Strain

Q. Figure shows two rods $A$ and $B$ of same length $L$ and same cross-sectional area $S$ but of different material having coefficient of linear expansion ${\alpha }_1$ and ${\alpha }_2$ respectively. They are clamped between two rigid walls, separated by a distance $2L$. This all refers to temp $t^{\circ }\text{C}$. Find the tension in each rod at temp $2t^{\circ }\text{C}$ (Take the young’s modulus for the two rods to be $Y_1$ and $Y_2$ respectively).

1. $\frac{S{Y}_1{Y}_2(2{\alpha }_1+{\alpha }_2)t}{(2Y_1+Y_2)}$
2. $\frac{S{Y}_1{Y}_2({\alpha }_1+2{\alpha }_2)t}{(Y_1+2Y_2)}$
3. $\frac{S{Y}_1{Y}_2({\alpha }_1+{\alpha }_2)t}{(Y_1+Y_2)}$
4. $\frac{S{Y}_1{Y}_2({\alpha }_1+2{\alpha }_2)t}{(Y_1+Y_2)}$

Q. A uniform cylindrical rod, of length $L$ and radius $r$, is made from a material, whose Young’s modulus of Elasticity equals $Y$. When this rod is heated by temperature $T$ and simultaneously subjected to a net longitudinal compressional force $F$, its length remains unchanged. The coefficient of volume expansion, of the material of the rod, is (nearly) equal to :

1. $\frac{9F}{\left(\pi r^{2}YT\right)}$
2. $\frac{F}{\left(3\pi r^{2}YT\right)}$
3. $\frac{3F}{\left(\pi r^{2}YT\right)}$
4. $\frac{6F}{\left(\pi r^{2}YT\right)}$

Q. Two metal rods of same length and same area of cross section are fixed, end to end between rigid supports. Young’s modulus of elasticity of two metal rods are $Y_1\text{and}{Y}_2$, and coefficient of linear expansion are ${\alpha }_1\text{and}{\alpha }_2$. When the rods are cooled the junction does not change its position, if

1. $\frac{Y_1}{Y_2}=\frac{{\alpha }_1}{{\alpha }_2}$
2. $\frac{Y_1}{Y_2}=\frac{{\alpha }_2^2}{{\alpha }_1^2}$
3. $\frac{Y_1}{Y_2}=\sqrt{\frac{{\alpha }_2}{{\alpha }_1}}$
4. $Y_1{\alpha }_1=Y_2{\alpha }_2$

Q. A spherical shell of radius $10cm$ is carrying a charge $q$. If the electric potential at distances $5cm,20cm$ and $15cm$ from the centre of the spherical shell is $V_1,V_2$ and $V_3$ respectively, then :

1. $V_1=V_2>V_3$
2. $V_1>V_2>V_3$
3. $V_1=V_2<V_3$
4. $V_2<V_3<V_1$

Q. Two rods of different materials having coefficients of linear expansion ${\alpha }_1$ and ${\alpha }_2$ and Young's moduli, $Y_1$ and $Y_2$, 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 rods. If ${\alpha }_1/{\alpha }_2=2/3$, then the thermal stresses developed in the two rods are equal, provided $Y_1/Y_2$ is equal to

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

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

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

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

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

Q. Two rods of different materials having coefficients of thermal expansion ${\alpha }_1,:{\alpha }_2$ and Young's moduli $Y_1,Y_2$ 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 ${\alpha }_1:{\alpha }_2=2:3,$ the thermal stresses developed in the two rods are equal provided $Y_1:Y_2$ is equal to

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

Q. Two metal rods of same length and same area of cross section are fixed, end to end between rigid supports. Young's modulus of elasticity of two metal rods are $Y_{1}and{Y}_2$, and coefficient of linear expansion are ${\alpha }_{l}and{\alpha }_2$. When the rods are cooled the junction does not change its position, if

1. $\frac{Y_1}{Y_2}=\sqrt{\frac{{\alpha }_1}{{\alpha }_2}}$
2. $\frac{Y_1}{Y_2}=\frac{{\alpha }_1}{{\alpha }_2}$
3. $\frac{Y_1}{Y_2}=\frac{{\alpha }_2^2}{{\alpha }_1^2}$
4. $Y_{1\alpha 1}=Y_{2\alpha 2}$

Q. Two rods identical in geometry but of different materials having coefficient of thermal expansion ${\alpha }_{1}and{\alpha }_2$ and Youngs moduli $Y_{1}and{Y}_2$ 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 : 6, the thermal stresses developed in the two rods are equal provided $Y_1:Y_2$ is equal to

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

Q. Coefficients of linear expansions of two metals are in the ratio 3:4. The ratio of initial lengths of rods so that the expansions may be equal when heated through the same range of temperature is

1. 3:4
2. 4:3
3. 1:1
4. 4:1

Q. One mole of a monatomic ideal gas undergoes a cyclic process as shown in the figure (where $V$ is the volume and $T$ is the temperature). Which of the statements below is (are) true?

