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Simple pendulum review

Overview of key terms, equations, and skills for simple pendulums, including how to analyze the forces on the mass. 

Key terms

Simple pendulumA mass suspended from a light string that can oscillate when displaced from its rest position.


EquationSymbolsMeaning in words
T, start subscript, p, end subscript, equals, 2, pi, square root of, start fraction, l, divided by, g, end fraction, end square rootT, start subscript, p, end subscript is period, l is pendulum length, and g is the acceleration due to gravityThe pendulum’s period is proportional to the square root of the pendulum’s length and inversely proportional to the square root of g

Analyzing the forces on a simple pendulum

An object is a simple harmonic oscillator when the restoring force is directly proportional to displacement.
Figure 1: A simple pendulum with length l, mass m, and displacement angle theta has a net restoring force of minus, m, g, sine, theta.
For the pendulum in Figure 1, we can use Newton's second law to write an equation for the forces on the pendulum. The only force responsible for the oscillating motion of the pendulum is the x-component of the weight, so the restoring force on a pendulum is:
F, equals, minus, m, g, sine, theta
For angles under about 15, degree, we can approximate sine, theta as theta and the restoring force simplifies to:
F, approximately equals, minus, m, g, theta
Thus, simple pendulums are simple harmonic oscillators for small displacement angles.

Common mistakes and misconceptions

Sometimes people think that a pendulum’s period depends on the displacement or the mass. Increasing the amplitude means that there is a larger distance to travel, but the restoring force also increases, which proportionally increases the acceleration. This means the mass can travel a greater distance at a greater speed. These attributes cancel each other, so amplitude has no effect on period. The pendulum’s inertia resists the change in direction, but it’s also the source of the restoring force. As a result, the mass cancels out too.

Learn more

For deeper explanations, see our video introducing pendulums.
To check your understanding and work toward mastering these concepts, check out the exercise on period and frequency of simple pendulums.

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