A metal bar of length \(L\) and area of cross-section \(A\) is clamped between two rigid supports. For the material of the rod, it's Young’s modulus is \(Y\) and the coefficient of linear expansion is \(\alpha.\) If the temperature of the rod is increased by \(\Delta t^{\circ} \text{C},\) the force exerted by the rod on the supports will be:
1. \(YAL\Delta t\)
2. \(YA\alpha\Delta t\)
3. \(\frac{YL\alpha\Delta t}{A}\)
4. \(Y\alpha AL\Delta t\)

The temperature of a wire of length \(1~\text{m}\) and an area of cross-section \(1~\text{cm}^2\) is increased from \(0^{\circ} \text {C}\) to \(100^{\circ} \text {C}.\) If the rod is not allowed to increase in length, the force required will be:
\((\alpha = 10^{-5}/ ^{\circ} \text {C} ~\text{and} ~Y = 10^{11} ~\text{N/m}^2)\)

A steel rod of length \(l,\) cross-sectional area \(A,\) Young’s modulus \(Y,\) and coefficient of linear expansion \(\alpha\) is rigidly fixed at both ends. It is heated through a temperature rise of \(t^{\circ}\text{C}\) so that expansion is fully prevented. If the rod is then allowed to perform work due to the stored elastic energy, the amount of work it can do is:
1. \((YA\alpha t)(l\alpha t)\)
2. \(\dfrac{1}{2}(YA\alpha t)(l\alpha t)\)
3. \(\dfrac{1}{2}(YA\alpha t) \times\dfrac{1}{2}(l\alpha t)\)
4. \(2(YA\alpha t)(l\alpha t)\)

A brass wire \(1.8~\text m\) long at \(27^\circ \text C\) is held taut with a little tension between two rigid supports. If the wire is cooled to a temperature of \(-39^\circ \text C,\) what is the tension created in the wire?
( Assume diameter of the wire to be \(2.0~\text{mm}\) , coefficient of linear expansion of brass \(=2.0 \times10^{-5}~\text{K}^{-1},\) Young's modulus of brass\(=0.91 \times10^{11}~\text{Pa}\) )
1. \(3.8 \times 10^3~\text N\) 
2. \(3.8 \times 10^2~\text N\) 
3. \(2.9 \times 10^{-2}~\text N\) 
4. \(2.9 \times 10^{2}~\text N\) 

The pressure applied from all directions on a cube is P. The volume elasticity of the cube is β and the coefficient of volume expansion is α. How much should its temperature be raised to maintain the original volume?

1. $\frac{P}{\alpha \beta }$                                 

2. $\frac{P\alpha }{\beta }$

3. $\frac{P\beta }{\alpha }$                                 

4. $\frac{\alpha \beta }{P}$