A small sphere falls from rest in a viscous liquid. Due to friction, heat is produced. Find the relation between the rate of production of heat and the radius of the sphere at terminal velocity.

Two spheres $P$ and $Q$ of equal radii have densities ${\mathrm{\rho }}_1$ and ${\mathrm{\rho }}_2$, respectively. The spheres are connected by a massless string and placed in liquids $L_1$ and $L_2$ of densities ${\mathrm{\sigma }}_1$ and ${\mathrm{\sigma }}_2$ viscosities ${\mathrm{\eta }}_1$ and ${\mathrm{\eta }}_2$, respectively. They float in equilibrium with the sphere $P$ in $L_1$ and sphere $Q$ in $L_2$ and the string is being taut (see figure). If sphere $P$ alone in $L_2$ has terminal velocity ${\stackrel{\mathit{\rightharpoonup }}{\mathit{V}}}_{\mathit{P}}$ and $Q$ alone in $L_1$ has terminal velocity ${\stackrel{\mathit{\rightharpoonup }}{\mathit{V}}}_{\mathit{Q}}$, then