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Grade 12Mechanics

a uniformly tapering conical wire is made from a material of youngs modulus Y and has a normal unextended lenght 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 fitted to a rigid support and a mass M is suspended from its lower end. the equilibrum extended length of this wire, would be

Profile image of aadhil
8 Years agoGrade 12
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1 Answer

Profile image of Arun
8 Years ago

As we learnt in

Young Modulus -

Ratio of normal stress to longitudnal 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 lenght

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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