Question

A small ball of mass $m$ and density $\rho$ is dropped in a viscous liquid of density $\rho_0$. After some time, the ball falls with constant velocity. The viscous force on the ball is:

• A. $mg \left( \frac{\rho_0}{\rho} - 1 \right)$
• B. $mg \left( 1 + \frac{\rho}{\rho_0} \right)$
• C. $mg (1 - \rho \rho_0)$
• D. $mg \left( 1 - \frac{\rho_0}{\rho} \right)$

Answer 1

The Correct Option is D

Applying force balance on the ball at constant velocity:

\[ mg - F_B - F_v = ma \]

Since acceleration $a = 0$ for constant velocity:

\[ \Rightarrow mg - F_B = F_v \]

The buoyant force is given by:

\[ F_B = v \rho_0 g \quad \text{where } v \text{ is the volume of the ball.} \]

Force balance equation becomes:

\[ F_v = mg - v \rho_0 g \]

Substituting $v = \frac{m}{\rho}$ (volume in terms of mass and density):

\[ \Rightarrow F_v = mg - \frac{m}{\rho} \rho_0 g \]

\[ F_v = mg \left( 1 - \frac{\rho_0}{\rho} \right) \]