Question

Two spheres P and Q of equal radii have densities ${\rho }_1$ and ${\rho }_2$ repectively. The spheres are connected by a massless string and placed in liquids $L_1$ and $L_2$ of densities ${\sigma }_1$ and ${\sigma }_2$ and viscosities ${\eta }_1$ and ${\eta }_2$ respectively. They float in equilibrium with the sphere P in $L_1$ and Q in $L_2$ and the string being taut (see figure). If sphere P alone in $L_2$ has terminal velocity ${\overrightarrow{V}}_P$ and Q alone in $L_1$ has terminal velocity ${\overrightarrow{V}}_Q$,
(Terminal velocity of a sphere having radius $r$ and density $\rho$ thrown in a liquid having density $\sigma$ and viscoscity $\eta$ is given by $=\frac{2r^2}{9\eta }(\rho -\sigma )$). Then which of the following are correct ?

• A. $\frac{\mid {\overrightarrow{V}}_P\mid }{\mid {\overrightarrow{V}}_Q\mid }=\frac{{\eta }_1}{{\eta }_2}$
• B. $\frac{\mid {\overrightarrow{V}}_P\mid }{\mid {\overrightarrow{V}}_Q\mid }=\frac{{\eta }_2}{{\eta }_1}$
• C. ${\overrightarrow{V}}_P.{\overrightarrow{V}}_Q>0$
• D. ${\overrightarrow{V}}_P.{\overrightarrow{V}}_Q<0$

Answer 1

Answer: (A), (D)

The correct options are
A $\frac{\mid {\overrightarrow{V}}_P\mid }{\mid {\overrightarrow{V}}_Q\mid }=\frac{{\eta }_1}{{\eta }_2}$
D ${\overrightarrow{V}}_P.{\overrightarrow{V}}_Q<0$

Free body diagram of P and Q when they are connected by string

For P: ${\sigma }_{1}V={\rho }_{1}V+T...\left(i\right)$
For Q: $T+{\sigma }_{2}V={\rho }_{2}V...\left(ii\right)$
Adding equations $\left(i\right)$ and $\left(ii\right)$
${\sigma }_1-{\rho }_2={\rho }_1-{\sigma }_2..\left(iii\right)$
If the sphere P alone in $L_2$
Terminal velocity of P, ${\overrightarrow{V}}_P=\frac{2r^2}{9{\eta }_2}({\rho }_1-{\sigma }_2)$


If the sphere Q alone in $L_1$
Terminal velocity of Q, ${\overrightarrow{V}}_Q=\frac{2r^2}{9{\eta }_1}({\rho }_2-{\sigma }_1)$

$\frac{{\overrightarrow{V}}_P}{{\overrightarrow{V}}_Q}=\frac{-{\eta }_1}{{\eta }_2}$
$\mid \frac{{\overrightarrow{V}}_P}{{\overrightarrow{V}}_Q}\mid =\frac{{\eta }_1}{{\eta }_2}$ and ${\overrightarrow{V}}_P.{\overrightarrow{V}}_Q<0$
as if ${\rho }_1-{\sigma }_2>0$
${\rho }_2-{\sigma }_1<0$