JEE Advanced · Physics · 11. Properties of Fluids

Two spheres $P$ and $Q$ of equal radii have densities $ρ_{1}$ and $ρ_{2}$ , respectively. The spheres are connected by a massless string and placed in liquids $L_{1}$ and $L_{2}$ of densities $σ_{1}$ and $σ_{2}$ and viscosities $η_{1}$ and $η_{2}$ , respectively. They float in equilibrium with the sphere $P$ in $L_{1}$ and sphere $Q$ in $L_{2}$ and the string being taut (see figure). If sphere $P$ alone in $L_{2}$ has terminal velocity ${\vec{V}}_{P}$ and $Q$ alone in $L_{1}$ has terminal velocity ${\vec{V}}_{Q}$ , then

A $\frac{|{\vec{V}}_{P}|}{|{\vec{V}}_{Q}|} = \frac{η_{1}}{η_{2}}$
B $\frac{|{\vec{V}}_{P}|}{|{\vec{V}}_{Q}|} = \frac{η_{2}}{η_{1}}$
C ${\vec{V}}_{P} . {\vec{V}}_{Q} > 0$
D ${\vec{V}}_{P} . {\vec{V}}_{Q} < 0$

Verified Solution

Answer & Solution

Correct Answer

(D) ${\vec{V}}_{P} . {\vec{V}}_{Q} < 0$

Step-by-step Solution

Detailed explanation

Consider a body of density

ρ_{b}

kept in density

ρ_{l}

whose viscosity is

η

and terminal velocity V. then

$\vec{F}_{\text {viscous }}+\vec{F}_{m g}+\vec{F}_{\text {Buoyancy }}=0$
$\vec{F}_{v i s c o u s}+\rho_b \frac{4}{3} \pi R^3(-\hat{j})+\rho_l \frac{4}{3} \pi R^3(\hat{j})=0$
$\therefore \vec{F}_{\text {viscous }}=\left(\rho_b-\rho_{\ell}\right) \frac{4}{3} \pi R^3(\hat{j}) \Rightarrow 6 \pi \eta R V=$ $\left(\rho_b-\rho_{\ell}\right) \frac{4}{3} \pi R^3$
$\therefore$ if $\rho_b>\rho_1$ then $\vec{F}_{\text {viscous }} \uparrow V \propto \frac{1}{\eta} \&$ if $\rho_b<\rho_{\ell}$
$\vec{F}_{\text {viscous }} \downarrow$
As per given diagram we can say

σ_{2} > σ_{1}; ρ_{1} < σ_{1} & σ_{2} > σ_{2}

\Rightarrow ρ_{2} > σ_{2} > σ_{1} > ρ_{1}

∴

if we put P in

L_{2}

where

|{\vec{V}}_{P}| \propto \frac{1}{η_{2}}

when

ρ_{1} < σ_{1}

∴ {\vec{F}}_{v i s c o u s} ↓

∴ {\vec{V}}_{P} ↑

∴

if we put Q in

L_{1}

where

|{\vec{V}}_{Q}| \propto \frac{1}{η_{1}}

when

ρ_{2} < σ_{1}

∴ {\vec{F}}_{v i s c o u s} ↑

∴ {\vec{V}}_{P} ↓

\Rightarrow \frac{|{\vec{V}}_{P}|}{|{\vec{V}}_{Q}|} = \frac{η_{1}}{η_{2}}

{\vec{V}}_{P} . {\vec{V}}_{Q} < 0

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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