Period of Pendula

I got to thinking about this due to the recent (September 2002) Scientific American article on Huygens’ clock (among others) by William J. H. Andrewes.

Briefly, Huygens developed the first working pendulum clock. As he was using a verge escapement, the pendulum needed to swing with a large angle. While the period of a simple pendulum swinging through a small angle will be independant of this amplitude, the precise size of “large” amplitudes will play a role in the pendulum’s period. As it was very difficult to regulate the size of the amplitude, variances in this amplitude would feed directly into variances of the period and ruin the clock as a timepiece.

He solved this problem by suspending the pendulum from wires, which would hit curved “cheeks” mounted in the plane of the swing, which would effectively decrease the length of the pendulum as the amplitude increased. This is called a cycloidal pendulum and its period is independant of the amplitude, problem solved (at least mathematically).

Eventually, the anchor escapement was developed, which didn’t need such large amplitudes to work, so it was possible to use longer pendula with smaller amplitudes and eliminate (within experimental error) the dependance upon amplitude that way.

I’m looking for two formulae:

  1. What is the period of a simple pendulum with a large amplitude?

  2. What is the period of a cycloidal pendulum?

All the sources I have been able to find discuss only the simple-small amplitude case. Can anyone do better? Otherwise I’m going to have to derive the damn things myself … and I’m not sure if I still have the mathematical skills to do that!

  1. I have no idea.

  2. I am not sure that this is a meaningful question. One can estimate the equation for a pendulum with large amplitude (i.e. the equation for angle as a function of time), but I don’t remember for sure whether it comes out strictly periodic or not.

These are, admittedly, not really helpful comments, and I’ll see if I can get something more useful later, if no one else has a solution first.

  1. Reference: Pendulum

The differential equation governing the motion of a simple pendulum is as follows:

d[sup]2[/sup]theta/dt[sup]2[/sup] + omega[sub]0[/sub][sup]2[/sup]sin(theta) = 0

This differential equation is not solvable in closed form, and so neither is the period of a pendulum. When theta is small, you make the small-angle approximation sin(theta) = theta, and that allows you to solve for the period.

You can solve for the period of a general pendulum in terms of elliptical integrals. See equations (25) and (28) on that page in the reference. It involves the Complete Elliptic Integral of the First Kind, K(k), which in turn can be defined in terms of an infinite series.

  1. Reference: Tautochrone Problem

There’s a pretty cool animated image of a cycloidal pendulum on that page. The period is given in the last equation on the page (18). As you can imagine, it’s quite a bit simpler:

T = pi × sqrt(a / g)

where “a” is a value used in the parametric equations describing the cycloid.

Erm, while what I said about the value “a” is true, it’s also the length of the string. a = l. I should have mentioned that. Sorry.

I meant… a = l/4. a = one-fourth of the string length. Also, the period P is actually four times the time T given in equation (18). The actual equation is of course:

P = 2pi × sqrt(l / g)

The same that you get for small-angle approximations.

I wonder how many of my posts are corrections…

Speak for yourself, buddy!

Achernar, when you post beautifully detailed answers citing high quality sites that that, within three hours of my posing the problem you just make all the corrections you want. Won’t bother me, uh-uh.

Thanks very much. That’s just great.