A body has mass M and radius 2r. If the body assumes four shapes, solid cylinder, solid sphere, disc and ring of same radius, then the moment of inertia will be maximum about axis through centre of mass for:

Options

A. ring
B. solid sphere
C. disc
D. solid cylinder

Answer

To solve the problem of finding which shape has the maximum moment of inertia about an axis through the center of mass, we will calculate the moment of inertia for each of the four shapes given: solid cylinder, solid sphere, disc, and ring.

### Step-by-Step Solution:

1. **Identify the Shapes and Their Formulas**:
- We have four shapes: solid cylinder, solid sphere, disc, and ring.
- The formulas for the moment of inertia (I) for each shape about an axis through the center of mass are:
- Solid Cylinder: \( I = \frac{1}{2} M R^2 \)
- Solid Sphere: \( I = \frac{2}{5} M R^2 \)
- Disc: \( I = \frac{1}{2} M R^2 \)
- Ring: \( I = M R^2 \)

2. **Substitute the Given Values**:
- The mass of the body is \( M \) and the radius is \( 2r \).
- We will substitute \( R = 2r \) into each formula.

3. **Calculate Moment of Inertia for Each Shape**:
- **Solid Cylinder**:
\[ I_{cylinder} = \frac{1}{2} M (2r)^2 = \frac{1}{2} M (4r^2) = 2 M r^2 \]

- **Solid Sphere**:
\[ I_{sphere} = \frac{2}{5} M (2r)^2 = \frac{2}{5} M (4r^2) = \frac{8}{5} M r^2 \]

- **Disc**:
\[ I_{disc} = \frac{1}{2} M (2r)^2 = \frac{1}{2} M (4r^2) = 2 M r^2 \]

- **Ring**:
\[ I_{ring} = M (2r)^2 = M (4r^2) = 4 M r^2 \]

4. **Compare the Moments of Inertia**:
- Now we have the moments of inertia for each shape:
- Solid Cylinder: \( 2 M r^2 \)
- Solid Sphere: \( \frac{8}{5} M r^2 \) (which is \( 1.6 M r^2 \))
- Disc: \( 2 M r^2 \)
- Ring: \( 4 M r^2 \)

5. **Determine the Maximum Moment of Inertia**:
- Comparing the values:
- \( 2 M r^2 \) (Solid Cylinder)
- \( 1.6 M r^2 \) (Solid Sphere)
- \( 2 M r^2 \) (Disc)
- \( 4 M r^2 \) (Ring)
- The maximum moment of inertia is \( 4 M r^2 \) for the ring.

### Conclusion:
The moment of inertia will be maximum about the axis through the center of mass for the **ring**.

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