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

Explanation

To find the moment of inertia of a solid cylinder about two different axes, we will use the following steps:

### Step 1: Determine the Moment of Inertia about the Center of Mass (I1)

The moment of inertia \( I_1 \) of a solid cylinder about an axis passing through its center of mass and perpendicular to its length can be calculated using the formula:

\[ I_1 = \frac{1}{12} M (3R^2 + L^2) \]

Where:
- \( M \) is the mass of the cylinder,
- \( R \) is the radius of the cylinder,
- \( L \) is the length of the cylinder.

Given that the length of the cylinder is \( 2R \), we substitute \( L = 2R \):

\[ I_1 = \frac{1}{12} M (3R^2 + (2R)^2) = \frac{1}{12} M (3R^2 + 4R^2) = \frac{1}{12} M (7R^2) = \frac{7}{12} MR^2 \]

### Step 2: Determine the Moment of Inertia about the End of the Cylinder (I2)

To find the moment of inertia \( I_2 \) about an axis passing through one end of the cylinder and perpendicular to its length, we can use the parallel axis theorem. The parallel axis theorem states:

\[ I = I_{cm} + Md^2 \]

Where:
- \( I_{cm} \) is the moment of inertia about the center of mass,
- \( d \) is the distance from the center of mass to the new axis.

For our case:
- The distance \( d \) from the center of mass to the end of the cylinder is \( R \) (half the length of the cylinder).

Using the value of \( I_1 \) we calculated earlier:

\[ I_2 = I_1 + M(R^2) \]

Substituting \( I_1 = \frac{7}{12} MR^2 \):

\[ I_2 = \frac{7}{12} MR^2 + M(R^2) = \frac{7}{12} MR^2 + \frac{12}{12} MR^2 = \frac{19}{12} MR^2 \]

### Summary of Results

- The moment of inertia about the center of mass \( I_1 \) is:

\[ I_1 = \frac{7}{12} MR^2 \]

- The moment of inertia about one end of the cylinder \( I_2 \) is:

\[ I_2 = \frac{19}{12} MR^2 \]