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 given that they have the same moment of inertia and their radii are in the ratio of 1:4.

### Step-by-Step Solution:

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

2. **Set Up the Given Information**:
Let:
- \(m_1\) and \(r_1\) be the mass and radius of the first ring.
- \(m_2\) and \(r_2\) be the mass and radius of the second ring.
According to the problem:
\[ \frac{r_1}{r_2} = \frac{1}{4} \]
This implies:
\[ r_2 = 4r_1 \]

3. **Equate the Moments of Inertia**:
Since the moment of inertia of both rings is the same:
\[ I_1 = I_2 \]
This gives us:
\[ m_1 r_1^2 = m_2 r_2^2 \]

4. **Substitute the Radius Ratio**:
Substitute \(r_2 = 4r_1\) into the equation:
\[ m_1 r_1^2 = m_2 (4r_1)^2 \]
Simplifying the right side:
\[ m_1 r_1^2 = m_2 \cdot 16 r_1^2 \]

5. **Cancel \(r_1^2\)**:
Since \(r_1^2\) is common on both sides (and not zero), we can cancel it:
\[ m_1 = 16 m_2 \]

6. **Find the Mass Ratio**:
Rearranging gives us the ratio of the masses:
\[ \frac{m_1}{m_2} = 16 \]
Therefore, the ratio of the masses \(m_1 : m_2\) is:
\[ 16 : 1 \]

### Final Answer:
The ratio of the masses of the two rings is \(16 : 1\).