Find the moment of inertia of a section of circular disc of radius R about an axis perpendicualr to its plane as shown in the figure. The section has mass m and subtends an angle `theta` at the centre.

Options

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

Answer

To find the moment of inertia of a section of a circular disc of radius \( R \) about an axis perpendicular to its plane, we can follow these steps:

### Step 1: Understand the Geometry
The section of the circular disc subtends an angle \( \theta \) at the center. The mass of this section is given as \( m \).

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

### Step 3: Relate the Mass of the Section to the Full Disc
The mass \( m \) of the section that subtends an angle \( \theta \) at the center can be related to the mass \( M \) of the full disc. The angle \( \theta \) represents a fraction of the full circle (which is \( 2\pi \) radians). Therefore, the mass of the section is:
\[ m = M \cdot \frac{\theta}{2\pi} \]

### Step 4: Moment of Inertia of the Section
Since the moment of inertia is proportional to the mass, the moment of inertia \( I_{\text{section}} \) of the section can be calculated as:
\[ I_{\text{section}} = \frac{\theta}{2\pi} I_{\text{full}} = \frac{\theta}{2\pi} \cdot \frac{1}{2} M R^2 \]

### Step 5: Substitute for Mass \( M \)
Substituting \( M \) from the equation for \( m \):
\[ M = \frac{2\pi m}{\theta} \]
Now, substituting this back into the equation for \( I_{\text{section}} \):
\[ I_{\text{section}} = \frac{\theta}{2\pi} \cdot \frac{1}{2} \cdot \frac{2\pi m}{\theta} R^2 \]

### Step 6: Simplify the Expression
This simplifies to:
\[ I_{\text{section}} = \frac{m R^2}{2} \]

### Final Result
Thus, the moment of inertia of the section of the circular disc about the axis perpendicular to its plane is:
\[ I_{\text{section}} = \frac{m R^2}{2} \]