Question

Two solid spheres of same metal but of mass M and 64 M fall simultaneously through a viscous liquid and their terminal velocities are V and nV respectively then value of n is ______.

Options

• 4

• 8

• 16

• 32

Answer 1

Two solid spheres of same metal but of mass M and 64 M fall simultaneously through a viscous liquid and their terminal velocities are V and nV respectively then value of n is 16.

Explanation:

\[\mathrm{Mass}=\mathrm{Volume}\times\mathrm{Density}\Rightarrow\mathrm{M}=\frac{4}{3}\pi\mathrm{r}^3\times\rho\]

As the density remains constant

\[\therefore\quad\mathrm{M}\propto\mathrm{r}^{3}\]

\[\therefore\quad\frac{\mathbf{r}_{1}}{\mathbf{r}_{2}}=\left(\frac{\mathbf{M}_{1}}{\mathbf{M}_{2}}\right)^{1/3}=\left(\frac{\mathbf{M}}{64\mathbf{M}}\right)^{1/3}=\frac{1}{4}\quad\ldots.(\mathbf{i})\]

Terminal velocity, \[\mathrm{v}={\frac{2}{9}}{\frac{\mathrm{r}^{2}(\rho-\sigma)\mathrm{g}}{\eta}}\]

\[\therefore v\propto r^{2}\]

\[\therefore\quad\frac{\mathrm{vT}_1}{\mathrm{vT}_2}=\left(\frac{\mathrm{r}_1}{\mathrm{r}_2}\right)^2\]

\[\frac{\mathrm{v}}{\mathrm{nv}}=\left(\frac{\mathrm{r}_1}{\mathrm{r}_2}\right)^2\mathrm{or}\frac{1}{\mathrm{n}}=\left(\frac{1}{4}\right)^2\]       ...[Using (i)]

\[\therefore\] n = 16