The M.I. of a rectangular plane lamina of mass M, length 'l' and breadth 'b' about an axis passing through its centre and perpendicular to plane of lamina is

Options

A. `(Ml^(2))/(12)`
B. `(Mb^(2))/(12)`
C. `(M(l^(2)+b^(2)))/(12)`
D. `(M)/(12) ((l^(2))/(b^(2))+(b^(2))/(4))`

Answer

To find the moment of inertia (M.I.) of a rectangular plane lamina of mass \( M \), length \( l \), and breadth \( b \) about an axis passing through its center and perpendicular to the plane of the lamina, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Geometry**:
- We have a rectangular lamina with length \( l \) and breadth \( b \). The mass of the lamina is \( M \).
- The axis of rotation is perpendicular to the plane of the lamina and passes through its center.

2. **Use the Formula for Moment of Inertia**:
- The moment of inertia \( I \) for a rectangular lamina about an axis perpendicular to its plane and passing through its center is given by the formula:
\[ I = \frac{1}{12} M (l^2 + b^2) \]
- This formula is derived from the integration of the mass distribution over the area of the rectangle.

3. **Substitute Values**:
- Substitute the values of \( M \), \( l \), and \( b \) into the formula:
\[ I = \frac{1}{12} M (l^2 + b^2) \]

4. **Final Result**:
- Therefore, the moment of inertia of the rectangular lamina about the specified axis is:
\[ I = \frac{1}{12} M (l^2 + b^2) \]