A small metal sphere of mass 'M' and density 'd $_{1}$ ', when dropped in a jar filled with liquid moves…

A small metal sphere of mass 'M' and density 'd $_{1}$ ', when dropped in a jar filled with liquid moves with terminal velocity after some time. The viscous force acting on the sphere is $\left(d_{2}=\right.$ density of liquid, $\mathrm{g}=$ gravitational acceleration $)$
  1. $\operatorname{Mg}\left(1-\frac{\mathrm{d}_{2}}{\mathrm{~d}_{1}}\right)$
  2. $\operatorname{Mg}\left(\frac{\mathrm{d}_{2}}{\mathrm{~d}_{1}}\right)$
  3. $\operatorname{Mg}\left(1-\frac{\mathrm{d}_{1}}{\mathrm{~d}_{2}}\right)$
  4. $\operatorname{Mg}\left(\frac{\mathrm{d}_{1}}{\mathrm{~d}_{2}}\right)$

Solution

$\mathrm{F}=\frac{4}{3} \pi \mathrm{r}^{3}\left(\mathrm{~d}_{1}-\mathrm{d}_{2}\right) \mathrm{g}$ $\mathrm{F}=\frac{4}{3} \pi \mathrm{r}^{3} \mathrm{~d}_{1}\left(1-\frac{\mathrm{d}_{2}}{\mathrm{~d}_{1}}\right) \mathrm{g}=\mathrm{M}\left(1-\frac{\mathrm{d}_{2}}{\mathrm{~d}_{1}}\right) \mathrm{g}$

Asked in: MHT CET 2020 (16 Oct Shift 1)

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