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## Homework Statement

Consider a bead of mass m that can slide freely on a circular hoop of radius a which lies in a vertical plane. The hoop rotates about a vertical axis through its centre at fixed angular velocity [tex]\omega[/tex]. The angle [tex]\theta[/tex] is the polar angle of the bead measured from the downward vertical.

1. Find the kinetic and potential energy of the bead.

2. Find the bead's equation of motion.

3. Show that for [tex]\omega^{2}a < g [/tex] the bead can remain in a stable equilibrium at the bottom of the hoop, otherwise this position is unstable.

4. In the case [tex]\omega^{2}a > g [/tex] find the new position [tex]\theta^{*}[/tex] of stable equilibrium and calculate the angular frequancy [tex]\Omega[/tex] of small oscillations about this position.

## Homework Equations

I worked out the answers for the first 2 questions myself (see below)

## The Attempt at a Solution

1. The kinetic energy is [tex]\stackrel{1}{2}ma^{2}(\omega^{2}sin^{2}(\theta)+ \stackrel{.}{\theta}^{2})[/tex]

The potential energy is mga[tex](1-cos(\theta))[/tex]

The above are just from the geometry of the situation

2. The equation of motion of the bead is [tex]\stackrel{..}{\theta} = (\omega^{2}cos(\theta) - g/a)sin(\theta)[/tex]

This just comes from using the Lagrangian L=T+V and treating it as a function of [tex]\theta[/tex] and [tex]\stackrel{.}{\theta}[/tex]

3 and 4. This is where I am getting stuck! When I set to zero [tex]dv/d\theta[/tex] I get [tex]mgasin(\theta)=0[/tex] which does not relate to [tex]\omega[/tex]! Would it be necessary to use minimise an effective potential instead? I've tried this but the algebra is not going anywhere and a little insight would be helpful.

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