# Bead on a wire, potential energy

1. Oct 7, 2013

### Habeebe

1. The problem statement, all variables and given/known data
A particle is constrained to move without friction on a circular wire rotating with constant speed ω about a vertical diameter. Find the equilibrium position of the particle, and calculate the frequency of small oscillations around this position. Find and interpret physically a critical angular velocity, ω = ωc, that divides the particle’s motion into two distinct types.

2. Relevant equations
U=-∫Fdr

3. The attempt at a solution
I set it up in spherical coordinates. The wire is $\rho^2+z^2=R^2$ and it is being rotated about the z axis (rho lies in x-y plane). I set $\theta$ to be the angle measured around the circle counterclockwise from the rho axis. The forces acting on the particle are gravity (mg) and centripetal force ($-m\rho\omega^2$). The potential due to gravity is then mgy and the potential due to centripetal force is:

$\int_{0}^{\rho}m\rho\omega^2 d\rho=\frac{1}{2}m\omega^2\rho^2=\frac{1}{2}m\omega^2R^2cos^2(\theta)$
Gravitational potential becomes mgy=mgRsin(θ). Now my total potential energy is:
$U=mgRsin(\theta)+\frac{1}{2}m\omega^2R^2cos^2(\theta)$

I differentiate with respect to θ and set equal to zero:
$U_\theta=mgRcos(\theta)-m\omega^2R^2cos(\theta)sin(\theta)=0$
$gcos(\theta)=\omega^2Rcos(\theta)sin(\theta)$
$sin(\theta)=\frac{g}{R\omega^2}$
$\theta=arcsin(\frac{g}{R\omega^2})$