Point Mass Inside Hoop: Parametric Equation & Euler-Lagrange E.o.m.

[Oskar Paulsson]

Homework Statement

A point mass is constrained to move along inside a hoop.
The hoop can roll frictionless and without slipping on a horizontal plane along a fixed direction.
At some initial time t = 0 the hoop is at rest and the particle is at the top of the hoop with velocity v_top

Homework Equations

Suitable parametric equation for the point mass and compute the Euler-Lagrange E.o.m.

The Attempt at a Solution

First some drawings.

Since the "ball" is a point mass I consider its own radius to be negliable, hence it only has translational energy.
The velocity of the ball itself:
$$ v = r \alpha $$
$$ \alpha = \frac{d\theta}{dt} $$
The hoop has clockwise rotational motion
$$\omega = \frac{d\phi}{dt}$$

The point mass must have some parametric co-ordinate
$$ x(r,\theta) $$
which I'm having trouble finding.
My reasoning on this is as follows; the point can be described in terms of height about the horizontal line of the hoop
$$ rsin(\theta) $$
With an additional term for the horizontal displacement of the hoop:
$$ r\omega t$$
$$ x(r,\theta) = rsin(\theta) + r\omega t $$
then
$$ \alpha = \dot{x}(r,\theta)$$

Is this right?

 

[mfb]

The velocity of the ball itself:

This is not the total velocity of the ball.

ω is not constant, you cannot calculate the horizontal displacement like that.

 

[TSny]

Welcome to PF, Oskar.

As mfb pointed out, the hoop will have a complicated motion as it rolls in response to the time varying forces exerted on it by the moving particle and the friction from the surface.

You might want to step back and first consider how many degrees of freedom the entire system has. Then judiciously choose a set of independent, generalized coordinates for specifying the configuration of the system. The number of generalized coordinates should equal the number of degrees of freedom. Then see if you can express the kinetic and potential energies of the system in terms of the generalized coordinates.