First slide

Stress and strain

Question

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 $∆$ l on applying a force F, how much force is needed to stretch the second wire by the same amount?

Moderate

A

9F
B

6F
C

4F
D

F

Solution

$Young's modulus, Y = \frac{F l}{A Δ l}$
Since initial volume of wires are same
Their areas of cross sections are A and 3 A and lengths are 3l and l respectively.
For wire 1,
$Δ l = (\frac{F}{A Y}) 3 l . . . . . . . . . . . (i)$
$For wire 2, let F^{'} force is applied$
$\frac{F^{'}}{3 A} = Y \frac{Δ l}{l}$
$\Rightarrow Δ l = (\frac{F^{'}}{3 A Y}) l . . . . . . . . . . (i i)$
From the equations (i) and (ii)
$Δ l = (\frac{F}{A Y}) 3 l = (\frac{F^{'}}{3 A Y}) l \Rightarrow F^{'} :$

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