The terminal speed of a sphere of gold (density = 19.5 kg `m^(-3)`) is 0.2 `ms^(-1)` in a viscous liquid (density = 1.5 kg `m^(-3)`). Then, the terminal speed of a sphere of silver (density = 10.5 kg `m^(-3)`) of the same size in the same liquid is

Options

A. ` 0.4 m//s `
B. `0.133 m//s `
C. `0.1 m//s `
D. `0.2 m//s `

Explanation

To find the terminal speed of a sphere of silver in a viscous liquid, we can use the relationship between the terminal speeds of two spheres of different materials but the same size and in the same liquid. The formula for terminal velocity (VT) is given by:

\[ V_T = \frac{2}{9} \frac{g R^2 (\rho - \sigma)}{\eta} \]

Where:
- \( V_T \) = terminal velocity
- \( g \) = acceleration due to gravity
- \( R \) = radius of the sphere
- \( \rho \) = density of the sphere
- \( \sigma \) = density of the liquid
- \( \eta \) = coefficient of viscosity

Since the size of the spheres and the liquid are the same, we can say that:

\[ V_{T, \text{gold}} \propto \rho_{\text{gold}} - \sigma \]
\[ V_{T, \text{silver}} \propto \rho_{\text{silver}} - \sigma \]

This allows us to set up a ratio:

\[ \frac{V_{T, \text{gold}}}{V_{T, \text{silver}}} = \frac{\rho_{\text{gold}} - \sigma}{\rho_{\text{silver}} - \sigma} \]

Given:
- \( V_{T, \text{gold}} = 0.2 \, \text{m/s} \)
- \( \rho_{\text{gold}} = 19.5 \, \text{kg/m}^3 \)
- \( \rho_{\text{silver}} = 10.5 \, \text{kg/m}^3 \)
- \( \sigma = 1.5 \, \text{kg/m}^3 \)

Now, substituting the values into the equation:

1. Calculate \( \rho_{\text{gold}} - \sigma \):
\[ \rho_{\text{gold}} - \sigma = 19.5 - 1.5 = 18 \, \text{kg/m}^3 \]

2. Calculate \( \rho_{\text{silver}} - \sigma \):
\[ \rho_{\text{silver}} - \sigma = 10.5 - 1.5 = 9 \, \text{kg/m}^3 \]

3. Set up the ratio:
\[ \frac{0.2}{V_{T, \text{silver}}} = \frac{18}{9} \]

4. Simplify the ratio:
\[ \frac{0.2}{V_{T, \text{silver}}} = 2 \]

5. Solve for \( V_{T, \text{silver}} \):
\[ V_{T, \text{silver}} = \frac{0.2}{2} = 0.1 \, \text{m/s} \]

Thus, the terminal speed of the sphere of silver in the same viscous liquid is **0.1 m/s**.