A solid sphere A and a hollow sphere B have the same mass, radius and same angular velocity are moving in the same direction. The angular momentum of sphere A will be?

Options

A. equal to B
B. less than B
C. more than B
D. insufficient data

Answer

To find the angular momentum of the solid sphere A in comparison to the hollow sphere B, we can follow these steps:

### Step 1: Understand the Moment of Inertia
The moment of inertia (I) is a measure of an object's resistance to changes in its rotation. For different shapes, the moment of inertia is calculated differently.

- For a solid sphere (Sphere A), the moment of inertia is given by:
\[ I_A = \frac{2}{5} m R^2 \]
- For a hollow sphere (Sphere B), the moment of inertia is given by:
\[ I_B = \frac{2}{3} m R^2 \]

### Step 2: Write the Formula for Angular Momentum
Angular momentum (L) is calculated using the formula:
\[ L = I \cdot \omega \]
where \( \omega \) is the angular velocity.

### Step 3: Calculate Angular Momentum for Both Spheres
- For Sphere A (solid):
\[ L_A = I_A \cdot \omega = \left(\frac{2}{5} m R^2\right) \cdot \omega \]
- For Sphere B (hollow):
\[ L_B = I_B \cdot \omega = \left(\frac{2}{3} m R^2\right) \cdot \omega \]

### Step 4: Take the Ratio of Angular Momenta
To compare the angular momenta of the two spheres, we can take the ratio:
\[ \frac{L_A}{L_B} = \frac{I_A \cdot \omega}{I_B \cdot \omega} \]
Since \( \omega \) is the same for both spheres, it cancels out:
\[ \frac{L_A}{L_B} = \frac{I_A}{I_B} \]

### Step 5: Substitute the Moments of Inertia
Substituting the values of \( I_A \) and \( I_B \):
\[ \frac{L_A}{L_B} = \frac{\frac{2}{5} m R^2}{\frac{2}{3} m R^2} \]
The mass \( m \) and radius \( R^2 \) cancel out:
\[ \frac{L_A}{L_B} = \frac{\frac{2}{5}}{\frac{2}{3}} = \frac{2}{5} \cdot \frac{3}{2} = \frac{3}{5} \]

### Step 6: Conclusion
Since \( \frac{3}{5} < 1 \), we conclude that:
\[ L_A < L_B \]
Thus, the angular momentum of sphere A (solid sphere) is less than that of sphere B (hollow sphere).

### Final Answer
The angular momentum of sphere A will be less than that of sphere B.

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