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 determine which shape has the maximum moment of inertia about an axis through its center of mass, we will calculate the moment of inertia for each of the four shapes: solid cylinder, solid sphere, disc, and ring. The mass \( M \) and radius \( 2r \) are given for all shapes.

### Step 1: Moment of Inertia of a Solid Cylinder
The formula for the moment of inertia of a solid cylinder about its central axis is given by:

\[ I_{\text{cylinder}} = \frac{1}{2} M R^2 \]

Substituting \( R = 2r \):

\[ I_{\text{cylinder}} = \frac{1}{2} M (2r)^2 = \frac{1}{2} M \cdot 4r^2 = 2M r^2 \]

### Step 2: Moment of Inertia of a Solid Sphere
The formula for the moment of inertia of a solid sphere about its central axis is:

\[ I_{\text{sphere}} = \frac{2}{5} M R^2 \]

Substituting \( R = 2r \):

\[ I_{\text{sphere}} = \frac{2}{5} M (2r)^2 = \frac{2}{5} M \cdot 4r^2 = \frac{8}{5} M r^2 \]

### Step 3: Moment of Inertia of a Disc
The formula for the moment of inertia of a disc about its central axis is:

\[ I_{\text{disc}} = \frac{1}{2} M R^2 \]

Substituting \( R = 2r \):

\[ I_{\text{disc}} = \frac{1}{2} M (2r)^2 = \frac{1}{2} M \cdot 4r^2 = 2M r^2 \]

### Step 4: Moment of Inertia of a Ring
The formula for the moment of inertia of a ring about its central axis is:

\[ I_{\text{ring}} = M R^2 \]

Substituting \( R = 2r \):

\[ I_{\text{ring}} = M (2r)^2 = M \cdot 4r^2 = 4M r^2 \]

### Step 5: Comparison of Moments of Inertia
Now we compare the moments of inertia calculated:

- Solid Cylinder: \( I_{\text{cylinder}} = 2M r^2 \)
- Solid Sphere: \( I_{\text{sphere}} = \frac{8}{5} M r^2 \)
- Disc: \( I_{\text{disc}} = 2M r^2 \)
- Ring: \( I_{\text{ring}} = 4M r^2 \)

### Conclusion
The maximum moment of inertia is for the ring, which is \( 4M r^2 \).

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

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