Question

Two wires made of the same material are clamped rigidly at one end and pulled by the same force on the other end. The length and the radius of the first wire are three times those of the second wire. If \(x\) is the increase in the length of the first wire, then the increase in the length of the second wire is

• A. \(\dfrac{x}{3}\)
• B. \(3x\)
• C. \(9x\)
• D. \(\sqrt{3}x\)

Hint:

For a stretched wire, \[ \Delta L=\frac{FL}{AY}. \] Extension is directly proportional to length and inversely proportional to cross-sectional area. Since area depends on \(r^2\), even a small change in radius significantly affects the extension.

Answer 1

The Correct Option is B

Step 1: Apply the extension formula.
\( \Delta L = \frac{FL}{AY} \). Same material (Y same), same force F. Wire 1: \( L_1 = 3L_2 \), \( r_1 = 3r_2 \Rightarrow A_1 = 9A_2 \).

Step 2: Find the ratio of extensions.
\[ \frac{\Delta L_1}{\Delta L_2} = \frac{L_1/A_1}{L_2/A_2} = \frac{3L_2/(9A_2)}{L_2/A_2} = \frac{1}{3} \] So \( \Delta L_2 = 3\Delta L_1 = 3x \). \[ \boxed{3x} \]