A uniformly tapering conical wire is made from a material of Young’s… | JEE Main 2016 (9 Apr, Shift 1)

Q26JEE Main 2016 · 9 APR · SHIFT 1Difficult

Wire Combinations & Special Cases

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 3 R, 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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Answer: (b)

Radius at distance $x$ from the top: $r = R\left(1 + \frac{2x}{L}\right)$ ; tension is $Mg$ throughout.
$\Delta L = \int_0^L \frac{Mg\,dx}{\pi r^2 Y} = \frac{Mg}{\pi Y R^2}\int_0^L \frac{dx}{(1 + 2x/L)^2}$
$\int_0^L \frac{dx}{(1 + 2x/L)^2} = \frac{L}{2}\left(1 - \frac{1}{3}\right) = \frac{L}{3}$ (same as $\Delta L = \frac{MgL}{\pi Y R_1R_2}$ with $R_1R_2 = 3R^2$ )
Extended length $= L\left(1 + \frac{1}{3}\frac{Mg}{\pi Y R^2}\right)$ , so the answer is (b).

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Wire Combinations & Special Cases question from JEE Main 2016 (9 Apr, Shift 1)

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