Question

Two metallic spheres $A$ and $B$ are made of the same material and have got an identical surface finish. The mass of sphere $A$ is four times that of $B$. Both the spheres are heated to the same temperature and placed in a room having lower temperature but thermally insulated from each other.

• A. The ratio of rate of heat loss of $A$ to that of $B$ is $2^{4/3}$.
• B. The ratio of heat loss of $A$ to that of $B$ is $2^{2/3}$.
• C. The ratio of the initial rate of cooling of $A$ to that of $B$ is $2^{-2/3}$
• D. The ratio of the initial rate of cooling of $A$ to that of $B$ is $2^{-4/3}$

Answer 1

Answer: (A), (C)

The correct options are
A The ratio of rate of heat loss of $A$ to that of $B$ is $2^{4/3}$.
C The ratio of the initial rate of cooling of $A$ to that of $B$ is $2^{-2/3}$

Heat loss, as per stephan boltzman law is given by the relation:

$q^{\prime }=e\sigma A{T}^4$

$q^{\prime }=e\sigma T^4\left(4\pi r^2\right)$

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

$r=k{m}^{1/3}$

Hence, $\frac{q_1^{{}^{\prime }}}{q_2^{{}^{\prime }}}=\frac{r_1^2}{r_2^2}={\left(\frac{m_1}{m_2}\right)}^{2/3}=4^{2/3}=2^{4/3}$

Also, rate of cooling is given by

$R=\frac{q^{\prime }}{mc}=\frac{A{q}^{\prime }}{m}$ where A is some constant.

Hence, $\frac{R_1}{R_2}={\left(\frac{m_2}{m_1}\right)}^{1/3}=2^{-2/3}$

Answer are options A and C.