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 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. \(Y\alpha AL\Delta t\)

A rod of length 40 cm is stretched between two rigid supports at zero tension. If the temperature is increased by 10°C, the stress produced in it is:

(Young's modulus, Y = 1.2x${10}^{12}<mspace\; width="1em"></mspace>\mathrm{N}/{\mathrm{m}}^2$, coefficient of linear expansion of rod = 2 x ${10}^{-5}$/°C)

(1) 2.4 x ${10}^8<mspace\; width="1em"></mspace>\mathrm{N}/{\mathrm{m}}^2$

(2) 1.2 x ${10}^8<mspace\; width="1em"></mspace>\mathrm{N}/{\mathrm{m}}^2$

(3) 3.6 × ${10}^8<mspace\; width="1em"></mspace>\mathrm{N}/{\mathrm{m}}^2$

(4) 4.8 x ${10}^8<mspace\; width="1em"></mspace>\mathrm{N}/{\mathrm{m}}^2$

An aluminum rod of length 2m held fixed in a horizontal position from its one end has the other end free. When the temperature of rod increases by 100°C, the thermal stress produced in the rod will be ($\mathrm{\alpha }$ = 1.6 x ${10}^{-6}$/°C)

(1) Zero

(2) 0.08

(3) 0.1

(4) 0.4

The metal rod (Y = 2 x ${10}^{12}$ dyne/sq. cm) of the coefficient of linear expansion 1.6 x ${10}^{-5}$ per °C has its temperature raised by 20°C. The linear compressive stress to prevent the expansion of the rod is:

(1) 2.4 x ${10}^8$ dyne/sq. cm

(2) 3.2 x ${10}^8$ dyne/sq. cm

(3) 6.4 x ${10}^8$ dyne/sq. cm

(4) 1.6 x ${10}^8$ dyne/sq. cm