A steel ring of radius r and cross section area A is fitted on to a wooden disc of radius `R(Rgtr)`. If Young's modulus be E, then the force with which the steel ring is expanded is

Options

A. `AE(R )/(r )`
B. `AE((R-r)/(r ))`
C. `(E)/(A)((R-r)/(A))`
D. `(Er)/(AR)`

Explanation

To find the force with which the steel ring is expanded when fitted onto a wooden disc, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Parameters**:
- Let the radius of the steel ring be \( r \).
- Let the radius of the wooden disc be \( R \).
- Let the cross-sectional area of the steel ring be \( A \).
- Let Young's modulus of the steel be \( E \).

2. **Determine the Change in Circumference**:
- The initial circumference of the steel ring is given by:
\[ C_{\text{initial}} = 2\pi r \]
- The circumference of the wooden disc is:
\[ C_{\text{final}} = 2\pi R \]
- The change in length (circumference) when the ring is fitted onto the disc is:
\[ \Delta C = C_{\text{final}} - C_{\text{initial}} = 2\pi R - 2\pi r = 2\pi (R - r) \]

3. **Calculate the Strain**:
- Strain is defined as the change in length divided by the original length. Here, the original length is the initial circumference of the ring:
\[ \text{Strain} = \frac{\Delta C}{C_{\text{initial}}} = \frac{2\pi (R - r)}{2\pi r} = \frac{R - r}{r} \]

4. **Apply Young's Modulus**:
- Young's modulus \( E \) is defined as the ratio of stress to strain:
\[ E = \frac{\text{Stress}}{\text{Strain}} \]
- Stress is defined as force per unit area:
\[ \text{Stress} = \frac{F}{A} \]
- Therefore, we can write:
\[ E = \frac{F/A}{(R - r)/r} \]

5. **Rearranging to Find Force**:
- Rearranging the equation to solve for force \( F \):
\[ F = E \cdot A \cdot \frac{(R - r)}{r} \]

6. **Final Expression**:
- Thus, the force with which the steel ring is expanded is:
\[ F = E \cdot A \cdot \frac{(R - r)}{r} \]