1. Process I is an isochoric process
2. In process II, gas absorbs heat
3. In process IV, gas releases heat
4. Processes I and II are not isobaric

Q. Two identical vessels are filled with equal amount of ice. The vessels are made from different materials. If the ice melts in the two vessels in times $t_1$ and $t_2$ respectively then their thermal conductivities are in the ratio:

1. $t_2:t_1$
2. $t_2^2:t_1^2$
3. $t_1:t_2$
4. $t_1^2:t_2^2$

Q. An equilateral triangle $ABC$ is formed by joining three rods of equal length and $D$ is the mid point of $AB$. The coefficient of linear expansion for $AB$ is ${\alpha }_1$and for $BC$ is ${\alpha }_2$. The relation between ${\alpha }_1$and ${\alpha }_2$, if distance $DC$ remains constant for small changes in temperature is:

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

Q. One end of the chain falls through a hole in its support and pulls the remaining links after it in a steady flow it the links which are initially at rest acquire the velocity of the chain suddenly and without frictional resistance or interference from the support or from adjacent links Choose the incorrect statement (when x = o, then v = 0) (length of the chain is L and p is the mass per unit length of the chain)

1. the acceferation of a of the falling chain as a function of x is $\frac{g}{3}$
2. the velocity v of the chain as a function of x is $\sqrt{\frac{2gx}{3}}$
3. the energy Q lost from the system as the last link leaves the plafform is $\frac{pg{L}^2}{6}$
4. tension at the middle point of falling chain is $\frac{pgx}{3}$

Q.

An isosceles triangle is formed with a rod of length $l_1$ and coefficient of linear expansion ${\alpha }_1$ for the base and two thin rods each of length $l_2$ and coefficient of linear expansion ${\alpha }_2$ for the two pieces, if the distance between the apex and the midpoint of the base remain unchanged as the temperatures varied, we get $\frac{l_1}{l_2}=x\sqrt{\frac{{\alpha }_2}{{\alpha }_1}}$. Find '$x$'.

Q. Explain the work done in an isothermal process.

Q. statement (1): To have the same difference between the lengths of two metallic rods their initial lengths of 0C should be in the inverse ratio of their coefficient of linear expansion.
statement (2): If the lengths of two metallic rods at 0 Care in the inverse ratio of their coefficient of linear expansion then the change in the lengths due to the same rise of temperature is the same.

1. statement 1 and statement 2 are correct and statement 2is correct explanation for statement 1
2. statement 1 and statement 2 are correct and statement 2 is not correct explanation for statement 1
3. statement 1 is true and statement 2 is false
4. statement 1 is wrong and statement 2 is true.

Q. $\frac{1}{1.2}+\frac{1}{2.3}+.........+\frac{1}{n(n+1)}=?$

1. $\frac{2n}{n+1}$
2. $\frac{n}{2n+1}$
3. $\frac{n}{n+1}$
4. None of these

Q. Two rods of different materials having coefficients of thermal expansion ${\alpha }_1,{\alpha }_2$ and Young's modulus $Y_1,Y_2$ 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 ${\alpha }_1:{\alpha }_2=2:3$, the thermal stresses developed in the two rods are equal provided $Y_1:Y_2$ is equal to.

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

Q. Figure shows two rods $A$ and $B$ of same length $L$ and same cross-sectional area $S$ but of different material having coefficient of linear expansion ${\alpha }_1$ and ${\alpha }_2$ respectively. They are clamped between two rigid walls, separated by a distance $2L$. This all refers to temp $t^{\circ }\text{C}$. Find the tension in each rod at temp $2t^{\circ }\text{C}$ (Take the young’s modulus for the two rods to be $Y_1$ and $Y_2$ respectively).

1. $\frac{S{Y}_1{Y}_2(2{\alpha }_1+{\alpha }_2)t}{(2Y_1+Y_2)}$
2. $\frac{S{Y}_1{Y}_2({\alpha }_1+2{\alpha }_2)t}{(Y_1+2Y_2)}$
3. $\frac{S{Y}_1{Y}_2({\alpha }_1+{\alpha }_2)t}{(Y_1+Y_2)}$
4. $\frac{S{Y}_1{Y}_2({\alpha }_1+2{\alpha }_2)t}{(Y_1+Y_2)}$

Q. At ${40}^{\circ }C$, a brass rod has a length $50\text{cm}$ and a diameter $3.0\text{mm}$, is joined in series to a steel rod of the same length and diameter at the same temperature. What is the change in the length of the composite rod close to (in $\text{mm}$), when it is heated to ${240}^{\circ }C$ ? (Round off to the nearest integer value)
The coefficients of linear expansion of brass and steel are $2.0\times {10}^{-5}{}^{\circ }{C}^{-1}$ and $1.2\times {10}^{-5}{}^{\circ }{C}^{-1}$ respectively:

Q. Two rods of different materials having coefficients of thermal expansion ${\alpha }_1$ and ${\alpha }_2$ and Youngs
moduli $Y_1$ and $Y_2$ 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 ${\alpha }_1$ and ${\alpha }_2$ are
in the ratio 2:3, the thermal stresses developed in the two rods are equal provided $Y_1:Y_2$ is equal to:

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