The velocity of a small ball of mass ' $M$ ' and density ' $\mathrm{d}_1$ ' when dropped in a container…

The velocity of a small ball of mass ' $M$ ' and density ' $\mathrm{d}_1$ ' when dropped in a container filled with glycerin becomes constant after some time. If the density of glycerin is ' $\mathrm{d}_2$ ', the viscous force acting on the ball is ( $\mathrm{g}$ = acceleration due to gravity)
  1. $\operatorname{Mg} \frac{\mathrm{d}_1}{\mathrm{~d}_2}$
  2. $\operatorname{Mgd}_1 \mathrm{~d}_2$
  3. $\operatorname{Mg}\left(\mathrm{d}_1-\mathrm{d}_2\right)$
  4. $\operatorname{Mg}\left(1-\frac{\mathrm{d}_2}{\mathrm{~d}_1}\right)$

Solution

Since the velocity is constant, the net force acting on the ball is zero. $\begin{aligned} & \mathrm{F}_{\mathrm{V}}+\mathrm{F}_{\mathrm{B}}=\mathrm{Mg} \\ & \mathrm{F}_{\mathrm{V}}+\mathrm{Vd}_2 \mathrm{~g}=\mathrm{Mg} \end{aligned}$ $\mathrm{F}_{\mathrm{v}}+\frac{\mathrm{M}}{\mathrm{d}_1} \mathrm{~d}_2 \mathrm{~g}=\mathrm{Mg}$ $\mathrm{F}_{\mathrm{V}}=\mathrm{Mg}\left(1-\frac{\mathrm{d}_2}{\mathrm{~d}_1}\right)$

Asked in: MHT CET 2021 (21 Sep Shift 1)

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