Question

A constant force is applied to a metal wire of length \( L \). Volume of the wire is constant. The extension produced is proportional to

• A. \( L^2 \)
• B. \( L^3 \)
• C. \( L \)
• D. \( L^{-2} \)

Hint:

When the volume of a wire is constant, its cross-sectional area is inversely proportional to its length. This leads to the extension being proportional to \( L^2 \).

Answer 1

The Correct Option is A

Step 1: Understanding the relationship.
The extension produced in a wire by applying a constant force is related to the wire's dimensions and material properties. The extension \( \Delta L \) is given by Hooke's Law, which is: \[ \Delta L = \frac{F L}{A Y} \] where \( F \) is the applied force, \( L \) is the length of the wire, \( A \) is the cross-sectional area, and \( Y \) is Young's modulus of the material.
Step 2: Using the constant volume condition.
Since the volume of the wire is constant, we have: \[ V = A L = \text{constant} \] Thus, \( A \propto \frac{1}{L} \).
Step 3: Proportionality of extension.
Substituting \( A \propto \frac{1}{L} \) into the expression for extension: \[ \Delta L \propto L^2 \]
Step 4: Conclusion.
Thus, the extension is proportional to \( L^2 \), which corresponds to option (A).