Two wheels are connected by a belt. The radius of larger wheel is three times that of the smaller one . What is the ratio of the rational inertia of larger wheel to the smaller wheel , when both wheels to the same angular momentum ?

Options

A. 3
B. 6
C. 9
D. 12

Explanation

To solve the problem, we need to find the ratio of the moment of inertia of the larger wheel to that of the smaller wheel, given that both wheels have the same angular momentum.

### Step-by-Step Solution:

1. **Define the Variables:**
- Let the radius of the smaller wheel be \( R_1 \).
- Then, the radius of the larger wheel \( R_2 = 3R_1 \).

2. **Angular Momentum Relationship:**
- The angular momentum \( L \) of a rotating object is given by the formula:
\[ L = I \omega \]
where \( I \) is the moment of inertia and \( \omega \) is the angular velocity.

3. **Set Up the Angular Momentum Equations:**
- For the smaller wheel:
\[ L_1 = I_1 \omega_1 \]
- For the larger wheel:
\[ L_2 = I_2 \omega_2 \]
- Given that \( L_1 = L_2 \), we have:
\[ I_1 \omega_1 = I_2 \omega_2 \]

4. **Relate Angular Velocities:**
- The linear velocities at the edges of both wheels must be equal since they are connected by a belt. Therefore:
\[ V_1 = V_2 \]
- This gives us:
\[ \omega_1 R_1 = \omega_2 R_2 \]
- Substituting \( R_2 = 3R_1 \):
\[ \omega_1 R_1 = \omega_2 (3R_1) \]
- Simplifying this, we find:
\[ \omega_1 = 3 \omega_2 \]

5. **Substitute Angular Velocities into Angular Momentum Equation:**
- Now substituting \( \omega_1 \) into the angular momentum equation:
\[ I_1 (3 \omega_2) = I_2 \omega_2 \]
- Dividing both sides by \( \omega_2 \) (assuming \( \omega_2 \neq 0 \)):
\[ 3 I_1 = I_2 \]

6. **Find the Ratio of Moments of Inertia:**
- Rearranging gives us:
\[ \frac{I_2}{I_1} = 3 \]

### Final Answer:
The ratio of the moment of inertia of the larger wheel to that of the smaller wheel is:
\[ \frac{I_2}{I_1} = 3 \]