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 :

Question

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 :

Infinity Learn · Accepted Answer

Take a $$cross-section$$ taken at a point x from the bottom; hence, at this point,$$r=3R-xL2RMg$$$$π$$$$r2=Ydydx=Mg$$$$π$$$$3R-2RxL2where$$ dy is the elongation of the portion dx.As we solve for x in the range of $$0$$ to L, we obtain,$$Y=L1+13Mg$$πγ$$R2Hence$$ the correct answer is $$L1+13Mg$$πγ$$R2$$.

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