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

Answer: 7

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
$\therefore W=\frac{1}{2}m{v}^2$
$\therefore W\propto \left(\frac{4}{3}{\pi }^3\rho \right)v^2$
$\therefore W\propto r^3{\left(r^2\right)}^2$
$\therefore W\propto r^7$
$\Rightarrow n=7$