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Harmonic motion occurs when a system of some kind vibrates about an equilibrium point. The main condition for harmonic motion is the presence of a restoring force that serves to return the system to its equilibrium point. However, due to the inertia of the masses present in the system, they overshoot equilibrium, and if there is no energy lost the system will oscillate indefinitely.

Countless examples of harmonic oscillators can be found everywhere, from the swinging of a pendulum, where gravity is the restoring force, to a mass bobbing on a spring, where Hooke's Law serves as the restoring force, an inductor-capacitor circuit in electronics, atoms in a crystal lattice, and so forth.

Harmonic Oscillators, as examples of Simple Harmonic Motion (SHM), exhibit certain common principles based upon the idea of a restoring force. This is best exhibited in the prototypical example of a harmonic oscillator, a spring (with constant “k”) and an attached mass (with mass “m”). When a mass on a spring is stretched or compressed beyond the equilibrium point a distance “x,” the restoring force (“F”) is simply calculated by the equation F=-kx. Now you may be asking yourselves, “Hey, why is the mass important?” Well, dear reader, Newton’s famous equation F=ma (where “a” is acceleration) explains just how quickly the attached mass accelerates back to the equilibrium point and out to the other side.

Let’s say we have a spring attached to a vertical surface at one end and a block sitting on a frictionless table at the other end. Because no energy will be lost to friction, the block, if displaced from equilibrium, will theoretically oscillate into infinity. Now, imagine that attached to this block is a pen facing upwards. Above this whole shindig is a roll of paper that scrolls at a constant speed and is written on as the block moves back and forth. If you looked at the roll of paper after it’d been written on, you’d see a sine curve, which can be shown otherwise by calculating the graph y=sin(x). Pretty cool, huh? This is the sinusoidal description of SHM.

While the principles of SHM always apply to the idea of a spring and an attached mass, they do not always apply to the pendulum (with length “l,” attached mass “m,” and angle of displacement “θ”). With a pendulum, the restoring force is given by the equation F=(mg)(sin(θ)). Now, a pendulum shares these similarities with SHM:

  • Displacement is zero at the equilibrium position
  • At the greatest displacement, the restoring force and tangential acceleration are at their maximum, the velocity is zero, and the potential energy is greatest
  • It’s kinetic energy (and, thus, it’s speed) is maximized when the hanging mass returns to the equilibrium position

Unfortunately, pendulum enthusiasts, I must inform you that pendulums technically do not operate on the same principles by which SHM operates. Harmonic Oscillators experience a restoring force proportional to their displacement. If you recall, the equation for the restoring force of a pendulum is F=(mg)(sin(θ)), which is not proportional to the displacement angle θ, so a pendulum is never technically harmonic.

There is hope, however. At small angles, usually less than about 15 ̊, sin(θ) is rather close numerically to θ. So, if θ < 15 ̊ then the restoring force is approximately equal to mgθ, which would mean, like the rest of the examples of SHM, the restoring force is (sort of) directly proportional to the displacement. But, whenever the displacement angle increases beyond around 15ish, a pendulum’s not really an example of SHM. How crazy is that?

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