question_answer

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 Rand 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) \[L\left( 1+\frac{2}{9}\frac{Mg}{\pi Y{{R}^{2}}} \right)\]

B) \[L\left( 1+\frac{1}{9}\frac{Mg}{\pi Y{{R}^{2}}} \right)\]

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

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

Correct Answer: C

Solution :

[c] Consider a small element dx of radius r,               \[r=\frac{2R}{L}x+R\]   At equilibrium change in length of the wire        \[\int\limits_{0}^{1}{dL}=\int{\frac{Mgdx}{\pi {{\left[ \frac{2R}{L}x+R \right]}^{2}}y}}\] Taking limit from 0 to L \[\Delta L=\frac{Mg}{\pi y}\left[ -\frac{1}{\left[ \frac{2Rx}{L}+R \right]_{0}^{L}}\times \frac{L}{2R} \right]=\frac{MgL}{3\pi {{R}^{2}}y}\] The equilibrium extended length of wire\[=L+\Delta L\] \[=L+\frac{MgL}{3\pi {{R}^{2}}Y}=L\left( 1+\frac{1}{3}\frac{Mg}{\pi Y{{R}^{2}}} \right)\]