Question

Two metallic sphere $A$ and $B$ are made of same material and have got identical surface finish. The mass of sphere $A$ is four times that of $B$. Both the spheres are heated to the same temperature$\left(T\right)$ and placed in a room having temperature$\left(T_0\right)$ but thermally insulated from each other. Then,
(Assume that $T>T_0$)

• A. The ratio of heat loss of $A$ to that of $B$ is $2^{\frac{2}{5}}$
• B. The ratio of heat loss of $A$ to that of $B$ is $2^{\frac{2}{3}}$
• C. The ratio of the initial rate of cooling of $A$ to that of $B$ is $2^{-\frac{2}{3}}$
• D. The ratio of the initial rate of cooling of $A$ to that of $B$ is $2^{\frac{4}{3}}$

Answer 1

The correct option is C The ratio of the initial rate of cooling of $A$ to that of $B$ is $2^{-\frac{2}{3}}$

Given that
$m_A=4m_B,\rho \times \frac{4}{3}\pi r_A^3=\rho \times \frac{4}{3}\pi r_B^3\times 4$
$\Rightarrow \frac{r_A}{r_B}=4^{\frac{1}{3}}=2^{\frac{2}{3}}$
Rate of heat loss $=\frac{dQ}{dt}=eA\sigma (T^4-T_0^4)$
Ratio $\frac{{\left(\frac{dQ}{dt}\right)}_A}{{\left(\frac{dQ}{dt}\right)}_B}=\frac{A_A}{A_B}={\left(\frac{r_A}{r_B}\right)}^2=2^{\frac{4}{3}}$
Rate of cooling $\frac{-dT}{dt}=\frac{\frac{dQ}{dt}}{ms}$
Ratio $\frac{{\left(\frac{-dT}{dt}\right)}_A}{{\left(\frac{-dT}{dt}\right)}_B}=\frac{{\left(\frac{dQ}{dt}\right)}_A}{{\left(\frac{dQ}{dt}\right)}_B}\times \frac{m_B}{m_A}$
$=2^{\frac{4}{3}}\times \frac{1}{4}=2^{-\frac{2}{3}}$