Question

A small sphere of radius r falls from rest in a viscous medium of coefficient of viscosity η. Heat is produced in sphere due to the viscous force of medium. The rate of heat produced in sphere when it has accquired terminal velocity is proportional to

(1) r³

(2) r⁴

(3) r⁵

(4) r⁶

Answer 1

(3) r⁵

Let v_T be the terminal velocity accquired by the sphere. We know v_T α r² . The viscous force, F, on sphere when it has accquired terminal velocity is

The rate of heat produced is rate of work done by sphere against viscous froce = F.vT . Since F is proportional to r³ and v_T is proportional to r₂ rate of heat produced is directly proportional to r⁵ .