Thermal Stress

Q. The temperature of a wire of length 1 m and area of cross section
$1c{m}^2$ is increased from $0^{\circ }C$ to ${100}^{\circ }C$. If the length of rod is not allowed to increase, then force developed will be $(\alpha ={10}^{-5}/CandY={10}^{11}\frac{N}{m^2})$

1. ${10}^3$ N
2. ${10}^4$ N
3. ${10}^5$ N
4. ${10}^9$ N

Q. A steel rod of length $5\text{m}$ and diameter $30\text{mm}$ is fixed between two rigid supports. Determine the thermal stress and change in diameter of the rod, when the temperature increases by $50{}^{\circ }\text{C}$. Take Young's modulus of elasticity $\left(Y\right)=2.0\times {10}^6{\text{kg/cm}}^2$, coefficient of thermal expansion $\left(\alpha \right)=12\times {10}^{-6}{/}^{\circ }\text{C}$ and Poisson's ratio $\left(\nu \right)=0.3$.

1. $120{\text{kg/cm}}^2,0.0234\text{mm}$
2. $1200{\text{kg/cm}}^2,0.018\text{mm}$
3. $12000{\text{kg/cm}}^2,0.0054\text{mm}$
4. $1200{\text{kg/cm}}^2,0.0234\text{mm}$

Q. There is some change in length when a $33000\text{N}$ tensile force is applied on a steel rod of area of cross section ${10}^{-3}{\text{m}}^2$. The change of temperature required to produce the same elongation, if steel rod is heated, is: (The modulus of elasticity is $3\times {10}^{11}{\text{N/m}}^2$ and the coefficient of linear expansion of steel is $1.1\times {{10}^{-5}}^{\circ }{C}^{-1}$)

1. ${10}^{\circ }C$
2. $1^{\circ }C$
3. ${20}^{\circ }C$
4. ${30}^{\circ }C$

Q. A wire of length $L_0$ is supplied heat to to raise its temperature by T. If $\gamma$ is the coefficient of volume ex the wire and Y is the young's modulus of the wire then the energy density stored in the wire is

1. $\frac{1}{2}{\gamma }^2{T}^{2}Y$
2. $\frac{1}{3}{\gamma }^2{T}^2{Y}^3$
3. $\frac{1}{18}\frac{{\gamma }^2{T}^2}{Y}$
4. $\frac{1}{18}{\gamma }^2{T}^{2}Y$

Q.

In the given figure, a rod is free at one end and other and is fixed. When we change the temperature of rod by $\Delta$T, then strain produced in the rod will be

1. $\alpha \Delta T$

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

3. zero
4. information incomplete

Q. A steel bar $2.5\text{cm}$ in diameter is rigidly attached to two parallel supports which are $5\text{m}$ apart. Find the stress and the change in the diameter of the bar when the temperature is increased by ${100}^{\circ }C$.
Take $\alpha =12\times {10}^{-6}/{}^{\circ }C,Y=210\text{GPa}$.

1. $252\text{MPa}$, $0.03\text{mm}$
2. $150\text{MPa}$, $0.13\text{mm}$
3. $59\text{MPa}$, $0.33\text{mm}$
4. $212\text{MPa}$, $0.13\text{mm}$

Q. An iron rod of length $1\text{m}$ is fixed between two walls as shown in figure. Find thermal stress and thermal strain developed in the rod if it is heated by $\Delta T={20}^{\circ }C$
(Given: Coefficient of linear expansion of iron is $1.2\times {10}^{-5}{}^{\circ }{C}^{-1}$; Young's modulus of iron is $2\times {10}^{11}{\text{N/m}}^2$).

1. $24\times {10}^7{\text{N/m}}^2,48\times {10}^{-5}$
2. $48\times {10}^6{\text{N/m}}^2,32\times {10}^{-4}$
3. $32\times {10}^7{\text{N/m}}^2,48\times {10}^{-4}$
4. $48\times {10}^6{\text{N/m}}^2,24\times {10}^{-5}$

Q. Two rods of lengths $L_1$ and $L_2$ are made of materials whose coefficients of linear expansion are ${\alpha }_1$ and ${\alpha }_2$ respectively. If the difference between the two lengths is independent of temperature, then choose the correct option.

1. $\left(L_1/L_2\right)=\left({\alpha }_1/{\alpha }_2\right)$
2. $\left(L_1/L_2\right)=\left({\alpha }_2/{\alpha }_1\right)$
3. $L_1^2{\alpha }_1=L_2^2{\alpha }_2$
4. ${\alpha }_1^2{L}_1={\alpha }_2^2{L}_2$