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## Homework Statement

A circular wire hoop rotates with constant angular velocity ! about a vertical diameter. A small bead moves, without friction, along the hoop. Find the equilibrium position of the particle and calculate the frequency of small oscillations about this position.

## Homework Equations

## The Attempt at a Solution

Let's take our reference inertial frame as a spherical co-ordinate system whose axis is along the axis of rotation of the who and whose origin is at the centre of the hoop.

Then position of M can be given by r, ## \theta, \phi ##.

Constraint: r= R,

## \dot \phi = \omega ## , constant.

There are two generalised coordinates ## \phi ## and ## \theta ##.

L = T - U

## T = \frac 1 2 mR^2 ( {\dot \theta}^2 + \sin ^2 \theta {\dot \phi}^2) ##

Taking U = 0 at the origin, U = mgR ## \cos \theta ##

So, L = ## \frac 1 2 mR^2 ( {\dot \theta}^2 + \sin ^2 \theta {\dot \phi}^2 ) - mgR ## ## \cos \theta ##

Lagrange's equation of motion gives,

## \ddot \theta = \frac g R \sin {\theta} + \sin {\theta}~ \cos { \theta}~{ \dot \phi}^2 ##

At eqbm. ## \ddot \theta = 0 ##

## \theta = 0, \Pi , \cos {\theta} = \frac {-g} { \omega^2 R} ##

What to do next?

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