The moment of inertia of an elliptical disc of uniform mass distribution of mass 'm' major axis 'r', minor axis 'd' about its axis is :

Options

A. `=(mr^2)/2`
B. `=(md^2)/2`
C. `gt(mr^2)/2`
D. `lt(mr^2)/2`

Answer

To find the moment of inertia of an elliptical disc of uniform mass distribution with mass 'm', major axis 'R', and minor axis 'd' about its axis, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- An elliptical disc has a major axis (R) and a minor axis (d). The semi-major axis is \( a = \frac{R}{2} \) and the semi-minor axis is \( b = \frac{d}{2} \).

2. **Moment of Inertia of a Circular Disc**:
- The moment of inertia \( I \) of a circular disc about its central axis is given by the formula:
\[ I = \frac{1}{2} m r^2 \]
- Here, \( r \) is the radius of the circular disc.

3. **Relating Elliptical Disc to Circular Disc**:
- To find the moment of inertia of the elliptical disc, we can compare it to a circular disc.
- The moment of inertia of an elliptical disc can be derived by considering it as a collection of infinitesimally thin circular discs stacked along the minor axis.

4. **Calculating Moment of Inertia for the Elliptical Disc**:
- The moment of inertia about the major axis (R) can be expressed as:
\[ I_{major} = \frac{1}{2} m a^2 \]
- The moment of inertia about the minor axis (d) can be expressed as:
\[ I_{minor} = \frac{1}{2} m b^2 \]

5. **Final Expression for Moment of Inertia**:
- For the elliptical disc, the moment of inertia about its axis can be approximated as:
\[ I_{ellipse} = \frac{1}{4} m (R^2 + d^2) \]
- This formula accounts for the distribution of mass in both axes.

6. **Conclusion**:
- Therefore, the moment of inertia of an elliptical disc of uniform mass distribution about its axis is:
\[ I = \frac{1}{4} m (R^2 + d^2) \]