Two wires are made of the same material and have the same volume. The first wire has cross-sectional area A and the second wire has cross-sectional area 3A. If the length of the first wire is increased by on applying a force F, how much force is needed t

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Solution

Q.
Two wires are made of the same material and have the same volume. The first wire has cross-sectional area A and the second wire has cross-sectional area 3A. If the length of the first wire is increased by $Δ l$ on applying a force F, how much force is needed to stretch the second wire by the same amount [2018]

1 $9 F$

2 $6 F$

3 $4 F$

4 $F$

Ans.
(1)

Young's modulus, $Y = \frac{F l}{A Δ l}$

Since initial volume of wires are same and their areas of cross-sections are A and 3A, so lengths are $3 l$ and $l$ respectively.

For wire 1,

$Δ l = (\frac{F}{A Y}) 3 l$ ...(i)

For wire 2, let $F'$ force be applied,

$\frac{F'}{3 A} = Y \frac{Δ l}{l}$

$\Rightarrow Δ l = (\frac{F'}{3 A Y}) l$ ...(ii)

From equations (i) and (ii), $(\frac{F}{A Y}) 3 l = (\frac{F'}{3 A Y}) l \Rightarrow F' = 9 F$

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