpha }_2$ and Young's modulii $Y_1,Y_2$ respectively are fixed between two rigid massive walls. The rods are heated such that they undergo the same increase in temperature. There is no bending of the rods. If ${\alpha }_1:{\alpha }_2=2:3$ and thermal stresses developed in the two rods are equal, then $Y_1:Y_2$ is equal to

1. $2:3$
2. $1:1$
3. $3:2$
4. $4:9$

Q. When the temperature of a rod increases from $T$ to $T+\Delta T$, the moment of inertia of the rod about an axis increases from $I$ to $I+\Delta I$. If the coefficient of linear expansion of rod is $\alpha$, then find the ratio $\frac{\Delta I}{I}$

1. $\frac{\Delta T}{T}$
2. $\frac{2\Delta T}{T}$
3. $\alpha \Delta T$
4. $2\alpha \Delta T$

Q. A hole is drilled in a copper sheet. The diameter of the hole is $4.24\text{cm}$ at $27{}^{\circ }\text{C}$. What is the change in the diameter of the hole when the sheet is heated to $227{}^{\circ }\text{C}$ ?
$[\alpha =1.70\times {10}^{-5}/{}^{\circ }\text{C}]$.

1. $1.44\times {10}^{-2}\text{cm}$
2. $1.96\times {10}^{-2}\text{cm}$
3. $1.78\times {10}^{-2}\text{cm}$
4. $1.28\times {10}^{-2}\text{cm}$

Q. A seconds pendulum clock has a steel wire. The clock shows correct time at ${25}^{\circ }\text{C}$. How much time does the clock lose or gain in one week, when the temperature is increased to ${35}^{\circ }\text{C}$ ? $({\alpha }_{steel}=1.2\times {10}^{-5}{/}^{\circ }\text{C})$

1. $5.18\text{s}$ gain
2. $5.18\text{s}$ lose
3. $36.28\text{s}$ gain
4. $36.28\text{s}$ lose

Q. A pendulum has time period T in air. When it is made to oscillate in water, it acquired a time period $T^{\prime }=\sqrt{2T}$. The specific gravity of the pendulum bob is equal to

1. 2
2. None of these
3. 
4. 

Q. Two rods of different materials and of equal cross-sectional areas and length $1\text{m}$ are joined face to face at one end and their free ends are fixed to rigid walls. If the temperature of the surrounding is increased by $30{}^{\circ }\text{C}$, the magnitude of the displacement of the joint of the rods is :
$[{\alpha }_1=2\times {10}^{-5}{}^{\circ }{\text{C}}^{-1},{\alpha }_2=5\times {10}^{-5}{}^{\circ }{\text{C}}^{-1},$
$Y_1=2\times {10}^{10}{\text{N/m}}^2,Y_2={10}^{10}{\text{N/m}}^2]$

1. $15\times {10}^{-5}\text{m}$
2. $12\times {10}^{-5}\text{m}$
3. $10\times {10}^{-5}\text{m}$
4. $18\times {10}^{-5}\text{m}$

Q. Two rods, one of aluminum and the other made of steel, having initial lengths $l_1$ and $l_2$ are connected together to form a single rod of length $l_1+l_2$. The co-efficients of linear expansion for aluminum and steel are ${\alpha }_a$ and ${\alpha }_s$ respectively. If the length of each rod increases by the same amount when their temperature are raised by $t^{\circ }C$, then find the ratio $\frac{l_1}{(l_1+l_2)}$

1. 
2. 
3. 
4. 

Q. Two straight metallic strips each of thickness $t$ and length $L$ are riveted together. Their coefficients of linear expansion are ${\alpha }_1$ and ${\alpha }_2$. If they are heated through temperature $\Delta \theta$, the bimetallic strip will bend to form an arc of radius

1. $\frac{t}{({\alpha }_1+{\alpha }_2)\Delta \theta }$
2. $\frac{t}{({\alpha }_2-{\alpha }_1)\Delta \theta }$
3. $\frac{t}{2({\alpha }_1+{\alpha }_2)\Delta \theta }$
4. $\frac{t}{2({\alpha }_2-{\alpha }_1)\Delta \theta }$

Q. A bimetallic strip is made of two metal strips $A$ and $B$, having coefficient of linear expansions ${\alpha }_A$ and ${\alpha }_B$ . If ${\alpha }_A>{\alpha }_B$, which of the following describes the behaviour of the bimetallic strip when heated?

1. It will bend but will not elongate.
2. It will bend with the strip $B$ on the outer side.
3. It will bend with the strip $A$ on the outer side.
4. No bending at all.

Q. A rod of length $20\text{cm}$ is made of metal. It expands by $0.075\text{cm}$ when its temperature is raised from $0^{\circ }\text{C}$ to ${100}^{\circ }\text{C}$. Another rod of a different metal B having the same length expands by $0.045\text{cm}$ for the same change in temperature. A third rod of the same length is composed of two parts one of metal A and the other of metal B. Now, the rod expands by $0.06\text{cm}$ for the same change in temperature. The portion made of metal A has the length (in $\text{cm}$)

Q.

