The moment of inertia of cylinder of radius a, mass M and height h about an axis parallel to the axis of the cylinder and distance b from its centre is :

Options

A. `1/2 M(a^(2) + 2b^(2))`
B. `1/2 M (2a^(2) + b^(2))`
C. `1/2 M (a^(2) + b^(2))`
D. `1/2 M(a^(2)/3 + b^(2)/12)`

Explanation

To find the moment of inertia of a cylinder of radius \( a \), mass \( M \), and height \( h \) about an axis parallel to the cylinder's axis and at a distance \( b \) from its center, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Type of Cylinder**:
- We need to determine if the cylinder is hollow or solid (rigid). The moment of inertia formulas differ for these two types.
- For a hollow cylinder (thin-walled), the moment of inertia about its central axis is given by:
\[ I_{\text{hollow}} = M a^2 \]
- For a solid cylinder (rigid), the moment of inertia about its central axis is given by:
\[ I_{\text{solid}} = \frac{1}{2} M a^2 \]

2. **Apply the Parallel Axis Theorem**:
- The parallel axis theorem states that if you know the moment of inertia about an axis through the center of mass, you can find the moment of inertia about any parallel axis a distance \( b \) away using the formula:
\[ I = I_{\text{cm}} + M b^2 \]
- Here, \( I_{\text{cm}} \) is the moment of inertia about the center of mass axis, and \( M \) is the mass of the cylinder.

3. **Calculate Moment of Inertia for Both Cases**:
- **For the hollow cylinder**:
\[ I_{\text{hollow}} = M a^2 + M b^2 = M (a^2 + b^2) \]
- **For the solid cylinder**:
\[ I_{\text{solid}} = \frac{1}{2} M a^2 + M b^2 = M \left(\frac{1}{2} a^2 + b^2\right) \]

4. **Conclusion**:
- The moment of inertia of the hollow cylinder about the given axis is:
\[ I_{\text{hollow}} = M (a^2 + b^2) \]
- The moment of inertia of the solid cylinder about the given axis is:
\[ I_{\text{solid}} = M \left(\frac{1}{2} a^2 + b^2\right) \]

### Final Answer:
- For a hollow cylinder: \( I = M(a^2 + b^2) \)
- For a solid cylinder: \( I = M\left(\frac{1}{2} a^2 + b^2\right) \)