The moment of inertia of a uniform semicircular disc of mass `M` and radius `r` about a line perpendicular to the plane of the disc through the center is

Options

A. `1/2 Mr^(2)`
B. `Mt^(2)`
C. `2/5 Mr^(2)`
D. `1/4 Mr^(2)`

Explanation

To find the moment of inertia of a uniform semicircular disc of mass \( M \) and radius \( r \) about a line perpendicular to the plane of the disc through the center, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Geometry**:
- We have a semicircular disc with radius \( r \) and mass \( M \). The axis of rotation is perpendicular to the plane of the disc and passes through the center.

2. **Moment of Inertia of a Full Circular Disc**:
- The moment of inertia \( I \) of a full circular disc about an axis perpendicular to its plane through the center is given by:
\[ I_{\text{full}} = \frac{1}{2} M r^2 \]

3. **Relate the Semicircular Disc to the Full Disc**:
- A semicircular disc can be considered as half of a full circular disc. Therefore, the moment of inertia of the semicircular disc about the same axis can be derived from the full disc's moment of inertia.
- Since the semicircular disc has half the mass of the full disc, we can express its moment of inertia as:
\[ I_{\text{semicircular}} = \frac{1}{2} I_{\text{full}} \]

4. **Substituting the Moment of Inertia of the Full Disc**:
- Now substituting the expression for the full disc's moment of inertia:
\[ I_{\text{semicircular}} = \frac{1}{2} \left(\frac{1}{2} M r^2\right) = \frac{1}{4} M r^2 \]

5. **Final Result**:
- Therefore, the moment of inertia of the uniform semicircular disc about the specified axis is:
\[ I = \frac{1}{4} M r^2 \]

### Conclusion:
The moment of inertia of a uniform semicircular disc of mass \( M \) and radius \( r \) about a line perpendicular to the plane of the disc through the center is \( \frac{1}{4} M r^2 \).