A semicircular disc of mass M and radius R is free to rotate about its diameter. The moment of inertia of semicircular disc about a line perpendicular to its plane through centre is

Options

A. `(3)/(4) MR^(2)`
B. `(MR^(2))/(2)`
C. `(MR^(2))/(3)`
D. `(MR^(2))/(4)`

Explanation

To find the moment of inertia of a semicircular disc of mass \( M \) and radius \( R \) about a line perpendicular to its plane 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 through its center and perpendicular to its plane is given by the formula:
\[ I_{\text{full}} = \frac{1}{2} M R^2 \]

### Step 2: Moment of Inertia of the Semicircular Disc
Since we are dealing with a semicircular disc, we need to find the moment of inertia for just half of the full disc. The mass of the semicircular disc is \( M \), and its radius is \( R \). The moment of inertia for the semicircular disc about an axis through its center and perpendicular to its plane is:
\[ I_{\text{semicircular}} = \frac{1}{2} I_{\text{full}} = \frac{1}{2} \left(\frac{1}{2} M R^2\right) = \frac{1}{4} M R^2 \]

### Step 3: Final Result
Thus, the moment of inertia of the semicircular disc about a line perpendicular to its plane through the center is:
\[ I = \frac{1}{4} M R^2 \]

### Summary
The moment of inertia of a semicircular disc about a line perpendicular to its plane through the center is:
\[ \boxed{\frac{1}{4} M R^2} \]