- #1

Saitama

- 4,243

- 93

## Homework Statement

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.

## Homework Equations

## 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.

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

Any help is appreciated. Thanks!