Question

A solid sphere and a hollow sphere of same material and size are heated to some temperature and allowed to cool in the same surroundings.If the temperature difference between each sphere and its surrounding is T, then:-

• A. The hollow sphere will cool at a faster rate for all values of T
• B. The solid sphere will cool at a faster rate for all values of T
• C. Both spheres will cool at the same rate for all values of T
• D. Both spheres will cool at the same rate only for small values of T

Answer 1

Answer: (C)

Both spheres will cool at the same rate for all values of T

We know that rate of change of heat is given by,

$\frac{dQ}{dt}=ms\frac{dT}{dt}\Rightarrow \frac{dT}{dt}=\frac{1}{ms}\frac{dQ}{dt}$

In this case the external radius of both the sphere is same.

The surface areas of both spheres are same.

If the two spheres are heated to the same temperature and allowed to cool in the same environment, then the rate of loss of heat $\frac{dQ}{dt}$ of both the sphere is same.

Therefore the rate of cooling $\frac{dT}{dt}$ is inversely propotional to mass of the sphere. The mass of the hollow sphere is less than that of solid sphere. Therefore the rate of cooling of hollowsphere is faster than that of solid sphere.