A pendulum clock gives correct time at ${20}^{\circ }C$ 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 $m{s}^{-2}$. At what temperature will it give correct time ? Coefficient of linear expansion of steel $=12\times {10}^{-6}{}^{\circ }{C}^{-1}$

Q. What is the effect of temperature on elasticity ? State reason .

Q. Two bodies of equal mass m are heated at a uniform rate under identical conditions. Their change in temperatures shown below. Then
(i) the ratio of melting points of the substances is $1.5$.
(ii) the ratio of their latent heats is $0.75$.
(iii) the ratio of specific heat of two substances is $0.33$ in solid state.
(iv) the ratio of specific heat of two substances is $0.5$ in liquid state.

1. $\left(iii\right),\left(iv\right)$
2. $\left(ii\right),\left(iii\right)$

3. $\left(i\right),\left(iii\right)$

4. All

Q. The length of wire, when $M_1$ is hung from it is $l_1$ and is $l_2$ with both $M_1$ and $M_2$ hanging. The natural length of wire is

1. $\frac{M_1}{M_2}(l_1-l_2)+l_1$
2. $\frac{M_2{l}_1-M_1{l}_2}{M_1+M_2}$
3. $\frac{l_1+L_2}{2}$
4. $\sqrt{l_1{L}_2}$

Q. A uniform copper rod of length $50\text{cm}$ and diameter $3\text{mm}$ is kept on a frictionless horizontal surface at $20{}^{\circ }\text{C}$. The coefficient of linear expansion of copper is $2\times {10}^{-5}{}^{\circ }{\text{C}}^{-1}$ and Young's modulus is $1.2\times {10}^{11}{\text{N/m}}^2$. The copper rod is heated to $100{}^{\circ }\text{C}$, Then, the tension developed in the copper rod is

1. $12\times {10}^3\text{N}$
2. $18\times {10}^3\text{N}$
3. Zero
4. $36\times {10}^3\text{N}$

Q. A uniform thermometer scale is at steady state with its $0\text{cm}$ mark at ${20}^{\circ }\text{C}$ and $100\text{cm}$ mark at ${100}^{\circ }\text{C}$. Temperature of the $60\text{cm}$ mark is

1. ${48}^{\circ }\text{C}$
2. ${68}^{\circ }\text{C}$
3. ${52}^{\circ }\text{C}$
4. ${58}^{\circ }\text{C}$

Q. A bimetallic strip consists of metals $X$ and $Y$. It is mounted rigidly at the base as shown in the figure. The metal $X$ has a higher coefficient of expansion than metal $Y$. When the bimetallic strip is placed in a cold bath,

1. It will bend towards right.
2. It will bend towards left.
3. It will not bend but shrink.
4. It will neither bend nor shrink

Q. A metal rod having linear expansion coefficient 2 $\times {10}^{-5}°C^{-1}$ has a length of 1 m at ${20}^{\circ }$C. The temperature at which it is shortened by 1 mm is

1. –15°C
2. –20°C
3. –30°C
4. –25°C

Q. Three rods of material $\text{X}$ and three of material $\text{Y}$ are connected as shown in figure. All the rods are identical in length and cross-sectional area. If the end $A$ is maintained at ${60}^{\circ }\text{C}$ and the junction $E$ at ${10}^{\circ }\text{C}$, calculate the temperature at the junction $B$. The thermal conductivity of $\text{X}$ is $800{\text{Wm}}^{-1}$${}^{\circ }{\text{C}}^{-1}$ and that of $\text{Y}$ is $400{\text{Wm}}^{-1}$${}^{\circ }{\text{C}}^{-1}$.

1. ${35}^{\circ }\text{C}$
2. ${40}^{\circ }\text{C}$
3. ${45}^{\circ }\text{C}$
4. ${50}^{\circ }\text{C}$

Q.

The coefficient of linear expansion of iron is ${10}^{-5}/{}^{\circ }C$. The volume of an iron cube, $5cm$ on edge will increase by what amount if it is heated from $10°\mathrm{C}$ to $60°\mathrm{C}$?

1. $0.1875{\mathrm{cm}}^3$

2. $0.00375{\mathrm{cm}}^3$

3. $0.0225{\mathrm{cm}}^3$

4. $0.0625{\mathrm{cm}}^3$

Q. A brass rod and a lead rod each $80\text{cm}$ long at $0{}^{\circ }\text{C}$ are clamped together at one end with their free ends coinciding. The separation the of free ends of the rods, if the system is placed in a steam bath is
$[{\alpha }_{\text{brass}}=18\times {10}^{-6}/{}^{\circ }\text{C}$ and ${\alpha }_{\text{lead}}=28\times {10}^{-6}/{}^{\circ }\text{C}\text{and temperature of steam}={100}^{\circ }\text{C}]$

1. $0.2\text{mm}$
2. $0.8\text{mm}$
3. $1.4\text{mm}$
4. $1.6\text{mm}$

Q. A uniform rod of length $2L$, area of cross section $A$ and Youngs modulus $Y$ is lying at rest under the action of three forces as shown. Choose the correct statement.

1. The stress at any cross section in section BC is $\frac{F}{A}$.
2. The stress at any cross section is $\frac{2F}{A}$.
3. The total extension is $\frac{3FL}{AY}$.
4. The total elongation is $\frac{6FL}{AY}$.