Two discs have same mass and thickness. Their materials are of densities `d_(1) and d_(2)`. The ratio of their moments of inertia about an axis passing through the centre and perpendicular to the plane is

Options

A. `d_(1) : d_(2)`
B. `d_(2):d_(1)`
C. `((d_(1))/(d_(2)))^(2)`
D. `((d_(2))/(d_(1)))^(2)`

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 1: Understand the relationship between mass, volume, and density
The mass \( m \) of an object can be expressed as:
\[ m = \text{Volume} \times \text{Density} \]
For a disc (which can be modeled as a cylinder), the volume \( V \) is given by:
\[ V = \pi r^2 t \]
where \( r \) is the radius and \( t \) is the thickness of the disc.

### Step 2: Set up the equations for the two discs
Let the densities of the two discs be \( d_1 \) and \( d_2 \), and their respective radii be \( r_1 \) and \( r_2 \). Since the mass of both discs is the same, we can write:
\[ m_1 = \pi r_1^2 t \cdot d_1 \]
\[ m_2 = \pi r_2^2 t \cdot d_2 \]
Since \( m_1 = m_2 \), we can equate the two expressions:
\[ \pi r_1^2 t \cdot d_1 = \pi r_2^2 t \cdot d_2 \]

### Step 3: Simplify the equation
We can cancel \( \pi \) and \( t \) from both sides (since they are equal for both discs):
\[ r_1^2 d_1 = r_2^2 d_2 \]
From this, we can derive the relationship between the radii:
\[ \frac{r_1^2}{r_2^2} = \frac{d_2}{d_1} \]

### Step 4: Write the moments of inertia for both discs
The moment of inertia \( I \) of a disc about an axis through its center and perpendicular to its plane is given by:
\[ I = \frac{1}{2} m r^2 \]
Thus, for the two discs:
\[ I_1 = \frac{1}{2} m_1 r_1^2 \]
\[ I_2 = \frac{1}{2} m_2 r_2^2 \]

### Step 5: Find the ratio of the moments of inertia
Now, we can find the ratio of the moments of inertia:
\[ \frac{I_1}{I_2} = \frac{\frac{1}{2} m_1 r_1^2}{\frac{1}{2} m_2 r_2^2} = \frac{m_1 r_1^2}{m_2 r_2^2} \]
Since \( m_1 = m_2 \), this simplifies to:
\[ \frac{I_1}{I_2} = \frac{r_1^2}{r_2^2} \]

### Step 6: Substitute the ratio of the radii
From the relationship we derived earlier:
\[ \frac{I_1}{I_2} = \frac{d_2}{d_1} \]

### Conclusion
Thus, the ratio of the moments of inertia of the two discs is:
\[ \frac{I_1}{I_2} = \frac{d_2}{d_1} \]

### Final Answer
The correct answer is option 2: \( d_2 : d_1 \).

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