A small sphere of radius‘r’ falls from rest in a viscous liquid. As a result, heat is produced due to viscous force. The rate of production of heat when the sphere attains its terminal velocity, is proportional to

The viscous drag force, $F=6\pi \eta rv$

where v= terminal velocity

$\therefore$The rate of production of heat = power 

                                                                    = force $\times$ terminal velocity

$\Rightarrow \text{ Power }=6\pi \eta rv·v=6\pi \eta r{v}^2$

$\because \text{ Terminal velocity }v=\frac{2r^2(\rho -\sigma )g}{9\eta };$

$\therefore v\propto r^2$

$\text{ Now, power }=6\pi \eta r\left[\frac{4r^4(\rho -\sigma {)}^2}{81{\eta }^2}{g}^2\right]\text{ or } \text{ Power }\propto r^5\text{ . }$