Calculate the moment of inertia of a solid sphere of mass 10 kg and radius 0.5 m, rotating about an axis 0.2 m from the centre of the sphere .

Answer

To calculate the moment of inertia of a solid sphere of mass 10 kg and radius 0.5 m, rotating about an axis 0.2 m from the center of the sphere, we can follow these steps:

### Step 1: Understand the Moment of Inertia for a Solid Sphere
The moment of inertia \( I \) of a solid sphere about its center is given by the formula:
\[ I_{\text{center}} = \frac{2}{5} m R^2 \]
where \( m \) is the mass of the sphere and \( R \) is its radius.

### Step 2: Calculate the Moment of Inertia about the Center
Given:
- Mass \( m = 10 \, \text{kg} \)
- Radius \( R = 0.5 \, \text{m} \)

Substituting the values into the formula:
\[ I_{\text{center}} = \frac{2}{5} \times 10 \times (0.5)^2 \]
Calculating \( (0.5)^2 \):
\[ (0.5)^2 = 0.25 \]
Now substituting this value:
\[ I_{\text{center}} = \frac{2}{5} \times 10 \times 0.25 = \frac{2 \times 10 \times 0.25}{5} = \frac{5}{5} = 1 \, \text{kg m}^2 \]

### Step 3: Use the Parallel Axis Theorem
To find the moment of inertia about an axis that is 0.2 m from the center, we use the Parallel Axis Theorem:
\[ I = I_{\text{center}} + m d^2 \]
where \( d \) is the distance from the center to the new axis of rotation.

Here, \( d = 0.2 \, \text{m} \).

### Step 4: Calculate the Moment of Inertia about the New Axis
Substituting the values into the equation:
\[ I = 1 + 10 \times (0.2)^2 \]
Calculating \( (0.2)^2 \):
\[ (0.2)^2 = 0.04 \]
Now substituting this value:
\[ I = 1 + 10 \times 0.04 = 1 + 0.4 = 1.4 \, \text{kg m}^2 \]

### Final Answer
The moment of inertia of the solid sphere about the axis 0.2 m from its center is:
\[ \boxed{1.4 \, \text{kg m}^2} \]