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 $=\frac{mc\theta }{t}$
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 ${\rho }_1:{\rho }_2=2:1$
$=\frac{1}{2}\times \frac{2}{1}=1:1$