Two discs of same mass and same thickness have densities as `17 g//cm^(3)` and `51 g//cm^(3)`. The ratio of their moment of inertia about their central axes is

Options

A. `(1)/(3)`
B. `(2)/(3)`
C. `(3)/(1)`
D. `(3)/(2)`

Answer

To find the ratio of the moment of inertia of two discs with different densities but the same mass and thickness, we can follow these steps:

### Step 1: Understand the formula for the moment of inertia of a disc
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.

### Step 2: Relate mass, density, and volume
The mass \( m \) of the disc can also be expressed in terms of its density \( \rho \) and volume \( V \):
\[ m = \rho V \]
The volume \( V \) of the disc can be calculated as:
\[ V = \text{Area} \times \text{Thickness} = \pi r^2 t \]
where \( t \) is the thickness of the disc.

### Step 3: Substitute the volume into the mass equation
Substituting the volume into the mass equation gives:
\[ m = \rho (\pi r^2 t) \]

### Step 4: Solve for \( r^2 \)
Rearranging the equation for mass, we can express \( r^2 \):
\[ r^2 = \frac{m}{\pi t \rho} \]

### Step 5: Substitute \( r^2 \) back into the moment of inertia formula
Now, substituting \( r^2 \) back into the moment of inertia formula:
\[ I = \frac{1}{2} m \left(\frac{m}{\pi t \rho}\right) = \frac{m^2}{2 \pi t \rho} \]

### Step 6: Find the ratio of the moments of inertia
Let \( I_1 \) be the moment of inertia of the first disc (density \( \rho_1 = 17 \, \text{g/cm}^3 \)) and \( I_2 \) be the moment of inertia of the second disc (density \( \rho_2 = 51 \, \text{g/cm}^3 \)):
\[ I_1 = \frac{m^2}{2 \pi t \rho_1}, \quad I_2 = \frac{m^2}{2 \pi t \rho_2} \]
Now, the ratio of the moments of inertia is:
\[ \frac{I_1}{I_2} = \frac{\frac{m^2}{2 \pi t \rho_1}}{\frac{m^2}{2 \pi t \rho_2}} = \frac{\rho_2}{\rho_1} \]

### Step 7: Substitute the densities to find the ratio
Substituting the given densities:
\[ \frac{I_1}{I_2} = \frac{51}{17} = 3 \]

### Conclusion
Thus, the ratio of the moments of inertia of the two discs is:
\[ \frac{I_1}{I_2} = 3 \]