Question

Rain drops fall from a certain height with a terminal velocity $v$ on the ground. The viscous force, $F =6 \pi \eta rv$ where, $\eta$ is coefficient of viscosity, $r$ is radius of rain drop and $v$ is the speed, then work done by all the forces acting on the rain drop till it reaches the ground is proportional to $r^{n}$. where $n=$_________.

Answer 1

As rain drops reach the ground with terminal velocity,
$\therefore F = mg$
$6 \pi \eta rv =\left(\frac{4}{3} \pi^{3} \rho\right) g$
$\therefore v \propto r ^{2}$
From work energy theorem, work done by all the forces $=$ change in kinetic energy of rain drop
$\therefore W =\frac{1}{2} mv ^{2}$
$\therefore W \propto\left(\frac{4}{3} \pi^{3} \rho\right) v ^{2}$
$\therefore W \propto r ^{3}\left( r ^{2}\right)^{2}$
$\therefore W \propto r ^{7}$
$\Rightarrow n =7$