The moment of inertia of a solid cylinder of mass M, length 2R and radius R about an axis passing through the centre of mass and perpendicular to the axis of the cylinder is `I_(1)` and about an axis passing through one end of the cylinder and perpendicular to the axis of cylinder is `I_(2)`

Options

A. `I_(2)-I_(1)=MR^(2)`
B. `I_(2)=I_(1)`
C. `(I_(2))/(I_(1))=(19)/(12)`
D. `I_(1)-I_(2)=MR^(2)`

Explanation

To solve the problem, we need to find the relationship between the moment of inertia \( I_1 \) of a solid cylinder about an axis passing through its center of mass and perpendicular to its axis, and \( I_2 \) about an axis passing through one end of the cylinder and perpendicular to its axis.

### Step-by-Step Solution:

1. **Identify the Moments of Inertia**:
- The moment of inertia \( I_1 \) is for the cylinder about its center of mass (CM).
- The moment of inertia \( I_2 \) is for the cylinder about an axis at one end.

2. **Use the Formula for \( I_1 \)**:
- For a solid cylinder of mass \( M \), length \( 2R \), and radius \( R \), the moment of inertia about its center of mass is given by:
\[ I_1 = \frac{1}{12} M (2R)^2 = \frac{1}{12} M \cdot 4R^2 = \frac{1}{3} MR^2 \]

3. **Apply the Parallel Axis Theorem for \( I_2 \)**:
- The parallel axis theorem states that:
\[ I_2 = I_{CM} + Md^2 \]
- Here, \( d \) is the distance from the center of mass to the new axis. Since the length of the cylinder is \( 2R \), the distance from the center of mass to one end is \( R \).
- Therefore, we have:
\[ I_2 = I_1 + M(R^2) \]

4. **Substitute \( I_1 \) into the Equation for \( I_2 \)**:
- Substitute \( I_1 = \frac{1}{3} MR^2 \) into the equation for \( I_2 \):
\[ I_2 = \frac{1}{3} MR^2 + MR^2 = \frac{1}{3} MR^2 + \frac{3}{3} MR^2 = \frac{4}{3} MR^2 \]

5. **Find the Relationship Between \( I_1 \) and \( I_2 \)**:
- Now we can find \( I_2 - I_1 \):
\[ I_2 - I_1 = \frac{4}{3} MR^2 - \frac{1}{3} MR^2 = \frac{3}{3} MR^2 = MR^2 \]

### Conclusion:
The relationship between the moments of inertia is:
\[ I_2 - I_1 = MR^2 \]

### Final Answer:
Thus, the correct option is that \( I_2 - I_1 = MR^2 \).

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