Question

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

Answer 1

Answer: (B)

$r=\frac{2R}{L}x+R$
$\int dl=\int \frac{Mgdx}{\pi {\left[\frac{2R}{L}x\times R\right]}^{2}Y}$
$\Delta L=\frac{Mg}{\pi y}\left[-\frac{1}{{\left[\frac{2Rx}{L}+R\right]}_{{}_{{}_0}}^L}\times \frac{L}{2R}\right]$
$=\frac{MgL}{3\pi R^{2}y}$