Moment of inertia of a cylindrical shell of mass M, radius R and length L about its geometrical axis would be -

Options

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

Answer

To find the moment of inertia of a cylindrical shell of mass \( M \), radius \( R \), and length \( L \) about its geometrical axis, we can follow these steps:

### Step 1: Understanding the Geometry
We have a cylindrical shell with:
- Mass \( M \)
- Radius \( R \)
- Length \( L \)

The moment of inertia \( I \) about the geometrical axis (the axis along the length of the cylinder) needs to be determined.

### Step 2: Consider a Differential Element
Consider a thin strip of the cylindrical shell at a distance \( R \) from the axis. This strip has a small mass \( dm \) and length \( L \). The moment of inertia \( dI \) of this differential strip about the axis is given by the formula:
\[ dI = R^2 \, dm \]

### Step 3: Integrate Over the Entire Mass
To find the total moment of inertia \( I \), we need to integrate \( dI \) over the entire mass \( M \) of the cylindrical shell. Since the radius \( R \) is constant for the entire shell, we can take \( R^2 \) out of the integral:
\[ I = \int dI = R^2 \int dm \]

### Step 4: Evaluate the Integral
The integral \( \int dm \) gives us the total mass \( M \) of the cylindrical shell:
\[ I = R^2 \cdot M \]

### Step 5: Final Result
Thus, the moment of inertia \( I \) of the cylindrical shell about its geometrical axis is:
\[ I = M R^2 \]

### Conclusion
The correct answer is \( M R^2 \), which corresponds to option 1.

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