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 to:

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

Answer 1

Answer: (A)

$L(1+\frac{1}{3}\frac{Mg}{\pi {\rm Y}{R}^2})$

Take a cross section at distance $x$ from the bottom, hence at this point,

$r=3R-\frac{x}{L}2R$.

$\frac{Mg}{\pi r^2}=Y\frac{dy}{dx}=\frac{Mg}{\pi {(3R-\frac{2Rx}{L})}^2}$ where $dy$ is elongation of the portion $dx$.

Integrating this equation for $x$ in 0 to L, we get,

$y=L(1+\frac{1}{3}\frac{Mg}{\pi Y{R}^2})$