Question

A spherical ball of mass 4m, density $\sigma$ and radius r is attached to a pulley-mass system as shown in the figure. The ball is released in a liquid of coefficient of viscosity $\eta$ and density $\rho (<\frac{\sigma }{2})$. If the length of the liquid column is sufficiently long, the terminal velocity attained by the ball is given by (assume all pulleys to be mass-less and string as mass-less and inextensible) :

• A. $\frac{2}{9}\frac{r^2(2\sigma -\rho )g}{\eta }$
• B. $\frac{1}{9}\frac{r^2(\sigma -2\rho )g}{\eta }$
• C. $\frac{2}{9}\frac{r^2(\sigma -4\rho )g}{\eta }$
• D. $\frac{1}{9}\frac{r^2(\sigma -3\rho )g}{\eta }$

Answer 1

Answer: (B)

$\frac{1}{9}\frac{r^2(\sigma -2\rho )g}{\eta }$

From the free body diagram of the sphere :
$F_V=4mg-2mg-F_B$
$\Rightarrow F_V=2mg-F_B$
$6\pi \eta rV=\frac{4}{3}\pi r^3(\frac{\sigma }{2}-\rho )g$
$($since $4m=\frac{4}{3}\pi r^3\times \sigma )$
$\pi \eta V=\frac{2}{9}\pi r^2\left(\frac{\sigma -2\rho }{2}\right)g$
$\Rightarrow V=\frac{1}{9}\frac{r^2(\sigma -2\rho )g}{\eta }$