Two wires are made of the same material and have the same volume. However wire 1 has cross-sectional area A and wire 2 has cross-sectional area 3A. If the length of wire 1 increases by \Delta x on applying force F, how much force is needed to stretch wire 2 by the same amount?

Properties of Matter PYQ — JEE Main 2009 · Shift 0 · Q72 with Solution

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Correct answer: C

For a wire under tension,

\Delta L = \frac{FL}{AY}

where:

Since both wires are made of the same material, $Y$ is same for both.

Volume of a wire is

V = AL

For wire 1:

V = A L_1

For wire 2:

V = 3A \cdot L_2

Since volumes are equal,

A L_1 = 3A L_2

L_1 = 3L_2

So,

L_2 = \frac{L_1}{3}

Given wire 1 extends by $\Delta x$ under force $F$ :

\Delta x = \frac{F L_1}{A Y}

Let the required force on wire 2 be $F_2$ . Then,

\Delta x = \frac{F_2 L_2}{(3A)Y}

Substitute $L_2 = \frac{L_1}{3}$ :

\Delta x = \frac{F_2 \left(\frac{L_1}{3}\right)}{3AY} = \frac{F_2 L_1}{9AY}

Since both extensions are equal,

\frac{F L_1}{AY} = \frac{F_2 L_1}{9AY}

Cancel common terms $L_1, A, Y$ :

F = \frac{F_2}{9}

F_2 = 9F

\boxed{9F}

So the correct option is C.