Question

Two metallic spheres $S_{1}and{S}_2$ are made of the same material and have identical surface finish. The mass of $S_1$ is three times to that of $S_2$. 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 $S_1$ to that $S_2$ is :

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

Answer 1

Answer: (B)

$(\frac{1}{3}{)}^{\frac{1}{3}}$

By the Stefan-Boltzmann law, the rate of loss of heat energy is

$Q=eA\sigma (T^4-T_s^4)=e\sigma \times 4\pi r^2(T^4-T_s^4)$ where $T$ is the temperature of body and $T_s$ is the temperature of surrounding.

As rate of heat radiated $Q=msd\theta /dt$, where $m$ is the mass and $d\theta /dt$ is the rate of change of temperature and $s$ is the specific heat capacity.

Mass, $m=\frac{4}{3}\pi r^3\times \rho \Rightarrow r\propto m^{1/3}$.

Using these value in equation of rate of heat loss, $d\theta /dt\propto 1/r\propto 1/m^{1/3}$.

Now, as masses are in ratio $1:3$, the rate of change of temperature is in ratio, $(1:3{)}^{\frac{1}{3}}$