When a man stands on a turn- table stretching with two equal loads in hand and rotates. Then he folds his arm. Which of the following statements is correct

Options

A. Linear momentum is conserved
B. Kinetic energy increases
C. Angular momentum increases
D. Angular velocity increases

Explanation

To solve the problem step-by-step, we will analyze the situation of a man standing on a turntable with two equal loads in hand, and what happens when he folds his arms.

### Step 1: Understand the System
The system consists of a man standing on a turntable, holding two equal weights. When he rotates, he has a certain angular velocity (ω). The moment of inertia (I) of the system is determined by the distribution of mass (the man and the weights) relative to the axis of rotation.

**Hint:** Consider how the distribution of mass affects the moment of inertia.

### Step 2: Initial Moment of Inertia
When the man stretches out his arms, the moment of inertia (I1) is larger because the weights are farther from the axis of rotation. The moment of inertia for point masses is given by \( I = m \cdot r^2 \), where \( r \) is the distance from the axis of rotation.

**Hint:** Remember that a larger distance from the axis increases the moment of inertia.

### Step 3: Folding Arms
When the man folds his arms, the weights come closer to the axis of rotation, which reduces the moment of inertia (I2). Therefore, I2 < I1.

**Hint:** Think about how moving mass closer to the axis affects the moment of inertia.

### Step 4: Conservation of Angular Momentum
Since there are no external torques acting on the system, angular momentum is conserved. This means that the initial angular momentum (L1) equals the final angular momentum (L2):
\[ L_1 = I_1 \cdot \omega_1 = I_2 \cdot \omega_2 = L_2 \]

**Hint:** Recall that angular momentum is conserved in the absence of external torques.

### Step 5: Relating Angular Velocities
From the conservation of angular momentum, we can derive:
\[ I_1 \cdot \omega_1 = I_2 \cdot \omega_2 \]
Since I1 > I2, it follows that ω2 > ω1. This indicates that when the man folds his arms, his angular velocity increases.

**Hint:** A decrease in moment of inertia leads to an increase in angular velocity to conserve angular momentum.

### Step 6: Analyzing Kinetic Energy
The rotational kinetic energy (K) of the system is given by:
\[ K = \frac{1}{2} I \omega^2 \]
As the moment of inertia decreases and the angular velocity increases, the overall kinetic energy may increase or decrease depending on the specific values of I and ω.

**Hint:** Consider how both I and ω change when the man folds his arms.

### Step 7: Evaluating the Statements
1. **Linear momentum is conserved** - Incorrect, as we are dealing with angular momentum.
2. **Kinetic energy increases** - This can be true if the increase in angular velocity compensates for the decrease in moment of inertia.
3. **Angular momentum increases** - Incorrect, as angular momentum is conserved.
4. **Angular velocity increases** - Correct, as shown by our analysis.

### Conclusion
The correct statements are:
- Kinetic energy may increase (depending on the specific values).
- Angular velocity increases.

### Final Answer
The correct options are **B (Kinetic energy increases)** and **D (Angular velocity increases)**.

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