A metal sphere of radius $R$, density $\rho_1$ moves with terminal velocity $V_1$ through a liquid of…

A metal sphere of radius $R$, density $\rho_1$ moves with terminal velocity $V_1$ through a liquid of density $\sigma$. Another sphere of same radius but density $\rho_2$ - moves through same liquid. Its terminal velocity is $\mathrm{V}_2$. The ratio $\mathrm{V}_1: \mathrm{V}_2$ is
  1. $\left(\rho_2+\sigma\right):\left(\rho_1-\sigma\right)$
  2. $\left(\rho_1+\sigma\right):\left(\rho_2-\sigma\right)$
  3. $\left(\rho_2-\sigma\right):\left(\rho_1-\sigma\right)$
  4. $\left(\rho_1-\sigma\right):\left(\rho_2-\sigma\right)$

Solution

Terminal velocity, $\mathrm{v}_1=\frac{2}{9} \frac{\left(\rho_1-\sigma\right) \mathrm{R}^2 \mathrm{~g}}{\eta}$ Similarly, $\begin{aligned} & v_2=\frac{2}{9} \frac{\left(\rho_2-\sigma\right) R^2 g}{\eta} \\ \therefore \quad & \frac{v_1}{v_2}=\frac{\left(\rho_1-\sigma\right)}{\left(\rho_2-\sigma\right)} \end{aligned}$

Asked in: MHT CET 2024 (10 May Shift 1)

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