Which of the following has the highest moment of inertia when each of them has the same mass and the same radius?

Options

A. A ring about any of its diameter
B. A disc about any of its diameter
C. A hollow sphere about any of its diameter
D. A solid sphere about any of its diameter

Explanation

To solve the question of which object has the highest moment of inertia when each has the same mass and radius, we need to calculate the moment of inertia for each of the given shapes: a ring, a disc, a hollow sphere, and a solid sphere.

### Step-by-Step Solution:

1. **Understanding Moment of Inertia**:
The moment of inertia (I) of an object depends on its mass distribution relative to the axis of rotation. The formula for moment of inertia is given by:
\[ I = m \cdot k^2 \]
where \( m \) is the mass and \( k \) is the radius of gyration.

2. **Moment of Inertia of a Ring**:
For a ring about an axis passing through its center and perpendicular to its plane:
\[ I_{\text{ring}} = m \cdot r^2 \]
Using the perpendicular axis theorem, we can find the moment of inertia about any diameter:
\[ I_x + I_y = I_z \implies 2I_x = I_z \implies I_x = \frac{1}{2} I_z = \frac{1}{2} m r^2 \]

3. **Moment of Inertia of a Disc**:
For a disc about an axis passing through its center and perpendicular to its plane:
\[ I_{\text{disc}} = \frac{1}{2} m r^2 \]
Using the perpendicular axis theorem:
\[ I_x + I_y = I_z \implies 2I_x = I_z \implies I_x = \frac{1}{4} m r^2 \]

4. **Moment of Inertia of a Solid Sphere**:
For a solid sphere about any diameter:
\[ I_{\text{solid sphere}} = \frac{2}{5} m r^2 \]

5. **Moment of Inertia of a Hollow Sphere**:
For a hollow sphere about any diameter:
\[ I_{\text{hollow sphere}} = \frac{2}{3} m r^2 \]

6. **Comparing the Moments of Inertia**:
Now we can summarize the moments of inertia for each shape:
- Ring: \( \frac{1}{2} m r^2 \)
- Disc: \( \frac{1}{4} m r^2 \)
- Solid Sphere: \( \frac{2}{5} m r^2 \)
- Hollow Sphere: \( \frac{2}{3} m r^2 \)

Converting these into decimal form for easier comparison:
- Ring: \( 0.5 m r^2 \)
- Disc: \( 0.25 m r^2 \)
- Solid Sphere: \( 0.4 m r^2 \)
- Hollow Sphere: \( 0.667 m r^2 \)

7. **Conclusion**:
Comparing these values, we find that the hollow sphere has the highest moment of inertia.

### Final Answer:
The object with the highest moment of inertia is the **hollow sphere**.