Question 74 of 118

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:

A$L\left(1+\dfrac{2}{9}\dfrac{Mg}{\pi Y{R}^{2}}\right)$B$L\left(1+\dfrac{1}{9}\dfrac{Mg}{\pi Y{R}^{2}}\right)$C$L\left(1+\dfrac{1}{3}\dfrac{Mg}{\pi Y{R}^{2}}\right)$D$L\left(1+\dfrac{2}{3}\dfrac{Mg}{\pi Y{R}^{2}}\right)$

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Answer: (C) $L\left(1+\dfrac{1}{3}\dfrac{Mg}{\pi Y{R}^{2}}\right)$

$r=\dfrac{2R}{L}x+R$

For a small section

$\dfrac{F}{A}=Y\dfrac{dl}{dx}$

$⟹$${\int }_{0}^{L}dL={\int }_{0}^{L}\dfrac{Mgdx}{\pi {\left[\dfrac{2R}{L}x\times R\right]}^{2}Y}$



$∆L=\dfrac{Mg}{\pi Y}\left[-\dfrac{1}{{\left[\dfrac{2Rx}{L}+R\right]}_{0}^{L}}\times \dfrac{L}{2R}\right]$

$=\dfrac{MgL}{3\pi {R}^{2}Y}$

${L}_{f}=L+\Delta L=L[1+\dfrac{mg}{3\pi {R}^{2}Y}]$