The curve for the moment of inertia of a sphere of constant mass M versus its radius will be:

Options

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B.
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D.

Answer

To determine the curve for the moment of inertia of a sphere of constant mass \( M \) versus its radius \( r \), we can follow these steps:

### Step 1: Understand the Formula for Moment of Inertia
The moment of inertia \( I \) of a solid sphere about an axis through its center is given by the formula:
\[ I = \frac{2}{5} M r^2 \]
where \( M \) is the mass of the sphere and \( r \) is its radius.

### Step 2: Identify the Variables
In this scenario, we have:
- Mass \( M \) is constant.
- Radius \( r \) is the variable.

### Step 3: Express Moment of Inertia in Terms of Radius
Since \( M \) is constant, we can express the moment of inertia as:
\[ I = k r^2 \]
where \( k = \frac{2}{5} M \) is a constant.

### Step 4: Analyze the Relationship
The equation \( I = k r^2 \) is a quadratic equation in terms of \( r \). This indicates that the moment of inertia \( I \) is proportional to the square of the radius \( r \).

### Step 5: Graph the Relationship
When plotting \( I \) (y-axis) against \( r \) (x-axis), the graph will be a parabola opening upwards, starting from the origin (0,0) since when \( r = 0 \), \( I \) will also be 0.

### Step 6: Conclusion
Thus, the curve for the moment of inertia of a sphere of constant mass \( M \) versus its radius \( r \) will be a parabola that passes through the origin.

### Final Answer
The correct answer is that the graph will be a parabola.

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