# Elastic potential energy of a hanging cone

1. The problem statement, all variables and given/known data
A cone of circular cross section having base radius R, mass M and height L is suspended from its base as shown in figure. The material of cone has Young’s modulus Y. If the elastic potential energy stored in the cone can be expressed as:
$$E=\frac{m^ag^bL^c}{d\pi^eY^fR^g}$$
Then find a+b+c+d+e+f+g.

2. Relevant equations

3. The attempt at a solution
Since the question doesn’t specify whether the cone is hollow or solid, I assumed it as solid and proceeding with this assumption gave me a close answer.

From a distance ##x## above the apex of the cone, I selected a disk (or frustum) of thickness ##dx##. The force responsible for the elongation of this small part is:
$$F=\rho\frac{1}{3}\pi r^2 xg$$
where ##r## is the radius of selected disk. Hence, elongation ##dl## is:
$$dl=\frac{Fdx}{AY}=\frac{\rho \pi r^2xg \,dx}{3Y \pi r^2}=\frac{\rho g}{3Y}x\,dx$$
The elastic potential energy stored in this part is:
$$dE=\frac{1}{2}\frac{YA}{dx}\,dl^2=\frac{1}{2}\frac{Y\pi r^2}{dx}\,dl^2$$
I plugged in ##r=x(R/L)## and the expression for ##dl## and integrated within ##x## from 0 to L. I got the following expression:
$$E=\frac{m^2g^2L}{10Y\pi R^2}$$
The above expression gives me 19 as the answer but the correct answer is 18. :confused:

I think there is something wrong with my initial assumptions and working.

Any help is appreciated. Thanks!

Attached Images hanging cone.png (12.7 KB)

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