Question

Two spheres $\text{P}$ and $\text{Q}$ of equal radii have densities ${\rho }_1$ and ${\rho }_2$, respectively. The spheres are connected by a massless string and placed in liquids ${\text{L}}_1$ and ${\text{L}}_2$ of densities ${\sigma }_1$ and ${\sigma }_2$ and viscosities ${\eta }_1$ and ${\eta }_2$, respectively. They float in equilibrium with the sphere $\text{P}$ in ${\text{L}}_1$ and sphere $\text{Q}$ in ${\text{L}}_2$ and the string being taut (see figure). If sphere $\text{P}$ alone in ${\text{L}}_2$ has terminal velocity ${\overrightarrow{v}}_P$ and $\text{Q}$ alone in ${\text{L}}_1$ has terminal velocity ${\overrightarrow{v}}_Q$, then

• A. $\frac{\left|{\overrightarrow{v}}_P\right|}{\left|{\overrightarrow{v}}_Q\right|}=\frac{{\eta }_1}{{\eta }_2}$
  
• B. $\frac{\left|{\overrightarrow{v}}_P\right|}{\left|{\overrightarrow{v}}_Q\right|}=\frac{{\eta }_2}{{\eta }_1}$
  
• C. ${\overrightarrow{v}}_P\cdot {\overrightarrow{v}}_Q>0$
  
• D. ${\overrightarrow{v}}_P\cdot {\overrightarrow{v}}_Q<0$
  

Answer 1

Answer: (B), (D)

${\overrightarrow{v}}_P\cdot {\overrightarrow{v}}_Q<0$



Consider a body of density ${\rho }_b$ kept in density ${\rho }_l$ whose viscosity is $\eta$ and terminal velocity $v$.

Then under the equilibrium, net force acting on the body must be zero.i.e.,

${\overrightarrow{F}}_{\text{viscous}}+{\overrightarrow{F}}_{mg}+{\overrightarrow{F}}_{\text{Buoyancy}}=0$

Substituting the values, from given problem,

${\overrightarrow{F}}_{\text{viscous}}+{\rho }_b\frac{4}{3}\pi R^{3}g(-\hat{j})+{\rho }_l\frac{4}{3}\pi R^{3}g\left(\hat{j}\right)=0$

Where,
${\rho }_b$ is the density of body,
${\rho }_l$ is the density of liquid.

$\therefore {\overrightarrow{F}}_{\text{viscous}}=({\rho }_b-{\rho }_l)\frac{4}{3}\pi R^{3}g\left(\hat{j}\right)....\left(1\right)$

Now, we know that viscous force on the body is,

$F_{\text{viscous}}=6\pi \eta Rv....\left(2\right)$

From equation $\left(1\right)$ and $\left(2\right)$ we have,

$6\pi \eta Rv=({\rho }_b-{\rho }_l)\frac{4}{3}\pi R^{3}g$

$\Rightarrow v=\frac{2({\rho }_b-{\rho }_l)\pi R^{2}g}{9\pi \eta }$

where,
$v$ is terminal velocity, $R$ is the radius of sphere.

Thus,
$v\propto \frac{{\rho }_b-{\rho }_l}{\eta }$

As per question, if sphere $\text{P}$ alone is in liquid $L_2$ has terminal velocity ${\overrightarrow{v}}_P$ and $\text{Q}$ alone in liquid $L_1$ has terminal velocity ${\overrightarrow{v}}_Q$, then

$\frac{{\overrightarrow{v}}_P}{{\overrightarrow{v}}_Q}=\frac{({\rho }_1-{\sigma }_2)/{\eta }_1}{({\rho }_2-{\sigma }_1)/{\eta }_2}....\left(1\right)$

As per given diagram we can say
${\sigma }_2>{\sigma }_1;{\rho }_1<{\sigma }_1$ and ${\rho }_2<{\sigma }_2$

$\Rightarrow {\rho }_2>{\sigma }_2>{\sigma }_1>{\rho }_1...\left(2\right)$

From equation (1) and (2), we can conclude that
${\overrightarrow{v}}_P$ is negative and ${\overrightarrow{v}}_Q$ is positive. So, we have
${\overrightarrow{v}}_P.{\overrightarrow{v}}_Q<0$

As the system is floating in equilibrium. Let $U_P$ and $U_Q$ are the upthrust on $\text{P}$ and $\text{Q}$ respectively.

From free body diagram:
$U_P-W_P=T=W_Q-U_Q$

$\Rightarrow U_P+U_Q=W_P+W_Q$

Substituting the values,
$\Rightarrow {\sigma }_{1}Vg+{\sigma }_{2}Vg={\rho }_{1}Vg+{\rho }_{2}Vg$

Here, $V$ is the volume of the spheres.

$\Rightarrow {\sigma }_1+{\sigma }_2={\rho }_1+{\rho }_2$

$\Rightarrow {\rho }_1-{\sigma }_2={\sigma }_1-{\rho }_2...\left(3\right)$

Now, from equation (2) and (3), we get

$\frac{{\overrightarrow{v}}_P}{{\overrightarrow{v}}_Q}=\frac{-({\rho }_2-{\sigma }_1)/{\eta }_1}{({\rho }_2-{\sigma }_1)/{\eta }_2}$

$\Rightarrow \frac{{\overrightarrow{v}}_P}{{\overrightarrow{v}}_Q}=\frac{-{\eta }_2}{{\eta }_1}$

$\therefore \frac{\left|{\overrightarrow{v}}_P\right|}{\left|{\overrightarrow{v}}_Q\right|}=\frac{{\eta }_2}{{\eta }_1}$

Hence, option (a) and (d) are the correct answers.

$\text{Note:}$ We can use the terminal velocity formula directly to solve this problem instead of deriving it. Here, derivation is done to make feel the concept of terminal velocity which is very useful even after if someone unable to recall the formula while solving the problem of floating bodies.