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A wire of cross-section A is stretched horizontally between two clamps located 2 l metre apart. A weight W kg is suspended from the mid point of the wire. If the mid pint sags vertically through a distance x < I. Find the Young's modulus of the material and the value of extension x.

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${(\frac{Wl}{YA})}^{\frac{1}{3}}$

${(\frac{YA}{Wl})}^{\frac{1}{3}}$

$\frac{1}{l} {[\frac{WA}{Y}]}^{\frac{1}{3}}$

$l {[\frac{W}{YA}]}^{\frac{1}{3}}$

answer is D.

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Detailed Solution

Tension in the wire $T = \frac{W}{2 cosθ} = \frac{Wl}{2 x}$

extension $e = 2 l' - 2 l = 2 (\frac{x^{2}}{2 l})$

strain $= \frac{e}{2 l} = \frac{x^{2}}{2 l^{2}}$

We know Y = $\frac{stress}{strain} = \frac{Wl}{2 Ax} \times \frac{2 l^{2}}{x^{2}}$

$Y = \frac{{Wl}^{3}}{{Ax}^{3}} or x = {(\frac{W}{AY})}^{\frac{1}{3}} \times l$

$∴$ Correct choice is (4).

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