Locus of all the points in a plane on which the moment of inertia about all mutually parallel axes of a rigid body is same throughout is

Options

A. a straight line
B. a circle
C. a parabola
D. an ellipse

Answer

To determine the locus of all points in a plane where the moment of inertia about all mutually parallel axes of a rigid body is the same, we can follow these steps:

### Step 1: Understand the Moment of Inertia
The moment of inertia \( I \) of a rigid body about an axis is defined as:
\[ I = \sum m_i r_i^2 \]
where \( m_i \) is the mass of the individual particles and \( r_i \) is the distance from the axis of rotation.

### Step 2: Apply the Parallel Axis Theorem
According to the parallel axis theorem, the moment of inertia about a parallel axis can be expressed as:
\[ I = I_{cm} + Mh^2 \]
where:
- \( I_{cm} \) is the moment of inertia about the center of mass,
- \( M \) is the total mass of the body,
- \( h \) is the distance between the two parallel axes.

### Step 3: Analyze the Condition for Constant Moment of Inertia
For the moment of inertia to remain constant for all parallel axes, the term \( Mh^2 \) must not change with different positions of the axis. This implies that the distance \( h \) must be constant.

### Step 4: Determine the Locus of Points
If \( h \) is constant, then the locus of points that maintain this constant distance from a fixed point (the center of mass) forms a circle. This is because all points on a circle are equidistant from the center.

### Conclusion
Thus, the locus of all points in a plane where the moment of inertia about all mutually parallel axes of a rigid body is the same is a **circle**.

### Final Answer
The correct option is a **circle**.

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