A uniformly tapering conical wire is made from a material of Young's modulus \(Y\) 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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Correct Answer

(C) \(L\left( {1 + \frac{1}{3}\frac{{Mg}}{{\pi Y{R^2}}}} \right)\)

Detailed explanation

Consider a small element \(dx\) of radius \(r\),

\(r = \frac{{2R}}{L}x + R\)

At equilibrium change in length of the wire

\(\int\limits_0^l {dL = \int {\frac{{Mgdx}}{{\pi {{\left[ {\frac{{2R}}{L}x + R} \right]}^2}y}}} } \)

Taking limit from \(0\) to \(L\)

\(\Delta L = \frac{{Mg}}{{\pi y}} - \frac{1}{{\left[ {\frac{{2Rx}}{L} + R} \right]_0^L}} \times \frac{L}{{2R}} = \frac{{MgL}}{{3\pi {R^2}y}}\)

The equilibrium extended length of wire

\( = L + \Delta L\)

\( = L + \frac{{MgL}}{{3\pi {R^2}Y}} = L\left( {1 + \frac{1}{3}\frac{{Mg}}{{\pi Y{R^2}}}} \right)\)

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