First slide

Elastic behaviour of solids

Question

A wire of cross section A is stretched horizontally between two clamps located 2l m apart. A weight W kg is suspended from the mid-point of the wire. If the mid-point sags vertically through a distance x < 1 the strain produced is

Moderate

A

$\frac{2 x^{2}}{l^{2}}$
B

$\frac{x^{2}}{l^{2}}$
C

$\frac{x^{2}}{2 l^{2}}$
D

None of these

Solution

Here change in length is
$\begin{array}{l} Δl = (AC + BC) - 2 l \\ = 2 {(l^{2} + x^{2})}^{1 / 2} - 2 l \\ = 2 l {(1 + \frac{x^{2}}{l^{2}})}^{1 / 2} - 2 l \\ = 2 l (1 + \frac{1}{2} \frac{x^{2}}{l^{2}}) - 2 l = \frac{x^{2}}{l} \\ ∴ Strain = \frac{Δl}{2 l} = \frac{x^{2}}{2 l^{2}} \end{array}$

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