A small steel shphere tied at the end of a string is whirled in a horizontal circle with uniform angular velocity `omega ` the string is suddently pulled so that the radius of the circle is halved .If `omega_(2)` is the angular velocity then

Options

A. `omega_(1)=omega_(2)`
B. `omega_(1) gt omega_(2)`
C. `omega_(1)lt omega_(2)`
D. `omgea_(1)=2omega_(2)`

Explanation

To solve the problem, we need to analyze the situation using the principles of angular momentum conservation. The key points to consider are:

1. **Initial and Final Conditions**:
- The initial radius of the circular path is \( r \).
- The final radius after the string is pulled is \( \frac{r}{2} \).
- The initial angular velocity is \( \omega \) and the final angular velocity is \( \omega_2 \).

2. **Conservation of Angular Momentum**:
- Angular momentum \( L \) is conserved when no external torque acts on the system.
- The angular momentum \( L \) can be expressed as:
\[ L = m \cdot v \cdot r \]
- In terms of angular velocity, \( v \) can be expressed as \( v = r \cdot \omega \), so:
\[ L = m \cdot (r \cdot \omega) \cdot r = m \cdot r^2 \cdot \omega \]

3. **Setting Up the Equation**:
- For the initial state, the angular momentum is:
\[ L_i = m \cdot r^2 \cdot \omega \]
- For the final state, when the radius is halved, the angular momentum becomes:
\[ L_f = m \cdot \left(\frac{r}{2}\right)^2 \cdot \omega_2 = m \cdot \frac{r^2}{4} \cdot \omega_2 \]

4. **Applying Conservation of Angular Momentum**:
- According to the conservation of angular momentum:
\[ L_i = L_f \]
- Substituting the expressions we derived:
\[ m \cdot r^2 \cdot \omega = m \cdot \frac{r^2}{4} \cdot \omega_2 \]
- We can cancel \( m \) and \( r^2 \) from both sides (assuming \( m \neq 0 \) and \( r \neq 0 \)):
\[ \omega = \frac{1}{4} \cdot \omega_2 \]

5. **Solving for \( \omega_2 \)**:
- Rearranging the equation gives:
\[ \omega_2 = 4 \cdot \omega \]

Thus, the final angular velocity \( \omega_2 \) is four times the initial angular velocity \( \omega \).

### Final Answer:
\[ \omega_2 = 4 \cdot \omega \]