Question

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 $V_2$. The ratio $V_1:V_2$ is

• A. $(\rho_2 + \sigma) : (\rho_1 - \sigma)$
• B. $(\rho_1 + \sigma) : (\rho_2 - \sigma)$
• C. $(\rho_2 - \sigma) : (\rho_1 - \sigma)$
• D. $(\rho_1 - \sigma) : (\rho_2 - \sigma)$

Answer 1

Answer: (D)

$(\rho_1 - \sigma) : (\rho_2 - \sigma)$

Answer 2

For a sphere moving through a liquid at terminal velocity, the net force is zero. Using Stokes' law for the viscous drag, the terminal velocity is given by:

$V = \frac{2}{9}\frac{(\rho_{\text{sphere}} - \sigma)gR^2}{\eta}$

Thus, for the two spheres:

$V_1 \propto (\rho_1 - \sigma)$ and $V_2 \propto (\rho_2 - \sigma)$

So, the ratio is:

$\frac{V_1}{V_2} = \frac{\rho_1 - \sigma}{\rho_2 - \sigma}$