What is moment of inertia of a cylinder of radius r, along its height?

Options

A. a.`mr^(2)`
B. b.`(mr^(2))/(2)`
C. c.`(2mr^(2))/(5)`
D. d.`(mr^(2))/(5)`

Answer

To find the moment of inertia of a solid cylinder of radius \( r \) along its height (which is the axis passing through the center and perpendicular to the height), we can follow these steps:

### Step 1: Understand the Moment of Inertia
The moment of inertia \( I \) of an object is a measure of its resistance to rotational motion about an axis. For a solid cylinder, the moment of inertia can be calculated using the formula for different axes.

### Step 2: Identify the Axis of Rotation
In this case, we need to find the moment of inertia of the cylinder along its height. This means we are considering the axis that runs along the height of the cylinder, which is perpendicular to the circular base.

### Step 3: Use the Formula for Moment of Inertia of a Cylinder
The moment of inertia \( I \) of a solid cylinder about its central axis (which is along its height) is given by the formula:
\[ I = \frac{1}{2} m r^2 \]
where:
- \( m \) is the mass of the cylinder,
- \( r \) is the radius of the cylinder.

### Step 4: Substitute the Values
Since we are looking for the moment of inertia along the height of the cylinder, we can directly substitute the values into the formula:
\[ I = \frac{1}{2} m r^2 \]

### Step 5: Conclusion
Thus, the moment of inertia of the cylinder along its height is:
\[ I = \frac{m r^2}{2} \]

### Final Answer
The correct answer is \( \frac{m r^2}{2} \).

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