A simple approximate expression is derived for the dependence of the period of a simple pendulum on the amplitude. The approximation is more accurate than other simple relations. Good agreement with experimental data is verified.

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It seems the only exception is the pendulum of antique astronomical clocks, whose amplitude is less than 1.5, as pointed out in
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11.

This error depends on θ0 implicitly (through k) and its absolute value increases rapidly with it. For instance, T0 underestimates the exact period with an error of 15.3% for an amplitude of θ0=π2.

12.

Of course, the cases with θ0>π2 are of less interest because most simple pendulum experiments in introductory physics laboratories are done with flexible cords instead of rigid rods, which prevents the pendulum bob from following a circular path soon after it is released. However, our approximate expression is more accurate than other ones even for θ0>π2.

13.

The error with respect to the exact period T for each amplitude is the quantity to be analyzed here instead of the error with respect to T0.

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19.
For a version of Ref. 5 that is richer in experimental details, see phys-0409086, available at arxiv.org
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21.

The lengths were measured after tying up the thread firmly to a hook in the ceiling laboratory, at one end, and to a small ring at the top of the lead cylinder, at the other end.

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