A square plate has a moment of inertia `l_(0)` about an axis lying in its plane, passing through its centre and making an angle `theta` with one of the sides. Which graph represents the variation of l with `theta` ?

Options

A.
B.
C.
D.

Answer

To solve the problem of how the moment of inertia \( I \) of a square plate varies with the angle \( \theta \) of the axis passing through its center, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Moment of Inertia**:
The moment of inertia \( I \) of an object about a given axis is a measure of how difficult it is to change its rotational motion about that axis. For a square plate, the moment of inertia depends on the distribution of mass relative to the axis of rotation.

2. **Defining the Square Plate**:
Consider a square plate with uniform mass distribution. Let the side length of the square be \( a \). The mass of the plate can be denoted as \( m \).

3. **Identifying the Axis of Rotation**:
The axis of rotation is in the plane of the square and passes through its center, making an angle \( \theta \) with one of the sides of the square.

4. **Using the Parallel Axis Theorem**:
The moment of inertia about an axis through the center of mass (CM) can be calculated using the formula:
\[ I_{CM} = \frac{1}{6} m a^2 \]
This is the moment of inertia about an axis perpendicular to the plane of the square.

5. **Considering the Angle \( \theta \)**:
When the axis is tilted at an angle \( \theta \), the distribution of mass relative to the new axis changes. However, due to the symmetry of the square, the moment of inertia remains constant regardless of the angle \( \theta \).

6. **Conclusion**:
Since the axis divides the square symmetrically and the mass distribution remains uniform, the moment of inertia \( I \) does not depend on the angle \( \theta \). Thus, we can conclude that:
\[ I(\theta) = I_0 \quad \text{(constant)} \]

7. **Graph Representation**:
The graph representing the variation of \( I \) with \( \theta \) will be a horizontal line, indicating that \( I \) remains constant as \( \theta \) varies. Therefore, the correct graph is option C.