Figure represents the moment of inertia of the solid sphere about an axis parallel to the diameter of the solid sphere and at a distance x from t. Which one of the following represents the variations of `I` with `x` ?

Options

A.
B.
C.
D.

Explanation

To solve the problem regarding the variation of the moment of inertia \( I \) of a solid sphere about an axis parallel to its diameter and at a distance \( x \) from it, we can follow these steps:

### Step 1: Understand the Moment of Inertia Formula
The moment of inertia \( I \) about an axis parallel to the diameter of the sphere can be expressed using the parallel axis theorem. The formula is given by:

\[ I = I_{CM} + m d^2 \]

where:
- \( I_{CM} \) is the moment of inertia about the center of mass,
- \( m \) is the mass of the sphere,
- \( d \) is the distance from the center of mass to the new axis of rotation.

### Step 2: Identify the Variables
In this case, the distance \( d \) is equal to \( x \) (the distance from the center of the sphere to the new axis). Thus, we can rewrite the equation as:

\[ I = I_{CM} + m x^2 \]

### Step 3: Determine the Moment of Inertia about the Center of Mass
For a solid sphere, the moment of inertia about its center of mass is given by:

\[ I_{CM} = \frac{2}{5} m r^2 \]

where \( r \) is the radius of the sphere.

### Step 4: Substitute \( I_{CM} \) into the Equation
Substituting \( I_{CM} \) into our equation for \( I \):

\[ I = \frac{2}{5} m r^2 + m x^2 \]

### Step 5: Analyze the Equation
The equation \( I = \frac{2}{5} m r^2 + m x^2 \) shows that \( I \) is a quadratic function of \( x \). The term \( m x^2 \) indicates that as \( x \) increases, \( I \) will increase quadratically.

### Step 6: Graph the Relationship
The graph of \( I \) versus \( x \) will be a parabola that opens upwards. Since the constant term \( \frac{2}{5} m r^2 \) is present, the parabola does not pass through the origin.

### Conclusion
Thus, the variation of the moment of inertia \( I \) with respect to the distance \( x \) is represented by a parabola that does not pass through the origin.

### Final Answer
The correct representation of the variation of \( I \) with \( x \) is a parabola not passing through the origin.

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