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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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a
L1+29MgπγR2
b
L1+19MgπγR2
c
L1+13MgπγR2
d
L1+23MgπγR2

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detailed solution

Correct option is A

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

r=3R-xL2R

Mgπr2=Ydydx=Mgπ3R-2RxL2

where dy is the elongation of the portion dx.

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

Y=L1+13MgπγR2

Hence the correct answer is L1+13MgπγR2.

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