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A uniformly tapering conical wire is made from a material of Youngs’s $γ$ and has a normal, unextended length L. The radii, at the upper and lower ends of this conical wire, have values R and 3R, respectively. The upper end of the wire is fixed to a rigid support and a mass M is suspended from its lower end. The equilibrium extended length, of this wire, would equal :

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

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

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

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

answer is A.

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Detailed Solution

Take a cross-section taken at a point x from the bottom; hence, at this point,

$r = 3 R - \frac{x}{L} 2 R$

$\frac{Mg}{{πr}^{2}} = Y \frac{dy}{dx} = \frac{Mg}{π {(3 R - \frac{2 Rx}{L})}^{2}}$

where dy is the elongation of the portion dx.

As we solve for x in the range of 0 to L, we obtain,

$Y = L (1 + \frac{1}{3} \frac{Mg}{{πγR}^{2}})$

Hence the correct answer is $L (1 + \frac{1}{3} \frac{Mg}{{πγR}^{2}}) .$

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