Two rings having same moment of inertia have their radii in the ratio 1 : 4. Their masses will be in the ratio

Options

A. `4 : 1`
B. `16 : 1`
C. `1 : 4`
D. `2 : 1`

Answer

To solve the problem, we need to find the ratio of the masses of two rings that have the same moment of inertia but different radii. Let's go through the solution step by step.

### Step-by-Step Solution:

1. **Understand the Moment of Inertia Formula**:
The moment of inertia \( I \) of a ring about an axis passing through its center and perpendicular to its plane is given by the formula:
\[ I = m \cdot r^2 \]
where \( m \) is the mass of the ring and \( r \) is its radius.

2. **Set Up the Equation for Two Rings**:
Let the masses of the two rings be \( m_1 \) and \( m_2 \), and their radii be \( r_1 \) and \( r_2 \) respectively. According to the problem, the moment of inertia of both rings is the same:
\[ I_1 = I_2 \]
This implies:
\[ m_1 \cdot r_1^2 = m_2 \cdot r_2^2 \]

3. **Express the Mass Ratio**:
Rearranging the equation gives us:
\[ \frac{m_1}{m_2} = \frac{r_2^2}{r_1^2} \]

4. **Use the Given Ratio of Radii**:
The problem states that the radii are in the ratio \( r_1 : r_2 = 1 : 4 \). This means:
\[ \frac{r_1}{r_2} = \frac{1}{4} \]
Therefore, we can express \( r_2 \) in terms of \( r_1 \):
\[ r_2 = 4 \cdot r_1 \]

5. **Substitute the Radius Ratio into the Mass Ratio**:
Now substituting \( r_2 \) into the mass ratio equation:
\[ \frac{m_1}{m_2} = \frac{(4 \cdot r_1)^2}{r_1^2} = \frac{16 \cdot r_1^2}{r_1^2} = 16 \]

6. **Final Ratio of Masses**:
Thus, the ratio of the masses \( m_1 : m_2 \) is:
\[ m_1 : m_2 = 16 : 1 \]

### Conclusion:
The masses of the two rings are in the ratio \( 16 : 1 \).