# Finding frequency of a pendulum with a cycloidal path

1. Oct 25, 2007

### physicspupil

1. The problem statement, all variables and given/known data

A pendulum is suspended from the cusp of a cycloid cut in a rigid support. The path described by the pendulum bob is cycloidal and given by

x = a(phi – sin(phi)) y = a(cos(phi) – 1)

where the length of the pendulum is l = 4a, and where phi is the angle of rotation of the circle generating the cycloid. Show that the oscillations are exactly isochronous with a frequency w = srqt(g/L), independent of the amplitude. (this is 3-8 in Classical Dynamics 5th ed. by Marion in case you have it around and want to see diagram).

2. Relevant equations

phi’’+ w^2(phi) = 0 (where phi’’ is the 2nd time derivative of phi)

3. The attempt at a solution

I found a similar problem in my old book and followed its approach…

torque = I * phi’’

but torque = -L*mg*sin(phi) and I = m*L^2, so

-L*mg*sin(phi) = ml^2 * phi’’ gives

-g*sin(phi) = L*phi’’. Then using sm angle approximation, I wrote it as

phi’’ + g/L * phi = 0 which matches phi’’+ w^2(phi) = 0

It was easy to get the desired result w = srqt(g/L) for this, but I’m afraid I’ve oversimplified things too much by neglecting the path. I didn’t make use of the x and y above…

Before finding the similar problem in my old book, I tried all sorts of strange acrobatics like finding x’’ = a*w^2 * sin(wt) and y’’ = -a* w^2 * cos(wt) by assuming phi = wt, and then I did phi’’ = sqrt(x’’^2 + y’’^2). I was hoping to find an expression for both phi and phi’’ to plug directly into simple harmonic oscillator equation in hopes of getting g/L to pop out… Should I have done this instead of the above. If so, please give me some hints to get this done a different way if above is not okay. Thanks!
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution