Moment of inertia (M.I.) of four bodies, having same mass and radius, are reported as ,
`I_1` = M.I. of thin circular ring about its diameter.
`I_2` = M.I. of circular disc about an axis perpendicular to the disc and going through the centre,
`I_3` = M.I. of solid cylinder about its axis and
`I_4` = M.I. of solid sphere about its diameter. Then :

Options

A. `I_2 + I_3 lt I_2 + I_4`
B. `I_1 + I_2 = I_3 + 5/2 I_4`
C. `I_1 = I_2 = I_3 gt I_4`
D. `I_1 =I_2 = I_3 lt I_4`

Answer

To solve the problem, we need to calculate the moment of inertia (M.I.) for each of the four bodies mentioned and compare them.

### Step 1: Calculate the Moment of Inertia for Each Body

1. **For the thin circular ring about its diameter (`I1`)**:
The moment of inertia of a thin circular ring about its diameter is given by:
\[ I_1 = \frac{1}{2} m r^2 \]

2. **For the circular disc about an axis perpendicular to the disc and going through the center (`I2`)**:
The moment of inertia of a circular disc about an axis perpendicular to the disc through the center is:
\[ I_2 = \frac{1}{2} m r^2 \]

3. **For the solid cylinder about its axis (`I3`)**:
The moment of inertia of a solid cylinder about its axis is:
\[ I_3 = \frac{1}{2} m r^2 \]

4. **For the solid sphere about its diameter (`I4`)**:
The moment of inertia of a solid sphere about its diameter is:
\[ I_4 = \frac{2}{5} m r^2 \]

### Step 2: Compare the Values of the Moments of Inertia

Now we have:
- \( I_1 = \frac{1}{2} m r^2 = 0.5 m r^2 \)
- \( I_2 = \frac{1}{2} m r^2 = 0.5 m r^2 \)
- \( I_3 = \frac{1}{2} m r^2 = 0.5 m r^2 \)
- \( I_4 = \frac{2}{5} m r^2 = 0.4 m r^2 \)

### Step 3: Analyze the Relationships

From the calculations:
- \( I_1 = I_2 = I_3 = 0.5 m r^2 \)
- \( I_4 = 0.4 m r^2 \)

Now we can compare:
- \( I_1 = I_2 = I_3 > I_4 \)

### Step 4: Check the Options

1. **Option 1**: \( I_2 + I_3 < I_2 + I_4 \) - This is incorrect since \( I_3 > I_4 \).
2. **Option 2**: \( I_1 + I_2 = I_3 + \frac{5}{4} I_4 \) - This is incorrect upon calculation.
3. **Option 3**: \( I_1 = I_2 \) and \( I_3 > I_4 \) - This is correct.
4. **Option 4**: \( I_1 = I_2 = I_3 \) and \( I_3 < I_4 \) - This is incorrect.

### Conclusion

The correct option is **Option 3**: \( I_1 = I_2 \) and \( I_3 > I_4 \).