A dancer is standing on a rotating platform taking two sphere on her hands. If she drops down the sphere on ground, then dancer's:

Options

A. angular velocity will increase
B. angular momentum & angular velocity both will unchange
C. angular momentum unchanged & angular velocity will increase
D. both will decrease

Answer

To solve the problem, we need to analyze the situation using the principles of angular momentum and moment of inertia. Here's a step-by-step breakdown:

### Step 1: Understand the System
The dancer is on a rotating platform holding two spheres. When she drops one of the spheres, we need to consider the effects on angular momentum and angular velocity.

### Step 2: Apply Conservation of Angular Momentum
In the absence of external torques, the angular momentum of the system must be conserved. This means that the initial angular momentum (L_initial) must equal the final angular momentum (L_final).

### Step 3: Define Initial and Final Angular Momentum
1. **Initial Angular Momentum (L_initial)**:
- Let \( I_0 \) be the moment of inertia of the dancer.
- Let \( I_1 \) and \( I_2 \) be the moments of inertia of the two spheres.
- The initial angular momentum can be expressed as:
\[ L_{\text{initial}} = (I_0 + I_1 + I_2) \cdot \omega_i \]
where \( \omega_i \) is the initial angular velocity.

2. **Final Angular Momentum (L_final)**:
- After dropping one sphere, the moment of inertia of the system changes. Now, only the dancer and one sphere contribute to the moment of inertia:
\[ L_{\text{final}} = (I_0 + I_2) \cdot \omega_f \]
where \( \omega_f \) is the final angular velocity.

### Step 4: Set Initial and Final Angular Momentum Equal
Using the conservation of angular momentum:
\[ (I_0 + I_1 + I_2) \cdot \omega_i = (I_0 + I_2) \cdot \omega_f \]

### Step 5: Solve for Final Angular Velocity
Rearranging the equation gives:
\[ \omega_f = \frac{(I_0 + I_1 + I_2)}{(I_0 + I_2)} \cdot \omega_i \]
Since \( I_1 \) is positive, it follows that:
\[ \omega_f > \omega_i \]
This indicates that the final angular velocity increases when the dancer drops the sphere.

### Step 6: Conclusion on Angular Momentum and Angular Velocity
- **Angular Momentum**: Unchanged (conserved).
- **Angular Velocity**: Increases.

Thus, the correct answer is that the angular momentum remains unchanged and the angular velocity increases.

### Final Answer
**Angular momentum unchanged and angular velocity increases.**

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