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

(a) $\large \frac{1}{3}$

(b) $\large \frac{1}{\sqrt{3}}$

(c) $\large \frac{\sqrt{3}}{1}$

(d) $\large (\frac{1}{3})^{1/3}$

Ans: (d)

Sol: $\large \frac{\Delta Q}{\Delta t} = e \sigma A T^4$

As , $\large \Delta Q = m s \Delta T$

$\large \frac{m s \Delta T}{\Delta t} = e \sigma A T^4$

$\large \frac{\Delta T}{\Delta t} = \frac{e \sigma A T^4}{m s} $

Since , $\large m = \frac{4}{3}\pi r^3 \rho$

$\large A = 4\pi r^2 = 4\pi(\frac{3m}{4\pi \rho})^{2/3}$

$\large \frac{\Delta T}{\Delta t} = \frac{e \sigma T^4}{m s}[4\pi(\frac{3m}{4\pi \rho})^{2/3}]$

$\large \frac{\Delta T}{\Delta t} = K (\frac{1}{m})^{1/3} $

For two bodies ,

$\large \frac{(\Delta T/\Delta t)_1}{(\Delta T/\Delta t)_2} = K (\frac{m_2}{m_1})^{1/3} = (\frac{1}{3})^{1/3}$