1.A black sphere of radius $R$ at temperature $T$ radiates energy in vacuum. Find time taken for its temperature to fall from $T_1$ to $T_2$ if heat capacity is $C$.
Sol: $-C \frac{dT}{dt} = 4\pi R^2 \sigma T^4 \implies \int_{T_1}^{T_2} \frac{dT}{T^4} = -\frac{4\pi R^2 \sigma}{C} t \implies t = \frac{C}{12\pi R^2 \sigma} \left(\frac{1}{T_2^3} - \frac{1}{T_1^3}\right)$.
2.A metal rod of length $L_0$ and linear expansion coefficient $\alpha$ is clamped rigidly between two rigid supports. If temperature increases by $\Delta T$, find thermal stress developed.
Sol: Strain prevented $\epsilon = \frac{\Delta L}{L_0} = \alpha \Delta T$. Thermal stress = $Y \epsilon = Y \alpha \Delta T$.
