A metal sphere of radius 'R', density ' $\varrho_{1}$ ' moves with terminal velocity ' $\mathrm{v}_{1}$ '…

A metal sphere of radius 'R', density ' $\varrho_{1}$ ' moves with terminal velocity ' $\mathrm{v}_{1}$ ' through a liquid of density ' $\sigma^{\prime}$. Another sphere of same radius but of density ' $\varrho_{2}$ ' moves through same liquid. Its terminal velocity will be
  1. $\left[\frac{\varrho_{1}-\varrho_{2}}{\sigma}\right] \mathrm{v}_{1}$
  2. $\left[\frac{\varrho_{2}+\sigma}{\varrho_{1}+\sigma}\right] \mathrm{v}_{1}$
  3. $\left[\frac{\varrho_{1}+\varrho_{2}}{\sigma}\right] \mathrm{v}_{1}$
  4. $\left[\frac{\mathrm{e}_{2}-\sigma}{\mathrm{e}_{1}-\sigma}\right] \mathrm{v}_{1}$

Solution

$6 \pi \eta R v_{1}=\frac{4}{3} \pi R^{3}(\rho_{1}-\sigma)$ $6 \pi \eta R v_{2}=\frac{4}{3} \pi R^{3}\left(\rho_{2}-\sigma\right)$ $\therefore \frac{v_{2}}{v_{1}}=\frac{\rho_{2}-\sigma}{\rho_{1}-\sigma}$ $v_{2}=\frac{\rho_{2}-\sigma}{\rho_{1}-\sigma} v_{1}$

Asked in: MHT CET 2020 (20 Oct Shift 1)

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