The moment of inertia of a door of mass `m`, length `2 l` and width `l` about its longer side is.

Options

A. `(11ml^(2))/24`
B. `(5ml^(2))/24`
C. `(ml^(2))/3`
D. none of these

Explanation

To find the moment of inertia of a door of mass \( m \), length \( 2l \), and width \( l \) about its longer side, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Geometry**:
- The door has a mass \( m \), length \( 2l \), and width \( l \).
- We will calculate the moment of inertia about the longer side, which is along the length \( 2l \).

2. **Set Up the Coordinate System**:
- Place the door in the coordinate system such that the longer side (length \( 2l \)) is along the x-axis.
- The center of the door will be at the origin, so the edges will be at \( -l \) and \( l \) along the x-axis, and the width \( l \) extends along the y-axis from \( -\frac{l}{2} \) to \( \frac{l}{2} \).

3. **Consider an Elemental Strip**:
- Take a vertical strip of width \( dx \) at a distance \( x \) from the origin. The height of this strip is \( l \).
- The elemental mass \( dm \) of this strip can be expressed as:
\[ dm = \frac{m}{\text{Area}} \cdot \text{Area of strip} = \frac{m}{2l \cdot l} \cdot (l \cdot dx) = \frac{m}{2l} \cdot dx \]

4. **Calculate the Moment of Inertia of the Strip**:
- The moment of inertia \( dI \) of this elemental strip about the axis (longer side) is given by:
\[ dI = dm \cdot x^2 = \left(\frac{m}{2l} \cdot dx\right) \cdot x^2 \]

5. **Integrate to Find Total Moment of Inertia**:
- The total moment of inertia \( I \) is obtained by integrating \( dI \) from \( -l \) to \( l \):
\[ I = \int_{-l}^{l} dI = \int_{-l}^{l} \left(\frac{m}{2l} \cdot x^2 \cdot dx\right) \]
- Factor out the constant:
\[ I = \frac{m}{2l} \int_{-l}^{l} x^2 \, dx \]

6. **Evaluate the Integral**:
- The integral \( \int_{-l}^{l} x^2 \, dx \) can be calculated as:
\[ \int x^2 \, dx = \frac{x^3}{3} \Big|_{-l}^{l} = \frac{l^3}{3} - \left(-\frac{l^3}{3}\right) = \frac{2l^3}{3} \]
- Substitute this back into the equation for \( I \):
\[ I = \frac{m}{2l} \cdot \frac{2l^3}{3} = \frac{ml^2}{3} \]

7. **Final Result**:
- The moment of inertia of the door about its longer side is:
\[ I = \frac{ml^2}{3} \]