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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 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 of ${S_2}$ is
$1/3$
${(1/3)^{1/3}}$
$1/\sqrt 3 $
$\sqrt 3 /1$
Solution
(b) Rate of cooling $(R) = \frac{{\Delta \theta }}{t} = \frac{{A \in \sigma ({T^4} – T_0^4)}}{{mc}}$
==> $R \propto \frac{A}{m} \propto \frac{{{\rm{Area}}}}{{{\rm{volume}}}} \propto \frac{{{r^2}}}{{{r^3}}} \propto \frac{1}{r}$
==> Rate $(R) \propto \frac{1}{r} \propto \frac{1}{{{m^{1/3}}}}$ $\left[ {\because \,\;m = \rho \times \frac{4}{3}\pi {r^3} \Rightarrow r \propto {m^{1/3}}} \right]$
==> $\frac{{{R_1}}}{{{R_2}}} = {\left( {\frac{{{m_2}}}{{{m_1}}}} \right)^{1/3}} = {\left( {\frac{1}{3}} \right)^{1/3}}$
