A ballet dancer, dancing on a smooth floor is spinning about a vertical axis with her arms folded with angular velocity of `20 rad//s`. When the stretches her arms fully, the spinning speed decrease in `10 rad//s`. If `I` is the initial moment of inertia of the dancer, the new moment of inertia is: a) 2I b) 3I c) I/2 d) I/3

Options

A. 2I
B. 3I
C. `I/2`
D. `I/3`

Explanation

To solve the problem, we will use the principle of conservation of angular momentum. The angular momentum of a system remains constant if no external torque acts on it.

### Step-by-Step Solution:

1. **Understand the Given Information:**
- Initial angular velocity, \( \omega_i = 20 \, \text{rad/s} \)
- Final angular velocity, \( \omega_f = 10 \, \text{rad/s} \)
- Initial moment of inertia, \( I \)

2. **Write the Conservation of Angular Momentum Equation:**
According to the conservation of angular momentum:
\[ L_i = L_f \]
where \( L \) is the angular momentum. Angular momentum is given by:
\[ L = I \cdot \omega \]
Therefore, we can write:
\[ I \cdot \omega_i = I' \cdot \omega_f \]
Here, \( I' \) is the new moment of inertia after the dancer stretches her arms.

3. **Substitute the Known Values:**
Substitute \( \omega_i \) and \( \omega_f \) into the equation:
\[ I \cdot 20 = I' \cdot 10 \]

4. **Solve for the New Moment of Inertia \( I' \):**
Rearranging the equation gives:
\[ I' = \frac{I \cdot 20}{10} \]
Simplifying this:
\[ I' = 2I \]

5. **Conclusion:**
The new moment of inertia when the dancer stretches her arms is \( 2I \).

### Final Answer:
The new moment of inertia is \( 2I \), which corresponds to option (a).