Question

Two spheres of radii in the ratio $1 : 2$ and densities in the ratio $2 : 1 $ and of same specific heat, are heated to same temperature and left in the same surrounding. Their rate of cooling will be in the ratio :

• A. 2 : 1
• B. 1 : 1
• C. 1 : 2
• D. 1 : 4

Answer 1

Answer: (B)

The formula for rate of cooling is given by $=\frac{m c \theta}{t}$
As, mass = volume $\times$ density Mass of sphere
$=\frac{4}{3} \pi r^{2} \times \rho$,
where $\rho$ is density Mass per unit area
$=\frac{4}{3} \pi r^{2}=\frac{\frac{4}{3} \pi r^{3} \times \rho}{4 \pi r^{2}}=\frac{1}{3} r \rho$
Hence, rate of cooling per unit area must be proportional to $r \rho$.
here $r$ is the radius of sphere and $p$ is the density.
Hence, ratio of rate of cooling for two spheres Is
$=\frac{r_{1} \rho_{1}}{r_{2} \rho_{2}}$
where $r_{1}: r_{2}=1: 2$
and $\rho_{1}: \rho_{2}=2: 1$
$=\frac{1}{2} \times \frac{2}{1}=1: 1$