A man spinning in free space can change the moment of inertia of his body (I) by changing its shape. In this process,

Options

A. He will have to expend some energy to increase I
B. He will have to expend some energy to decrease I
C. He does not have to expend any energy to change I
D. Either (A) or (B) depending on the initial value of I

Explanation

To solve the problem, let's analyze the situation step by step.

### Step 1: Understand the Concept of Moment of Inertia
The moment of inertia (I) is a measure of an object's resistance to changes in its rotation. A man spinning in free space can change his moment of inertia by changing his body shape (e.g., pulling his arms in or extending them out).

**Hint:** Remember that the moment of inertia depends on the mass distribution relative to the axis of rotation.

### Step 2: Conservation of Angular Momentum
In free space, there are no external torques acting on the man. Therefore, the angular momentum (L) of the system is conserved. The relationship between angular momentum, moment of inertia, and angular velocity (ω) is given by:
\[ L = I \omega \]
Since L is constant, any change in I must result in a corresponding change in ω.

**Hint:** Angular momentum is conserved when no external torques are acting on the system.

### Step 3: Relationship Between Kinetic Energy and Moment of Inertia
The rotational kinetic energy (KE) of the man can be expressed as:
\[ KE = \frac{1}{2} I \omega^2 \]
Substituting the expression for ω from the angular momentum equation:
\[ \omega = \frac{L}{I} \]
We can rewrite the kinetic energy as:
\[ KE = \frac{1}{2} I \left(\frac{L}{I}\right)^2 = \frac{L^2}{2I} \]
This shows that kinetic energy is inversely proportional to the moment of inertia (I).

**Hint:** Kinetic energy decreases as the moment of inertia increases, and vice versa.

### Step 4: Analyzing Energy Expenditure
From the relationship derived, we can conclude:
- If the man increases his moment of inertia (I), his kinetic energy (KE) decreases.
- Conversely, if he decreases his moment of inertia, his kinetic energy increases.

To change the moment of inertia, the man must exert energy. Specifically:
- To **increase** I, he must expend energy to slow down (since KE decreases).
- To **decrease** I, he must expend energy to speed up (since KE increases).

**Hint:** Consider how changing the shape of the body affects the rotational speed and energy.

### Step 5: Conclusion
From the analysis, we can conclude:
- The correct statement is that the man will have to expend energy to **decrease** his moment of inertia (option B). He does not need to expend energy to increase his moment of inertia, as this would result in a decrease in kinetic energy.

**Final Answer:** The man will have to expend energy to decrease I (option B).