Two spheres P and Q of equal radii have densities ρ _ 1 and ρ _ 2 ,...

Consider a body of density ${ρ}_{b}$ kept in density ${ρ}_{l}$ whose viscosity is $η$ and terminal velocity V. then



${\overset{\to}{F}}_{viscous} + {\overset{\to}{F}}_{mg} + {\overset{\to}{F}}_{Buoyancy} = 0$
\overset{\to}{F}viscous + {ρ}_{b}\dfrac{4}{3}\pi{R}^{3}\left(- \overset{^}{j}\right) + {ρ}_{l}\dfrac{4}{3}\pi{R}^{3}\left(\overset{^}{j}\right) = 0
∴ \overset{\to}{F}viscous = \left({ρ}_{b} - {ρ}_{l}\right)\dfrac{4}{3}\pi{R}^{3}\left(\overset{^}{j}\right) ⇒ 6\piηRV = \left({ρ}_{b} - {ρ}_{l}\right)\dfrac{4}{3}\pi{R}^{3}
$∴$ if ${ρ}_{b} > {ρ}_{1}$ then ${\overset{\to}{F}}_{viscous} ↑ V ∝ \dfrac{1}{η}$ & if ${ρ}_{b} < {ρ}_{l}$
${\overset{\to}{F}}_{viscous} ↓$
As per given diagram we can say
${\sigma}_{2} > {\sigma}_{1}; {ρ}_{1} {\sigma}_{2}$
$⇒ {ρ}_{2} > {\sigma}_{2} > {\sigma}_{1} > {ρ}_{1}$
$∴$ if we put P in ${L}_{2}$ where \left|{\overset{\to}{V}}_{P}\right| ∝ \dfrac{1}{{η}_{2}}$when${ρ}_{1} < {\sigma}_{1}$<br >$∴ {\overset{\to}{F}}_{viscous} ↓$<br >$∴ {\overset{\to}{V}}_{P} ↑$<br >$∴$if we put Q in${L}_{1}$where \left|{\overset{\to}{V}}_{Q}\right| ∝ \dfrac{1}{{η}_{1}}$ when ${ρ}_{2} < {\sigma}_{1}$
$∴ {\overset{\to}{F}}_{viscous} ↑$
$∴ {\overset{\to}{V}}_{P} ↓$
$⇒ \dfrac{\left|{\overset{\to}{V}}_{P}\right|}{\left|{\overset{\to}{V}}_{Q}\right|} = \dfrac{{η}_{1}}{{η}_{2}}$ & ${\overset{\to}{V}}_{P} . {\overset{\to}{V}}_{Q} < 0$