History Graphs

This is a history graph of an oscillation, because it shows how the displacement $y$ changes with time $t$. You can learn a lot about an oscillation from a history graph:

• The equilibrium point is the horizontal line across the middle, where $y=0$
• The amplitude of the oscillation is the distance from the equilibrium point to the highest point on the graph. (For a sinusoidal oscillation, the amplitude is half the total height of the graph.)
• The period $T$ of the oscillation is the time difference between two maxima (or two minima, or between any two points which look identical).
• To find the frequency of the oscillation, we first find the period and then take the reciprocal. In this case, $f=\frac{1}{T}=\frac{1}{4\mathrm{s}}=0.25\mathrm{Hz}$
• You can find the initial phase ${\varphi }_0$ of the graph by matching it with the four options on Simple Harmonic Motion. This particular graph matches ${\varphi }_0=\frac{3\pi }{2}$
• Once we have $A$, $T$, and ${\varphi }_0$, we can write the complete expression for the displacement as a function of time. For this example, we have $$y(t)=3\cos (2\pi \frac{t}{4}+\frac{3\pi }{2})$$