Simple Pendulum - how small is small?

The equation for the time period of a simple pendulum is standard fare in many school Physics courses:



Derivations of this equation are often not required these days, but it's nice to include to show pupils where it comes from: see this pdf for two different ways of doing it.

If you do derive it, you will have to rely on an assumption, namely that the initial displacement of the pendulum is small. Like me, you may have wondered (or been asked):

  1. What exactly is meant by 'small' in this context?


  1. What exactly do pendulums do when this isn't the case?

The short but less helpful answer to the first question is 'it depends how accurate a solution you want'. A more specific answer would be to say that if you want your results to be accurate to better than 1%, the initial angular displacement must be less than 23°. If your pupils' pendulums are swinging with an initial angular displacement of 80° - younger pupils in particular seem be especially fond of huge amplitudes of swing - their time period results will be about 14% longer than the above equation will predict. This should be noticeable when analysing results, even at school level.

The answer to the second question is that the time period of a pendulum increases as the amplitude of swing increases. The following approximation series allows you to calculate exactly how much it increases by:

Deriving Time period of a simple pendulum.pdf

...where θ is the angular amplitude of the pendulum's swing. Derivations of this can be found online, but basically it involves setting up a Legendre polynomial for the pendulum's motion and substituting in a Maclaurin series. Your more mathematical pupils might enjoy seeing this, especially if they've covered those topics in further maths lessons.

To check this empirically, I swung a 2.00 m pendulum with amplitudes of 10° and 70°. The simple formula just gives T = 2.84s, irrespective of amplitude. However, I measured T for the smaller amplitude as 2.85 s and, for the larger amplitude, = 3.10 s. Both of  these results are nicely in line with the more complex equation above, so it's definitely not that difficult to see a difference.

The Excel file below shows you what happens when you progressively add more terms to the series. Notice that the series converges quite quickly: on the more detailed graph, the lines for the third and fourth approximations are indistinguishable from each other, and the second approximation isn't terribly different.


The PDF file shows you a fairly standard derivation of the above equation, such as you would find in most textbooks, but also derives the same thing in a different way using angular motion formulas rather than the usual linear motion ones. It's comforting to see the result is the same (and the same assumption needs to be made) no matter which approach you take.