Pendulum With Small Swings

## Question:

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,

## Answer:

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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