Pendulum With Small Swings


I visited your site in search for the small amplitude theory. I am currently completely a physics lab and was wondering if you could provide me with some insight regarding the small amplitude theory. Is period independent of amplitude? My results show that it is. but there is a question that asks which graph shows how period is dependent on amplitude. how can it be both? is it only independent when the angle is relatively small? which would be apprx. 50 degress according to my graph.And then would it become dependent on amplitude above those small angles? and why?

I have been reading informaion on it and have not been satisified. I thank you before hand for your extra knowledge.

Thank you for your time,


I assume you are talking about the simple pendulum when you speak of a small amplitude theory. Here's the deal with that. For period to be independent of amplitude the restoring force would have to depend linearly on displacement. In the case of a Hooke's law simple harmonic oscillator this is true. F=-k*x. In the case of a simple pendulum the restoring force is proportional to the sine of the displacement angle, not the angle itself F=-m*g*sin(q). For small angles the sine is approximately equal to the angle itself. In that case we may substitute q for sin(q) in the restoring force formula so that F=-m*g*q, reducing the formula to the same form as the simple harmonic oscillator where mg plays the role of k and q plays the role of x.

The heart of your question deals with below what angle is the sine of the angle close enough to the angle itself so that the motion of the pendulum appears simple harmonic. If your experimental setup only provides for accuracy to 3 decimal places for example, then when the sine and angle differ only in the fourth place, the pendulum period appears independent of amplitude. The better your experiment, the smaller the angle at which period appears to be independent of amplitude.

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