A solid sphere, of radius R acquires a terminal velocity ` v_1 ` when falling (due to gravity) through a viscous fluid having a coefficient of viscosity ` eta ` he sphere is broken into 27 identical solid spheres. If each of these spheres acquires a terminal velocity ` v _2 ` when falling through the same fluid, the ratio ` (v_1//v_2 ) ` equals:

Options

A. (a)`1//27 `
B. (b)`27`
C. (c)9
D. (d)`1//9`

Explanation

To solve the problem, we need to find the ratio of terminal velocities \( \frac{v_1}{v_2} \) for a solid sphere and for smaller spheres after the original sphere is broken into 27 identical spheres.

### Step 1: Understand Terminal Velocity
The terminal velocity \( v \) of a sphere falling through a viscous fluid is given by the formula:
\[ v = \frac{2}{9} \frac{r^2 (\rho_s - \rho_f) g}{\eta} \]
where:
- \( r \) = radius of the sphere
- \( \rho_s \) = density of the solid sphere
- \( \rho_f \) = density of the fluid
- \( g \) = acceleration due to gravity
- \( \eta \) = coefficient of viscosity of the fluid

### Step 2: Terminal Velocity of the Original Sphere
For the original solid sphere of radius \( R \), the terminal velocity \( v_1 \) is:
\[ v_1 = \frac{2}{9} \frac{R^2 (\rho_s - \rho_f) g}{\eta} \]

### Step 3: Determine the Radius of Smaller Spheres
When the original sphere is broken into 27 identical smaller spheres, each smaller sphere has a radius \( r \). Since the volume of the original sphere is equal to the total volume of the smaller spheres, we have:
\[ \frac{4}{3} \pi R^3 = 27 \left( \frac{4}{3} \pi r^3 \right) \]
This simplifies to:
\[ R^3 = 27 r^3 \]
Taking the cube root of both sides gives:
\[ R = 3r \]

### Step 4: Terminal Velocity of Smaller Spheres
Now, we can find the terminal velocity \( v_2 \) for one of the smaller spheres:
\[ v_2 = \frac{2}{9} \frac{r^2 (\rho_s - \rho_f) g}{\eta} \]
Substituting \( r = \frac{R}{3} \):
\[ v_2 = \frac{2}{9} \frac{\left(\frac{R}{3}\right)^2 (\rho_s - \rho_f) g}{\eta} = \frac{2}{9} \frac{\frac{R^2}{9} (\rho_s - \rho_f) g}{\eta} = \frac{2}{81} \frac{R^2 (\rho_s - \rho_f) g}{\eta} \]

### Step 5: Find the Ratio \( \frac{v_1}{v_2} \)
Now, we can find the ratio of the terminal velocities:
\[ \frac{v_1}{v_2} = \frac{\frac{2}{9} \frac{R^2 (\rho_s - \rho_f) g}{\eta}}{\frac{2}{81} \frac{R^2 (\rho_s - \rho_f) g}{\eta}} \]
The terms \( \frac{2}{9} \), \( \frac{R^2 (\rho_s - \rho_f) g}{\eta} \) cancel out:
\[ \frac{v_1}{v_2} = \frac{81}{9} = 9 \]

### Final Answer
Thus, the ratio \( \frac{v_1}{v_2} \) equals \( 9 \).

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