Question

Two metallic spheres S₁ and S₂ are made of the same material and have identical surface finish. The mass of S₁ is three times that of S₂. Both the spheres are heated to the same high temperature and placed in the same room having lower temperature but are thermally insulated from each other. The ratio of the initial rate of cooling of S₁ to that of S₂ is

• A. 1/ 3
• B. (1/ 3)^{1/ 3}
• C. $1/\sqrt{3}$
• D. $\sqrt{3}/1$

Answer 1

Answer: (B)

(1/ 3)^{1/ 3}

Answer 2

$\frac{dT}{dt} = \frac{eA\sigma}{mc}(T^{4} - T_{0}^{4})$

∴ Rate of cooling $R \propto \frac{4\pi r^{2}}{\frac{4}{3}\pi r^{3} \times \rho} \propto \frac{1}{r}$ ⇒ $\frac{R_{1}}{R_{2}} = \frac{r_{2}}{r_{1}}$

But according to problem m₁ = 3m₂ ⇒

$\frac{4}{3}\pi r_{1}^{3} \times \rho = 3\left( \frac{4}{3}\pi r_{2}^{3} \times \rho \right)$ ⇒ $r_{1}^{3} = 3r_{2}^{3}$ ⇒ $\left( \frac{r_{2}}{r_{1}} \right) = \left( \frac{1}{3} \right)^{1/3}$∴ Ratio of rate of cooling $\frac{R_{1}}{R_{2}} = \left( \frac{1}{3} \right)^{1/3}$.