The mass and radius of a solid cylinder be M and R respectively. Its M.I. about a generator line will be:

Options

A. `M((L^(2))/(12)+(R^(2))/(4))`
B. `M((L^(2))/(3)+(R^(2))/(4))`
C. `(MR^(2))/(2)`
D. `(3MR^(2))/(2)`

Answer

To find the moment of inertia (M.I.) of a solid cylinder about its generator line, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Cylinder**:
- A solid cylinder has a mass \( M \) and a radius \( R \). The generator line is an axis that runs along the length of the cylinder and is perpendicular to its circular faces.

2. **Identifying the Moment of Inertia**:
- The moment of inertia (I) is a measure of an object's resistance to changes in its rotation. For a solid cylinder rotating about its axis (the generator line), we can use the known formula for the moment of inertia of a solid cylinder.

3. **Using the Formula**:
- The moment of inertia of a solid cylinder about its central axis (which is the generator line) is given by the formula:
\[ I = \frac{1}{2} M R^2 \]
- This formula is derived from integrating the mass distribution of the cylinder.

4. **Final Result**:
- Therefore, the moment of inertia of the solid cylinder about its generator line is:
\[ I = \frac{1}{2} M R^2 \]

### Conclusion:
The moment of inertia of a solid cylinder about its generator line is \( \frac{1}{2} M R^2 \).

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