1.A body cools from $80^\circ\text{C}$ to $60^\circ\text{C}$ in $5\text{ minutes}$ in surroundings at $20^\circ\text{C}$. Find time taken to cool from $60^\circ\text{C}$ to $40^\circ\text{C}$.
Sol: Case 1: $\frac{80-60}{5} = K(70-20) \implies \frac{20}{5} = 50 K \implies K = \frac{4}{50} = 0.08$. Case 2: $\frac{60-40}{t} = K(50-20) \implies \frac{20}{t} = 0.08 \times 30 = 2.4 \implies t = \frac{20}{2.4} = 8.33\text{ minutes}$.
2.Two rods of equal length $L$ and area $A$ with thermal conductivities $K_1$ and $K_2$ are connected in series. Derive the effective thermal conductivity $K_{eq}$.
Sol: Thermal resistance in series: $R_{eq} = R_1 + R_2 \implies \frac{2L}{K_{eq} A} = \frac{L}{K_1 A} + \frac{L}{K_2 A} \implies K_{eq} = \frac{2 K_1 K_2}{K_1 + K_2}$.
