Two spheres of the same material, but of radii R and 3R are allowed to fall vertically downwards through a liquid of density o. The ratio of their terminal velocities is

Options

A. `1:3`
B. `1:6`
C. `1:9`
D. `1:1`

Explanation

To find the ratio of the terminal velocities of two spheres of the same material but different radii, we can follow these steps:

### Step 1: Understand the concept of terminal velocity
Terminal velocity is the constant speed that a freely falling object eventually reaches when the resistance of the medium prevents further acceleration. For a sphere falling through a fluid, the terminal velocity \( V_t \) can be expressed as:
\[ V_t = \frac{2}{9} \frac{R^2 (\rho - \sigma) g}{\eta} \]
where:
- \( R \) is the radius of the sphere,
- \( \rho \) is the density of the liquid,
- \( \sigma \) is the density of the sphere,
- \( g \) is the acceleration due to gravity,
- \( \eta \) is the viscosity of the liquid.

### Step 2: Identify the radii of the spheres
Let the radius of the first sphere be \( R \) and the radius of the second sphere be \( 3R \).

### Step 3: Write the expressions for terminal velocities
Using the formula for terminal velocity:
- For the first sphere (radius \( R \)):
\[ V_{t1} = \frac{2}{9} \frac{R^2 (\rho - \sigma) g}{\eta} \]

- For the second sphere (radius \( 3R \)):
\[ V_{t2} = \frac{2}{9} \frac{(3R)^2 (\rho - \sigma) g}{\eta} = \frac{2}{9} \frac{9R^2 (\rho - \sigma) g}{\eta} \]

### Step 4: Calculate the ratio of terminal velocities
Now, we can find the ratio of the terminal velocities:
\[ \frac{V_{t1}}{V_{t2}} = \frac{\frac{2}{9} \frac{R^2 (\rho - \sigma) g}{\eta}}{\frac{2}{9} \frac{9R^2 (\rho - \sigma) g}{\eta}} \]
The constants \( \frac{2}{9} \), \( g \), \( \eta \), and \( (\rho - \sigma) \) cancel out:
\[ \frac{V_{t1}}{V_{t2}} = \frac{R^2}{9R^2} = \frac{1}{9} \]

### Step 5: Final ratio of terminal velocities
Thus, the ratio of the terminal velocities of the two spheres is:
\[ V_{t1} : V_{t2} = 1 : 9 \]

### Conclusion
The ratio of their terminal velocities is \( 1 : 9 \).

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