Question

For a cube and sphere made of same material, what should be the ratio of side of the cube to the radius of the sphere should be for both of them to have same thermal capacity?

Answer 1

We know,

Thermal capacity=mass$\times$specific heat.

According to the question, the cube and sphere are of same material their specific heats are same.

To have same thermal capacity there masses should be same.

This gives,

mass of sphere = mass of cube

(density $\times$ volume) of sphere =(density $\times$ volume) of cube

There density will also be same.

Hence,

volume of sphere should be equal to volume of cube.

That means

$V_{sphere}=V_{cube}$

$\frac{4}{3}\pi r^3=a^3$, where a=side of cube, r= radius of the sphere

$\frac{a^3}{r^3}=\frac{4\pi }{3}⟹\frac{a}{r}={\left(\frac{4\pi }{3}\right)}^{\frac{1}{3}}$