# 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 :

As we learned in

Young Modulus -

The ratio of normal stress to longitudinal strain

it denoted by Y

$Y= \frac{Normal \: stress}{longitudnal\: strain}$

- wherein

$Y=\frac{F/A}{\Delta l/L}$

F -  applied force

A -  Area

$\Delta l$ -  Change in length

l - original length

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

$r=R\left(1+\frac{2x}{L} \right )$

$Y=\frac{mg}{\pi R^{2}\frac{dL}{dx}}\ \; \Rightarrow\ \; dL=\frac{mg}{\pi R^{2}}\frac{dx}{\left(1+\frac{2x}{L} \right )^{2}}$

$\Delta L=\frac{mg}{Y\pi R^{2}}\int_{0}^{L}\frac{dx}{\left(1+\frac{2x}{L} \right )^{2}}\ \; \Rightarrow\ \; \frac{mgL}{\left(1+\frac{2x}{L} \right )^{2}}$

Now, $L'=L+\Delta L=L+\frac{mgL}{\left(1+\frac{2x}{L} \right )^{2}}$

$L'=L\left(1+\frac{1}{3}\frac{mg}{\pi R^{2}Y} \right )$

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