A solid sphere is rotating about a diameter at an angular velocity `omega`. If it cools so that its radius reduces to `1//n` of its original value, its angular velocity becomes

Options

A. `eta omega_(0)`
B. `(omega_(0))/(eta)`
C. `(omega_(0))/(eta^(2))`
D. `eta^(2)omega_(0)`

Answer

To solve the problem, we will use the principle of conservation of angular momentum. The angular momentum of a rotating object is given by the product of its moment of inertia and its angular velocity.

### Step-by-Step Solution:

1. **Identify Initial Parameters:**
- Let the initial radius of the sphere be \( R \).
- The initial angular velocity is \( \omega \).
- The mass of the sphere is \( m \).

2. **Calculate Initial Moment of Inertia:**
- The moment of inertia \( I \) of a solid sphere about its diameter is given by:
\[ I = \frac{2}{5} m R^2 \]
- Thus, the initial angular momentum \( L_i \) can be expressed as:
\[ L_i = I \cdot \omega = \frac{2}{5} m R^2 \cdot \omega \]

3. **Determine Final Parameters:**
- The radius of the sphere reduces to \( \frac{R}{n} \).
- The final angular velocity is \( \omega_f \).

4. **Calculate Final Moment of Inertia:**
- The final moment of inertia \( I_f \) when the radius is \( \frac{R}{n} \) is:
\[ I_f = \frac{2}{5} m \left(\frac{R}{n}\right)^2 = \frac{2}{5} m \frac{R^2}{n^2} \]
- The final angular momentum \( L_f \) can be expressed as:
\[ L_f = I_f \cdot \omega_f = \frac{2}{5} m \frac{R^2}{n^2} \cdot \omega_f \]

5. **Apply Conservation of Angular Momentum:**
- Since there is no external torque acting on the system, we can set the initial angular momentum equal to the final angular momentum:
\[ L_i = L_f \]
- Substituting the expressions for \( L_i \) and \( L_f \):
\[ \frac{2}{5} m R^2 \cdot \omega = \frac{2}{5} m \frac{R^2}{n^2} \cdot \omega_f \]

6. **Simplify the Equation:**
- The \( \frac{2}{5} m R^2 \) terms cancel out from both sides:
\[ \omega = \frac{1}{n^2} \cdot \omega_f \]

7. **Solve for Final Angular Velocity:**
- Rearranging gives:
\[ \omega_f = n^2 \cdot \omega \]

### Final Answer:
The final angular velocity \( \omega_f \) becomes:
\[ \omega_f = n^2 \cdot \omega \]