Two small spheres of radii $R$ and $2R$ fall from rest in the same viscous liquid. When they attain

Question

Two small spheres of radii $R$ and $2R$ fall from rest in the same viscous liquid. When they attain their respective terminal velocities, the ratio of the rates of heat produced due to viscous force for the two spheres ($P_R / P_{2R}$) is

Professor PC Thomas Classes · Answer

(b) : The viscous drag force is $F = 6 \pi \eta r v$, where $v$ is the terminal velocity. The rate of production of heat (power) is $P = F 	imes v = 6 \pi \eta r v^2$. Since terminal velocity $v \propto r^2$, substituting this gives $P \propto r(r^2)^2 = r^5$. For two spheres of radii $R$ and $2R$, the ratio of heat production rates is $\frac{P_R}{P_{2R}} = \left(\frac{R}{2R}ight)^5 = \frac{1}{32}$.