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. `Mr^(2)`
B. `(1)/(2) MR^(2)`
C. `(1)/(4) Mr^(2)`
D. `(2)/(5) 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 1: Understand the Geometry
The semicircular disc can be visualized as half of a full circular disc. The moment of inertia of a full circular disc about an axis perpendicular to its plane through its center is given by the formula:
\[ I_{\text{full}} = \frac{1}{2} M r^2 \]

### Step 2: Consider the Semicircular Disc
Since we are dealing with a semicircular disc, we need to find the moment of inertia for this half-disc. The mass of the semicircular disc is \( M \), and its radius is \( r \).

### Step 3: Use the Parallel Axis Theorem
To find the moment of inertia of the semicircular disc about the desired axis, we can use the fact that the moment of inertia of the semicircular disc can be derived from the full disc. The moment of inertia of the semicircular disc about the same axis through the center can be expressed as:
\[ I_{\text{semicircular}} = \frac{1}{2} I_{\text{full}} = \frac{1}{2} \left( \frac{1}{2} M r^2 \right) = \frac{1}{4} M r^2 \]

### Step 4: Final Calculation
However, we need to consider the distribution of mass in the semicircular disc. The moment of inertia for a semicircular disc about the axis perpendicular to the plane through the center is actually:
\[ I = \frac{1}{2} M r^2 \]

### Conclusion
Thus, 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:
\[ I = \frac{1}{2} M r^2 \]