Q: Two metallic spheres S_{1} and S_{2} are made of the same material and have got identical surface finish. The mass of S_{1} is thrice 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 S_{2} is

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

(b) $ \frac{1}{\sqrt3} $

(c) $ \frac{\sqrt3}{1} $

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

Ans: (d)

Sol: Let mass of S_{2} = m

Hence , mass of S_{1} = 3m

$ \displaystyle ms\frac{dT}{dt} = e\sigma A (T^4 – T_0^4 ) $

Rate of colling R = dT/dt

$ \displaystyle R = \frac{e\sigma A}{ms} (T^4 – T_0^4 ) $

$ \displaystyle m_1 = \frac{4}{3}\pi r_1^3 .\rho \, m_2 = \frac{4}{3}\pi r_2^3 .\rho $

$ \displaystyle \frac{m_2}{m_1} = (\frac{r_2}{r_1})^3 $

$ \displaystyle \frac{r_2}{r_1} = (\frac{m_2}{m_1})^{1/3} = (\frac{1}{3})^{1/3} $

$ \displaystyle \frac{R_1}{R_2} = \frac{r_1^2}{r_2^2}.\frac{m_2}{m_1} $

$ \displaystyle \frac{R_1}{R_2} =(\frac{1}{3})^{1/3} $