A pendulum is a simple device composed of a weight suspended on a string, wire, metal or other material that swings back and forth. Pendulums are used in everything from grandfather clocks to metronomes. Scientific principles govern what affects the swing rate of the pendulum, and there are surprisingly few variables that affect a pendulum’s swing. Pendulums have fascinated scientists for a long time, with Newton and Galileo both studying them extensively, and pendulums continue to be an object of interest today.

The swing can be described by the period of the pendulum (T) which is one full swing back and forth to return to its starting position. In its simplest form, it can be described only with the gravitational force (g) and length of the pendulum arm (L). This equation is without considering friction or other compounding factors, just simple harmonic motion:

The interesting thing to notice with this equation is actually the absence of certain variables like mass of the pendulum. This will be expanded on further, but it is good to remember that the period of a pendulum relies only on the length of the string and the gravitational acceleration (excluding the constant of pi, which is the same no matter what).

Another useful way to describe pendulums can be to view them as oscillators, where an oscillation is a repeated motion. In this way the motion of a pendulum can be viewed like a wave, similar to a spring bouncing back and forth.

Pull a pendulum back and release it. You can let the pendulum swing back and forth on its own, or in the case of a pendulum clock, have it swing powered by the gears. Either way the principle of periodic motion affects the pendulum. The force of gravity pulls the weight, or bob, down as it swings. The pendulum acts like a falling body, moving toward the center of motion at a steady rate and then returning.

The swing rate, or frequency, of the pendulum is determined by its length from the pivot point to the end of the string. The longer the pendulum, whether it is a string, metal rod or wire, the slower the pendulum swings. Conversely the shorter the pendulum the faster the swing rate. This represents an absolute principle that will always work no matter the type of design. On grandfather clocks with long pendulums or smaller clocks with shorter ones, the swing rate depends upon the pendulum's length.

Amplitude refers to the angle of swing, or how far back the pendulum swings. A resting pendulum has an angle of 0 degrees; pull it back halfway between resting and parallel to the ground and you have a 45-degree angle. Start a pendulum and you determine the amplitude. Experiment with different starting points and you discover that the amplitude does not affect the swing rate. It will take the pendulum the same amount of time to return to its starting point. One exception involves a very large angle, one beyond any reasonable swing for a clock or any other device. In that case the swing rate will be affected as the pendulum goes faster.

Energy relates directly to amplitude and mass. At each starting point, the pendulum has the maximum potential energy, and in the center, the pendulum has maximum kinetic energy. The motion of the pendulum is not necessarily linked to energy, but both descriptors play a role in a pendulum’s properties. Additionally, under conservation of energy, the maximum kinetic energy is equal to the maximum potential energy.

One factor that does not affect swing rate is the weight of the bob. Increase the weight on the pendulum and gravity just pulls harder, evening out the extra weight. As School for Champions points out, the force of gravity on any falling object is the same no matter the mass of the bob.

In a real-world application air resistance affects the swing rate. Each swing encounters that resistance and it slows down the swing, although it might not be enough to be noticeable during one swing. Friction also slows down the swing. If the pendulum is swinging based upon inertia from the initial release eventually it will come to a stop.

Given two simple pendulums, the swing of the pendulum adjusts when placed in close proximity to the other pendulum. This phenomenon is called sympathetic vibration. The pendulums pass motion and energy back and forth. This transfer will eventually cause the swing rate of one pendulum to be identical with that of the other pendulum.

When we start to consider all the factors of pendulum motion, it becomes increasingly difficult to model a pendulum accurately. One way that physicists address this is with a small angle approximation that says pendulums oscillate more simply under a small amplitude, and we can approximate it as such.

There are additional configurations of pendulums that can add even more variation. Double pendulums (where there are two bobs chained together) create chaotic motion, and a Foucault pendulum with approximately 2d motion (instead of one dimensional motion) introduces another axis of complicated calculations.