The M.I. of a ring and disc of same mass and radius about their geometric axes will be:

Options

A. more for the disc
B. more for the ring
C. eual in both cases
D. none of these.

Answer

To solve the problem of comparing the moment of inertia (M.I.) of a ring and a disc of the same mass and radius about their geometric axes, we can follow these steps:

### Step 1: Understand the Definitions
The moment of inertia (I) is a measure of an object's resistance to changes in its rotation. It depends on the mass distribution of the object relative to the axis of rotation.

### Step 2: Identify the Objects
We have two objects:
- A ring with mass \( m \) and radius \( r \).
- A disc with mass \( m \) and radius \( r \).

### Step 3: Moment of Inertia of the Ring
The formula for the moment of inertia of a ring about an axis passing through its center and perpendicular to its plane is given by:
\[ I_{\text{ring}} = m r^2 \]

### Step 4: Moment of Inertia of the Disc
The formula for the moment of inertia of a disc about an axis passing through its center and perpendicular to its plane is given by:
\[ I_{\text{disc}} = \frac{1}{2} m r^2 \]

### Step 5: Compare the Two Moments of Inertia
Now, we can compare the two calculated moments of inertia:
- For the ring: \( I_{\text{ring}} = m r^2 \)
- For the disc: \( I_{\text{disc}} = \frac{1}{2} m r^2 \)

### Step 6: Conclusion
From the above expressions, it is clear that:
\[ I_{\text{ring}} > I_{\text{disc}} \]
Thus, the moment of inertia of the ring is greater than that of the disc.

### Final Answer
The moment of inertia of the ring is greater than that of the disc.