A spherical ball of density $\rho$ is dropped from rest into a long column of a highly viscous liqui

Question

A spherical ball of density $\rho$ is dropped from rest into a long column of a highly viscous liquid of density $\sigma$ ($\rho > \sigma$). What does the initial acceleration of the ball at $t=0$ evaluate to?

Professor PC Thomas Classes · Answer

(b) : At $t=0$, the velocity of the ball is zero, so the viscous drag force ($F_v = 6\pi \eta r v$) is zero. The net force acting on the ball is the downward force of gravity minus the upward buoyant force: $F_{net} = mg - F_B = V\rho g - V\sigma g = V(\rho - \sigma)g$. Using Newton's second law, acceleration $a = \frac{F_{net}}{m} = \frac{V(\rho - \sigma)g}{V\rho} = g\left(1 - \frac{\sigma}{\rho}\right)$.