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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):

where

• $A$ is the amplitude of the oscillation
• $T$ is the period of the oscillation
• ${\varphi }_0$ 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 ${\varphi }_0=0$ for simplicity.

For example, if ${\varphi }_0=0$, then $$y(t=0)=A\cos \left(2\pi \frac{0}{T}\right)=A\cos 0=A$$ because $\cos 0=1$ (see Trigonometric Functions). At $t=T$, it will have completed exactly one cycle, and its displacement is now $$y(t=T)=A\cos \left(2\pi \frac{T}{T}\right)=A\cos 2\pi =A$$ because $\cos 2\pi =1$. (We must work with radians in this section!)

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

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