Question

A spherical ball, of mass m and radius r, starts from rest and falls vertically through a medium of negligible density and having a coefficient of viscosity η. The time (T) taken by the ball to acquire a velocity V, is given by

(1) \(T=\frac{6\piη r}{m}ln[\frac{mg}{mg-6\pi ηrV}]\)

(2) \(T=\frac{m}{6\piη r}ln[\frac{mg-6\pi ηrV}{mg}]\)

(3) \(T=\frac{6\piη r}{m}ln[\frac{mg-6\pi ηrV}{mg}]\)

(4) \(T=\frac{m}{6\piη r}ln[\frac{mg}{mg-6\pi ηrV}]\)

Answer 1

(4) \(T=\frac{m}{6\piη r}ln[\frac{mg}{mg-6\pi ηrV}]\)

Let the velocity of the ball be v after a time t. The net downward force, acting on to ball, at this instant, is

F (t) mg - 6πηrv

The acceleration, a(t), at this instant, is