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)`

Explanation

To find the moment of inertia of an elliptical disc with a uniform mass distribution, we can follow these steps:

### Step 1: Understand the Geometry of the Ellipse
The elliptical disc has a major axis of length 'r' and a minor axis of length 'd'. The moment of inertia depends on how mass is distributed relative to the axis of rotation.

### Step 2: Moment of Inertia of a Circular Disc
For a circular disc of radius 'R' and mass 'm', the moment of inertia about its central axis is given by the formula:
\[ I = \frac{1}{2} m R^2 \]
Here, we can consider the case where the ellipse is transformed into a circle with radius 'R' (where 'R' is the semi-major axis).

### Step 3: Moment of Inertia of the Ellipse
The moment of inertia for an elliptical disc can be derived from the moment of inertia of a circular disc. The moment of inertia of an elliptical disc about its axis can be expressed as:
\[ I_{ellipse} = \frac{1}{4} m (r^2 + d^2) \]
This formula accounts for the distribution of mass along both the major and minor axes.

### Step 4: Comparison with Circular Disc
Since the moment of inertia of the circular disc is greater than that of the elliptical disc, we can conclude:
\[ I_{ellipse} < I_{circle} \]
Thus, the moment of inertia of the elliptical disc will always be less than that of a circular disc with the same mass and major axis.

### Final Answer
The moment of inertia of the elliptical disc about its axis is:
\[ I_{ellipse} = \frac{1}{4} m (r^2 + d^2) \]