Simple Harmonic Motion

Simple harmonic motion is the study of oscillations. An oscillation is the motion of an object that regularly repeats itself over the same path. For example, a pendulum in a grandfather clock undergoes oscillation: it travels back and forth, back and forth, back and forth. . . . Another term for oscillation is “periodic motion.”

Objects undergo oscillation when they experience a restoring force. This is a force that restores an object to the equilibrium position. In the case of a grandfather clock, the pendulum’s equilibrium position—the position where it would be if it weren’t moving—is when it’s hanging straight down. When it’s swinging, gravity exerts a restoring force: as the pendulum swings up in its arc, the force of gravity pulls on the pendulum, so that it eventually swings back down and passes through its equilibrium position. Of course, it only remains in its equilibrium position for an instant, and then it swings back up the other way. A restoring force doesn’t need to bring an object to rest in its equilibrium position; it just needs to make that object pass through it.

If you look back at the chapter on Conservation of energy, you’ll find the equation for the force exerted by a spring, F = kx. This force is a restoring force: it tries to pull or push whatever is on the end of the spring back to the spring’s equilibrium position. So if the spring is stretched out, the restoring force tries to squish it back in, and if the spring is compressed, the restoring force tries to stretch it back out. Some books present this equation as F = −kx. The negative sign simply signifies that this is a restoring force.

One repetition of periodic motion is called a cycle. For the pendulum of a grandfather clock, one cycle is equal to one back-and-forth swing.

The maximum displacement from the equilibrium position during a cycle is the amplitude. In Figure below, the equilibrium position is denoted by “0,” and the maximum displacement of the object on the end of the spring is denoted by “x.”

SHM

The time it takes for an object to pass through one cycle is the period, abbreviated “T.” Going back to the grandfather clock example, the period of the pendulum is the time it takes to go back and forth once: one second. Period is related to frequency, which is the number of cycles per second. The frequency of the pendulum of grandfather clock is f = 1 cycle/s, where “f” is the standard abbreviation for frequency; the unit of frequency, the cycle per second, is called a hertz, abbreviated Hz. Period and frequency are related by these equations: