- #1

jfy4

- 649

- 3

## Homework Statement

A particle of mass [itex]m[/itex] is placed on top of a vertical hoop of radius [itex]R[/itex] and mass [itex]M[/itex]. The particle is free slide on the outside of the hoop without friction while the hoop is free to roll in a vertical place without slipping. Use the method of Lagrange multipliers to determine the height at which the particle falls off of the hoop.

## Homework Equations

[tex]

\frac{d}{dt}\frac{\partial L}{\partial \dot{q}_j}-\frac{\partial L}{\partial q_j}=\lambda_i \frac{\partial f_i}{\partial q_j}

[/tex]

## The Attempt at a Solution

Let me first venture the Lagrangian, since that is a signifigant portion of the problem, then EL equations can move things along quickly once I have it. The way I read it, the hoop is free to move in the 2D vertical plane, and the bead just rests on top and the whole of dynamics is just between the two, all in a gravitational field.

I then have for the kinetic energy of the hoop

[tex]

T_h=\frac{1}{2}M\dot{X}^2 + \frac{1}{2}I\dot{\phi}^2

[/tex]

with the additional constraint that [itex]f_1 = X-R\phi=0[/itex] with [itex]X[/itex] the distance in the x direction the hoop has traveled, and [itex]\phi[/itex] the angle subtended by the hoop. For the bead I have

[tex]

T_b=\frac{1}{2}m(\dot{x}^2+\dot{y}^2) + \frac{1}{2}I_0 \dot{\theta}^2

[/tex]

with the constraint [itex]f_2 = r_2-R=0[/itex] with [itex]r_2[/itex] the radius of the hoop. Then I have

[tex]

x = r_1 + r_2 \cos\theta \quad y=r_2 \sin\theta

[/tex]

with [itex]x,y[/itex] the location of the bead. By placing the center of the hoop on the x-axis, [itex]r_1[/itex] points to the center of the hoop, and [itex]r_2[/itex] from the center to the bead, [itex]\theta[/itex] is the angle drawn from the x-axis at the center of the hoop to the bead. The potential for the hoop is zero, and for the bead

[tex]

V=-mgy=-mg r_2 \sin\theta

[/tex]

The Lagrangian is [itex]L=T_b+T_h -V[/itex]. Also [itex]I = Mr_{2}^{2}[/itex] and [itex]I_0 = m r_{2}^{2}[/itex]

Does this seem like a good set-up for the problem's lagrangian? Thanks.