Two discs have same mass and thickness. Their materials are of densities `rho_(1)` and `rho_(2)` . The ratio of their moment of inertia about central axis will be

Options

A. `rho_(1):rho_(2)`
B. `rho_(1)rho_(2):1`
C. `1:rho_(1)rho_(2)`
D. `rho_(2):rho_(1)`

Explanation

To solve the problem of finding the ratio of the moments of inertia of two discs with the same mass and thickness but different densities, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Moment of Inertia**:
The moment of inertia \( I \) of a disc about its central axis is given by the formula:
\[ I = \frac{1}{2} m r^2 \]
where \( m \) is the mass of the disc and \( r \) is its radius.

2. **Expressing Mass in Terms of Density**:
The mass \( m \) of the disc can be expressed in terms of its density \( \rho \), volume \( V \), and thickness \( t \):
\[ m = \rho \cdot V \]
The volume \( V \) of a disc is given by:
\[ V = \pi r^2 t \]
Thus, we can write:
\[ m = \rho \cdot (\pi r^2 t) \]

3. **Finding the Radius**:
Since both discs have the same mass and thickness, we can equate their masses:
\[ m_1 = \rho_1 \cdot (\pi r_1^2 t) \quad \text{and} \quad m_2 = \rho_2 \cdot (\pi r_2^2 t) \]
Since \( m_1 = m_2 \), we have:
\[ \rho_1 \cdot (\pi r_1^2 t) = \rho_2 \cdot (\pi r_2^2 t) \]
The thickness \( t \) and \( \pi \) cancel out, leading to:
\[ \rho_1 r_1^2 = \rho_2 r_2^2 \]

4. **Expressing Radius in Terms of Density**:
Rearranging the above equation gives us:
\[ r_1^2 = \frac{\rho_2}{\rho_1} r_2^2 \]

5. **Substituting Back into Moment of Inertia**:
Now we can substitute \( r_1^2 \) into the moment of inertia formula for both discs:
\[ I_1 = \frac{1}{2} m_1 r_1^2 = \frac{1}{2} m \left(\frac{\rho_2}{\rho_1} r_2^2\right) \]
\[ I_2 = \frac{1}{2} m_2 r_2^2 = \frac{1}{2} m r_2^2 \]

6. **Finding the Ratio of Moments of Inertia**:
Now we can find the ratio of the moments of inertia:
\[ \frac{I_1}{I_2} = \frac{\frac{1}{2} m \left(\frac{\rho_2}{\rho_1} r_2^2\right)}{\frac{1}{2} m r_2^2} \]
The \( \frac{1}{2} m \) and \( r_2^2 \) cancel out:
\[ \frac{I_1}{I_2} = \frac{\rho_2}{\rho_1} \]

### Final Answer:
The ratio of the moments of inertia of the two discs is:
\[ \frac{I_1}{I_2} = \frac{\rho_2}{\rho_1} \]