Consider three bodies : a ring, a disc and a sphere. All the bodies have same mass and radius. All rotate about their axis through their respective centre of mass and perpendicular to the plane. What is the ratio of moment of inertia ?

Answer

To find the ratio of the moment of inertia of a ring, a disc, and a sphere, we will follow these steps:

### Step 1: Write down the formulas for the moment of inertia for each body.

1. **Moment of Inertia of a Ring (I_ring)**:
\[ I_{\text{ring}} = m r^2 \]

2. **Moment of Inertia of a Disc (I_disc)**:
\[ I_{\text{disc}} = \frac{1}{2} m r^2 \]

3. **Moment of Inertia of a Sphere (I_sphere)**:
\[ I_{\text{sphere}} = \frac{2}{5} m r^2 \]

### Step 2: Set up the ratio of the moments of inertia.

We need to find the ratio \( I_{\text{ring}} : I_{\text{disc}} : I_{\text{sphere}} \).

Substituting the formulas we have:
\[ I_{\text{ring}} : I_{\text{disc}} : I_{\text{sphere}} = m r^2 : \frac{1}{2} m r^2 : \frac{2}{5} m r^2 \]

### Step 3: Simplify the ratio.

To simplify, we can factor out \( m r^2 \) from each term:
\[ 1 : \frac{1}{2} : \frac{2}{5} \]

### Step 4: Eliminate the fractions.

To eliminate the fractions, we can multiply each term by a common multiple. The least common multiple of the denominators (1, 2, and 5) is 10. Thus, we multiply each term by 10:
\[ 10 \cdot 1 : 10 \cdot \frac{1}{2} : 10 \cdot \frac{2}{5} = 10 : 5 : 4 \]

### Step 5: Write the final ratio.

So, the final ratio of the moments of inertia is:
\[ I_{\text{ring}} : I_{\text{disc}} : I_{\text{sphere}} = 10 : 5 : 4 \]

### Final Answer:
The ratio of the moment of inertia of the ring, disc, and sphere is \( 10 : 5 : 4 \).

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