Question

A small sized spherical ball of radius r, mass M is dropped in a viscous medium of coefficient of viscosity η. Neglect the bouyant force on ball. v_T is the terminal velocity acquired by

ball. t₁ is time taken by ball to acquire a speed v_T/4 from initial state. t₂ is time taken by ball to increase its speed from v_T/4 to v_T/2 . t₁/t₂ is

(1) In (8/9)

(2) \(\frac{In(\frac{4}{3})}{In(\frac{2}{3})}\)

(3) \(\frac{In(\frac{3}{4})}{In(\frac{2}{3})}\)

(4) In (9/8)

Answer 1

 (2) \(\frac{In(\frac{4}{3})}{In(\frac{2}{3})}\)

When ball accquires terminal velocity v_T ; net force on it (ball) is zero. In other words viscous force equals weight of ball, i.e.

6πrv_T = Mg ....(1)

Let ‘v’ and ‘a’ denote instantaneous speed and acceleration of ball. The equation of motion of ball is