Physics - Fluid Question with Solution | TestHub

When the velocity of the ball becomes constant, it has achieved terminal velocity. At this stage, the net force acting on the ball is zero.

The downward force is the weight of the ball, $W=mg$.

The upward forces are the buoyant force $F_B$ and the viscous force $F_v$.

$W=F_B+F_v$

$F_v=W-F_B$

The buoyant force is given by $F_B=V\sigma g$, where $V$ is the volume of the ball and $\sigma$ is the density of glycerin.

The mass of the ball is $m=V\rho$, where $\rho$ is the density of steel. Thus, $V=\frac{m}{\rho }$.

Substituting $V$ in the expression for $F_B$:

$F_B=\left(\frac{m}{\rho }\right)\sigma g=mg\left(\frac{\sigma }{\rho }\right)$

The viscous force is:

$F_v=mg-mg\left(\frac{\sigma }{\rho }\right)=mg\left(1-\frac{\sigma }{\rho }\right)$

Given $\rho =7.85$ g/cm${}^3$ and $\sigma =1.26$ g/cm${}^3$:

$F_v=mg\left(1-\frac{1.26}{7.85}\right)$

$F_v=mg\left(\frac{7.85-1.26}{7.85}\right)$

$F_v=mg\left(\frac{6.59}{7.85}\right)=mg\left(\frac{659}{785}\right)$

Answer: $mg\left(\frac{659}{785}\right)$