A uniformly tapering conical wire is made from a material of Young’s modulus Y and a normal, unexpended length L. The radii, at the upper and lower ends of the conical wire, have rules R and 3R, respectively. The upper end of the wire is fixed to the rigid support and a mass M is suspended from its lower end. The equilibrium extended length, of the wire ,would equal:

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$L (1 + \frac{2}{3} \frac{Mg}{{πYR}^{2}})$

$L (1 + \frac{2}{9} \frac{Mg}{{πYR}^{2}})$

$L (1 + \frac{1}{9} \frac{Mg}{{πYR}^{2}})$

$L (1 + \frac{1}{3} \frac{Mg}{{πYR}^{2}})$

answer is C.

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Detailed Solution

Take a cross section at distance x from the bottom, hence at this point,
$\begin{array}{l} r = 3 R - \frac{x}{L} 2 R \\ \frac{Mg}{{πr}^{2}} = Y \frac{dy}{dx} = \frac{Mg}{π {(3 R - \frac{2 Rx}{L})}^{2}} \end{array}$
where dy is the elongation of dx
Integrating this equation for x in 0 to L, we get,
$y = L (1 + \frac{1}{3} \frac{Mg}{{πYR}^{2}})$