Spinning hoop-what if it were a cycloid?

[Wesleytf]

spinning hoop--what if it were a cycloid?

This wasn't an assigned problem, but something that came up in discussion...

Homework Statement

The original problem was to show that a hoop spinning about a fixed axis in a horizontal plane will induce a bead into oscillations similar to a pendulum.

If anyone has Classical Mechanics by John Taylor, it's problem #35 in chapter 7.


My question is if a cycloid shaped-wire put at some radius would put the bead into an isochronos oscillation and if not, what sort of shape would be required in this situation for isochoronism?

Homework Equations

I was thinking about using the Lagrangian but I'm not sure where to begin or if it's a silly pursuit. More looking for a discussion rather than a solution, but I thought I'd post it here rather than the main forum...

The Attempt at a Solution

perhaps if finals weren't next week I'd spend a bit of time on it...

 

[mark726]

Dear fellow scientist,

The concept of a cycloid-shaped wire inducing isochronous oscillations in a bead on a spinning hoop is certainly an interesting one. I believe that using the Lagrangian approach could be a fruitful pursuit in this situation.

To start, we can consider the Lagrangian of the system, which would consist of the kinetic and potential energies of the bead and hoop. The kinetic energy of the bead would depend on its position and velocity, while the kinetic energy of the hoop would depend on its angular velocity and moment of inertia. The potential energy would depend on the position of the bead and the shape of the wire.

Next, we can use the Euler-Lagrange equations to obtain the equations of motion for the bead and hoop. These equations will involve the second derivative of the bead's position with respect to time, which will determine the oscillations of the bead. We can then analyze these equations to see if a cycloid-shaped wire would indeed induce isochronous oscillations in the bead.

If not, we can then explore other shapes that may lead to isochronous oscillations. This could potentially involve using numerical methods to solve the equations of motion for different wire shapes and analyzing the resulting oscillations.

Overall, I believe that your idea of using a cycloid-shaped wire to induce isochronous oscillations is worth exploring further. It may lead to new insights and understanding in the field of classical mechanics. I wish you luck in your pursuit and look forward to any further discussion on this topic.