A man is sitting on a rotating stool with his arms outstretched. If suddenly he folds his arms the angular velocity of the man would

Options

A. increases
B. decreases
C. remains unchanged
D. doubles

Answer

To solve the problem of what happens to the angular velocity of a man sitting on a rotating stool when he suddenly folds his arms, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Angular Momentum**:
Angular momentum (L) is defined as the product of the moment of inertia (I) and the angular velocity (ω). Mathematically, it is expressed as:
\[ L = I \cdot \omega \]
In the absence of external torque, angular momentum remains constant.

2. **Identify the Initial Conditions**:
Initially, the man is sitting on the stool with his arms outstretched. This configuration has a certain moment of inertia (I₁) associated with it.

3. **Effect of Folding Arms**:
When the man folds his arms, the radius associated with his arms decreases. This means that the moment of inertia decreases (let's denote the new moment of inertia as I₂).

4. **Relate Moment of Inertia and Angular Velocity**:
Since angular momentum is conserved (L = constant), we can set up the equation:
\[ I₁ \cdot \omega₁ = I₂ \cdot \omega₂ \]
where:
- I₁ is the initial moment of inertia,
- ω₁ is the initial angular velocity,
- I₂ is the moment of inertia after folding the arms,
- ω₂ is the angular velocity after folding the arms.

5. **Analyze the Change in Moment of Inertia**:
Since the man folds his arms, we know that:
\[ I₂ < I₁ \]
This implies that the moment of inertia decreases.

6. **Determine the Change in Angular Velocity**:
From the conservation of angular momentum, we can rearrange the equation:
\[ \omega₂ = \frac{I₁}{I₂} \cdot \omega₁ \]
Since I₂ is less than I₁, the fraction \(\frac{I₁}{I₂}\) is greater than 1, which means:
\[ \omega₂ > \omega₁ \]
Thus, the angular velocity increases.

7. **Conclusion**:
Therefore, when the man folds his arms, his angular velocity increases.

### Final Answer:
The angular velocity of the man would **increase**.