Moment of inertia of an ellipse shaped wire of mass 'm', semimajor axis 'a' about an axis passing through the centre and perpendicular to the plane of wire can be :

Options

A. `ma^(2)`
B. `1.2 ma^(2)`
C. `1.4 ma^(2)`
D. `0.6 ma^(2)`

Explanation

To find the moment of inertia of an ellipse-shaped wire of mass 'm' and semi-major axis 'a' about an axis passing through the center and perpendicular to the plane of the wire, we can follow these steps:

### Step 1: Understand the Geometry
The ellipse has a semi-major axis 'a' and a semi-minor axis 'b'. For the moment of inertia calculation, we need to visualize the ellipse and how it relates to a circle of radius 'a'.

### Step 2: Moment of Inertia of a Circle
The moment of inertia \( I \) of a circular wire of mass \( m \) and radius \( a \) about an axis perpendicular to its plane through its center is given by:
\[ I_{\text{circle}} = m a^2 \]

### Step 3: Compare the Ellipse with the Circle
The ellipse can be thought of as a stretched circle. The points on the ellipse are at a distance less than or equal to 'a' from the center. Therefore, the moment of inertia of the ellipse will be less than that of the circle because the mass distribution is not as far from the axis of rotation compared to the circle.

### Step 4: Moment of Inertia of the Ellipse
The moment of inertia \( I \) of the ellipse about the same axis can be expressed in terms of the moment of inertia of the circle:
\[ I_{\text{ellipse}} < I_{\text{circle}} = m a^2 \]
This implies that the moment of inertia of the ellipse is less than \( m a^2 \).

### Step 5: Conclusion
Since the moment of inertia of the ellipse is less than that of the circle, we can conclude that the moment of inertia of the ellipse-shaped wire is:
\[ I_{\text{ellipse}} < m a^2 \]

### Final Answer
The moment of inertia of the ellipse-shaped wire of mass 'm' and semi-major axis 'a' about an axis passing through the center and perpendicular to the plane of the wire can be represented as:
\[ I_{\text{ellipse}} < m a^2 \]