Q:

How does amplitude affect the period of a pendulum?

A:

Quick Answer

For larger amplitudes, the amplitude does affect the period of the pendulum, with a larger amplitude leading to a larger period. However, for small amplitudes (typically around a few degrees), the amplitude has no effect on the period of a pendulum.

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Full Answer

In a simple pendulum, which can be modeled as a point mass at the end of a string of negligible mass and a given length, the amplitude is normally only a few degrees. When the amplitude is this small, it does not affect the period of the pendulum. The period simply equals two times pi times the square root of the length of the pendulum divided by the gravitational constant (9.81 meters per second per second).

For a real pendulum, however, the amplitude is larger and does affect the period of the pendulum. When the amplitude is larger than a few degrees, the period of the pendulum becomes an elliptical integral, which can be approximated by an infinite series. The series includes terms with the amplitude squared, the amplitude to the fourth power, the amplitude to the sixth power, and so on. Therefore, the larger the amplitude, the more nonnegligible terms appear in the series. As the amplitude of the pendulum increases, the period increases.

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