Question

F denotes force, A denotes area, L denotes length, and \( \Delta L \) is the change in length due to the applied force. Assuming linear elasticity, select a relationship where the constant of proportionality is a material property independent of the dimensions of the body.

• A. \( F \propto \Delta L \)
• B. \( F \propto \frac{\Delta L}{L} \)
• C. \( F \propto A \times \frac{\Delta L}{L} \)
• D. \( F \propto \frac{\Delta L \times L}{A} \)

Hint:

Hooke’s Law: \( F = E \cdot A \cdot \frac{\Delta L}{L} \). Here, \( E \) is a material constant and independent of the object's dimensions.

Answer 1

The Correct Option is C

Step 1: Applying Hooke’s Law.
In linear elasticity, Hooke’s Law relates stress and strain: \[ {Stress} = E \times {Strain} \quad \Rightarrow \quad \frac{F}{A} = E \cdot \frac{\Delta L}{L} \] Step 2: Rearranging the expression. \[ F = E \cdot A \cdot \frac{\Delta L}{L} \] This shows that force \( F \) is proportional to \( A \cdot \frac{\Delta L}{L} \), where \( E \) (Young’s modulus) is a material property independent of body dimensions.