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.

Options

A. `2 I`
B. `3 I`
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 torques act on it.

### Step-by-Step Solution:

1. **Identify Initial Conditions**:
- Initial angular velocity, \( \omega_i = 20 \, \text{rad/s} \)
- Initial moment of inertia, \( I_i = I \) (given)
- Final angular velocity, \( \omega_f = 10 \, \text{rad/s} \)

2. **Apply Conservation of Angular Momentum**:
- According to the conservation of angular momentum:
\[ L_i = L_f \]
- Where \( L \) is the angular momentum, given by the product of moment of inertia and angular velocity:
\[ L = I \cdot \omega \]
- Thus, we can write:
\[ I_i \cdot \omega_i = I_f \cdot \omega_f \]

3. **Substitute Known Values**:
- Substitute the known values into the equation:
\[ I \cdot 20 = I_f \cdot 10 \]

4. **Solve for Final Moment of Inertia**:
- Rearranging the equation to solve for \( I_f \):
\[ I_f = \frac{I \cdot 20}{10} \]
- Simplifying gives:
\[ I_f = 2I \]

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

### Final Answer:
The new moment of inertia is \( 2I \).

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