Simple pendulum

Also called Pendulum

A simple pendulum is a point mass hanging from a light string of fixed length, swinging under gravity. Displaced by a small angle it moves in simple harmonic motion with a period set only by the string length and g.

AP Physics C: Mechanics EK 7.5.A.3 defines it as a special case of a physical pendulum in which the hanging object can be modeled as a point mass a distance \ell from the pivot. The period is

Tp=2πgT_p = 2\pi\sqrt{\frac{\ell}{g}}

which is printed on all four AP Physics equation sheets. With g=9.8 m/s2g = 9.8 \text{ m/s}^2, a 1.0 m pendulum has Tp=2.0T_p = 2.0 s.

Two things it does not depend on. Mass, because mm cancels: the same gravitational force that restores the bob also resists its acceleration. And amplitude, by EK 7.3.A.2.

The small-angle condition is part of the formula, not a footnote. AP Physics 1 EK 7.2.A.1.ii gives TpT_p for a pendulum displaced by a small angle, and EK 7.1.A.2.iii says a pendulum's motion can be modeled as simple harmonic motion because the restoring torque is proportional to the angular displacement, which only holds for small displacements. C: Mechanics writes the approximation out as sinθθ\sin\theta \approx \theta (EK 7.5.A.2.ii). Swing the bob through 60 degrees and the true period runs longer than the formula returns.

Note the gg in the formula: a pendulum clock keeps different time on the Moon, while a mass on a spring does not care.

Worked pendulum problems live in the SHM guide and the SHM practice set.

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