Question

A spherical ball falls through viscous medium with terminal velocity $v$. If this ball is replaced by another ball of the same mass but half the radius, then the terminal velocity will be (neglect the effect of buoyancy)

• A. $v$
• B. $2v$
• C. $4v$
• D. $8v$

Answer 1

The correct option is B $2v$

In case of spherical ball

$m=\frac{4}{3}\pi r^3\rho$

keeping $m$ constant, if $r$ is halved, $\rho$ will increased by a factor of $8$;

If buoyancy is neglected,

$V_T\propto r^2\rho ;\text{otherwise}{V}_T\propto r^2(\rho -{\rho }_o)\Rightarrow \frac{V_T^{\prime }}{V_T}=\frac{{r^{\prime }}^2{\rho }^{\prime }}{r^2\rho }\Rightarrow \frac{V_T^{\prime }}{V_T}=\frac{{\left(\frac{r}{2}\right)}^{2}8\rho }{r^2\rho }\Rightarrow \frac{V_T^{\prime }}{V_T}=\frac{2r^2\rho }{r^2\rho }\Rightarrow \frac{V_T^{\prime }}{V_T}=2\Rightarrow V_T^{\prime }=2V_T\Rightarrow V_T^{\prime }=2v$