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 but are thermally insulated from each other. The ratio of the initial rate of cooling of S1 to S2 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 S2 = m
Hence , mass of S1 = 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} $