If a uniform solid sphere of diameter 0.2 m and mass 10 kg is rotated about its diameter with an angular velocity of 2 rad/s, then the its angular momentum in kg `m^(2)//s` will be

Options

A. 0.01
B. 0.02
C. 0.08
D. 0.04

Explanation

To find the angular momentum of a uniform solid sphere rotating about its diameter, we can follow these steps:

### Step 1: Identify the given values
- Diameter of the sphere (d) = 0.2 m
- Mass of the sphere (m) = 10 kg
- Angular velocity (ω) = 2 rad/s

### Step 2: Calculate the radius of the sphere
The radius (r) is half of the diameter:
\[ r = \frac{d}{2} = \frac{0.2 \, \text{m}}{2} = 0.1 \, \text{m} \]

### Step 3: Use the formula for the moment of inertia (I) of a solid sphere
The moment of inertia (I) for a solid sphere rotating about its diameter is given by:
\[ I = \frac{2}{5} m r^2 \]

### Step 4: Substitute the values into the moment of inertia formula
Substituting the mass and radius into the formula:
\[ I = \frac{2}{5} \times 10 \, \text{kg} \times (0.1 \, \text{m})^2 \]
\[ I = \frac{2}{5} \times 10 \times 0.01 = \frac{2}{5} \times 0.1 = 0.04 \, \text{kg m}^2 \]

### Step 5: Calculate the angular momentum (L)
The angular momentum (L) is given by the product of the moment of inertia and the angular velocity:
\[ L = I \cdot \omega \]
Substituting the values we found:
\[ L = 0.04 \, \text{kg m}^2 \times 2 \, \text{rad/s} = 0.08 \, \text{kg m}^2/\text{s} \]

### Final Answer
The angular momentum of the sphere is:
\[ L = 0.08 \, \text{kg m}^2/\text{s} \]

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