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A metal sphere of mass ' $m$ ' and density ' $\sigma_1$ ' falls with terminal velocity through a container containing liquid. The density of liquid is ' $\sigma_2$ '. The viscous force acting on the sphere is
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The correct answer is:
$\operatorname{mg}\left(1-\frac{\sigma_2}{\sigma_1}\right)$
Given: Mass of sphere $=\mathrm{m}$, Density of sphere $=\sigma_1$, Density of liquid $=\sigma_2$.
$\mathrm{At}=\mathrm{v}=\mathrm{v}_{\mathrm{t}}$,
Weight of sphere $(\mathrm{W})=$ Viscous Force $\left(\mathrm{F}_{\mathrm{V}}\right)+$ Buoyant Force due to the medium $\left(\mathrm{F}_{\mathrm{B}}\right)$
$\begin{aligned}
\Rightarrow \mathrm{W} & =\mathrm{F}_{\mathrm{V}}+\mathrm{F}_{\mathrm{B}} \\
\mathrm{Mg} & =\mathrm{F}_{\mathrm{V}}+\left(\sigma_2 \mathrm{~V}\right) \mathrm{g} \quad \ldots .(\because \mathrm{m}=\mathrm{D} . \mathrm{V}) \\
\therefore \quad \mathrm{F}_{\mathrm{V}} & =\mathrm{mg}-\left(\sigma_2 \mathrm{~V}\right) \mathrm{g} \\
& =\mathrm{mg}\left[1-\frac{\sigma_2 \mathrm{~V}}{\mathrm{~m}}\right] \\
& =\mathrm{mg}\left[1-\frac{\sigma_2 \mathrm{~V}}{\sigma_1 \mathrm{~V}}\right] \\
& =\mathrm{mg}\left[1-\frac{\sigma_2}{\sigma_1}\right]
\end{aligned}$
$\mathrm{At}=\mathrm{v}=\mathrm{v}_{\mathrm{t}}$,
Weight of sphere $(\mathrm{W})=$ Viscous Force $\left(\mathrm{F}_{\mathrm{V}}\right)+$ Buoyant Force due to the medium $\left(\mathrm{F}_{\mathrm{B}}\right)$
$\begin{aligned}
\Rightarrow \mathrm{W} & =\mathrm{F}_{\mathrm{V}}+\mathrm{F}_{\mathrm{B}} \\
\mathrm{Mg} & =\mathrm{F}_{\mathrm{V}}+\left(\sigma_2 \mathrm{~V}\right) \mathrm{g} \quad \ldots .(\because \mathrm{m}=\mathrm{D} . \mathrm{V}) \\
\therefore \quad \mathrm{F}_{\mathrm{V}} & =\mathrm{mg}-\left(\sigma_2 \mathrm{~V}\right) \mathrm{g} \\
& =\mathrm{mg}\left[1-\frac{\sigma_2 \mathrm{~V}}{\mathrm{~m}}\right] \\
& =\mathrm{mg}\left[1-\frac{\sigma_2 \mathrm{~V}}{\sigma_1 \mathrm{~V}}\right] \\
& =\mathrm{mg}\left[1-\frac{\sigma_2}{\sigma_1}\right]
\end{aligned}$
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