For a disc of given density and thickness, its moment of inertia varies with radius of the disc as :

Options

A. `1propR^(2)`
B. `1propR^(4)`
C. `1propR^(3)`
D. `1propR`

Answer

To find how the moment of inertia of a disc varies with its radius, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Variables**:
- Let the density of the disc be \( \rho \).
- Let the thickness of the disc be \( d \).
- Let the radius of the disc be \( R \).

2. **Consider an Elemental Ring**:
- Take an elemental ring at a distance \( r \) from the center of the disc with a thickness \( dr \).
- The circumference of this ring is \( 2\pi r \).

3. **Calculate the Mass of the Elemental Ring**:
- The mass \( dm \) of the elemental ring can be expressed as:
\[ dm = \rho \cdot d \cdot (2\pi r \cdot dr) \]
- Here, \( \rho \cdot d \) is the volume density times the thickness, giving the mass per unit area.

4. **Moment of Inertia of the Elemental Ring**:
- The moment of inertia \( dI \) of the elemental ring about the axis of rotation (which is through the center of the disc) is given by:
\[ dI = r^2 \cdot dm = r^2 \cdot (\rho \cdot d \cdot 2\pi r \cdot dr) \]
- Simplifying this, we have:
\[ dI = \rho \cdot d \cdot 2\pi r^3 \cdot dr \]

5. **Integrate to Find the Total Moment of Inertia**:
- To find the total moment of inertia \( I \) of the disc, integrate \( dI \) from \( r = 0 \) to \( r = R \):
\[ I = \int_0^R dI = \int_0^R \rho \cdot d \cdot 2\pi r^3 \cdot dr \]
- Since \( \rho \), \( d \), and \( 2\pi \) are constants, they can be taken out of the integral:
\[ I = \rho \cdot d \cdot 2\pi \int_0^R r^3 \cdot dr \]

6. **Calculate the Integral**:
- The integral \( \int_0^R r^3 \cdot dr \) can be calculated as:
\[ \int_0^R r^3 \cdot dr = \left[ \frac{r^4}{4} \right]_0^R = \frac{R^4}{4} \]

7. **Substitute Back to Find \( I \)**:
- Substituting the result of the integral back into the equation for \( I \):
\[ I = \rho \cdot d \cdot 2\pi \cdot \frac{R^4}{4} \]
- This simplifies to:
\[ I = \frac{\rho \cdot d \cdot \pi}{2} R^4 \]

8. **Conclusion**:
- Thus, the moment of inertia \( I \) of the disc varies with the radius \( R \) as:
\[ I \propto R^4 \]

### Final Answer:
The moment of inertia of a disc of given density and thickness varies with the radius \( R \) as \( I \propto R^4 \).