A small ball of mass $\mathrm{M}$ and density $\rho$ is dropped in a viscous liquid of density $\rho_{0}$. After some time, the ball falls with a constant velocity. What is the viscous force on the ball ?

<p>When the ball is falling with a constant velocity, it means the net force acting on the ball is zero. This is because it&#39;s in a state of dynamic equilibrium - the downward force equals the upward force.</p> <p>The downward force is the gravitational force (weight of the ball), which is given by $F_g = Mg$.</p> <p>The upward force is the sum of buoyant force and the viscous drag. The buoyant force is the weight of the fluid displaced by the ball, which is given by $F_b = Vg\rho_0 = Mg\rho_0/\rho$ where $V = M/\rho$ is the volume of the ball.</p> <p>The viscous force, $F_v$, is the force that we need to find.</p> <p>Since the net force is zero, we have:</p> <p>$F_g = F_b + F_v$</p> <p>or</p> <p>$Mg = Mg\rho_0/\rho + F_v$</p> <p>which simplifies to</p> <p>$F_v = Mg - Mg\rho_0/\rho = Mg(1 - \rho_0/\rho)$</p>