Simple Harmonic Motion

When the restoring force on an oscillating object is proportional to its displacement (that is, when

F \sim y (t)

), then the oscillation is called simple harmonic motion. If we graph the displacement of such an oscillation as a function of time, we find that the displacement is sinusoidal-- it looks like a sine or cosine function.

We can write the displacement as a cosine (in radians mode):

y (t) = A \cos (2 π \frac{t}{T} + ϕ_{0})

where

$A$ is the amplitude of the oscillation
$T$ is the period of the oscillation
$ϕ_{0}$ is a number called the initial phase, which depends on where in the cycle the oscillation is at $t = 0$ . Most times we'll assume that $ϕ_{0} = 0$ for simplicity.

For example, if $ϕ_{0} = 0$ , then $y (t = 0) = A \cos (2 π \frac{0}{T}) = A \cos 0 = A$ because $\cos 0 = 1$ (see Trigonometric Functions). At $t = T$ , it will have completed exactly one cycle, and its displacement is now $y (t = T) = A \cos (2 π \frac{T}{T}) = A \cos 2 π = A$ because $\cos 2 π = 1$ . (We must work with radians in this section!)

Because $f = \frac{1}{T}$ , we can also write the displacement as

y (t) = A \cos (2 π f t + ϕ_{0})

Physicists like this version because we don't like writing fractions if we can avoid it.