An iron bar of length L, cross-section area A and Young's modulus Y is pulled by a force F from both ends so as to produce an elongation l . Which of the following statements is correct?

Options

A. `l prop Y`
B. `l prop 1/A`
C. `l prop A`
D. `lprop 1/L`

Explanation

To solve the problem, we need to analyze the relationship between the force applied to the iron bar, its elongation, cross-sectional area, and Young's modulus.

### Step-by-Step Solution:

1. **Understanding Young's Modulus**:
Young's modulus (Y) is defined as the ratio of longitudinal stress to longitudinal strain. Mathematically, it can be expressed as:
\[ Y = \frac{\text{Stress}}{\text{Strain}} \]
where:
- Stress = \(\frac{F}{A}\) (Force per unit area)
- Strain = \(\frac{\Delta L}{L}\) (Change in length per original length)

2. **Expressing Stress and Strain**:
- Stress can be calculated as:
\[ \text{Stress} = \frac{F}{A} \]
- Strain can be expressed as:
\[ \text{Strain} = \frac{l}{L} \]
where \(l\) is the elongation and \(L\) is the original length of the bar.

3. **Substituting Stress and Strain into Young's Modulus**:
Substituting the expressions for stress and strain into the formula for Young's modulus, we get:
\[ Y = \frac{\frac{F}{A}}{\frac{l}{L}} = \frac{F \cdot L}{A \cdot l} \]

4. **Rearranging the Equation**:
Rearranging the equation gives us:
\[ F = \frac{Y \cdot A \cdot l}{L} \]
This shows that the force \(F\) is directly proportional to the elongation \(l\) and Young's modulus \(Y\), and inversely proportional to the length \(L\) and the cross-sectional area \(A\).

5. **Analyzing the Relationship**:
From the rearranged equation, we can conclude that:
- If the cross-sectional area \(A\) increases, the force \(F\) required to produce the same elongation \(l\) decreases.
- If the length \(L\) increases, the force \(F\) required to produce the same elongation \(l\) also decreases.

6. **Conclusion**:
The correct statement is that the elongation \(l\) is inversely proportional to the cross-sectional area \(A\) of the bar, given a constant force \(F\) and Young's modulus \(Y\).