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, we will follow these steps:

### Step 1: Understand the concept of angular momentum
Angular momentum (L) of a rotating object is given by the formula:
\[ L = I \cdot \omega \]
where \( I \) is the moment of inertia and \( \omega \) is the angular velocity.

### Step 2: Identify the moment of inertia for both spheres
For a solid sphere, the moment of inertia \( I_A \) is given by:
\[ I_A = \frac{2}{5} m r^2 \]
For a hollow sphere, the moment of inertia \( I_B \) is given by:
\[ I_B = \frac{2}{3} m r^2 \]

### Step 3: Substitute the values into the angular momentum formula
Since both spheres have the same mass \( m \), radius \( r \), and angular velocity \( \omega \), we can write the angular momentum for both spheres:
- For sphere A (solid sphere):
\[ L_A = I_A \cdot \omega = \left(\frac{2}{5} m r^2\right) \cdot \omega \]
- For sphere B (hollow sphere):
\[ L_B = I_B \cdot \omega = \left(\frac{2}{3} m r^2\right) \cdot \omega \]

### Step 4: Compare the angular momentum of both spheres
We can compare the two angular momentum expressions:
- \( L_A = \frac{2}{5} m r^2 \cdot \omega \)
- \( L_B = \frac{2}{3} m r^2 \cdot \omega \)

Since \( \frac{2}{5} < \frac{2}{3} \), it follows that:
\[ L_A < L_B \]

### Conclusion
Thus, the angular momentum of sphere A (the solid sphere) will be less than that of sphere B (the hollow sphere).

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

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