# Simple Harmonic Motion

For small swings of a pendulum, the displacement $x$ satisfies the differential equation
$\Large\frac {d^2 x}{dt^2}\normalsize = -k^2 x$
where $t$ is time and we ignore frictional resistance.

This is called Simple Hamonic Motion or SHM.

This has a solution (which we may as well assume is 0 at $t = 0$) of the form
$x = A \sin(kt)$ with $A$ constant.
So the graph of $x$ against $t$ is a sine curve.

If we plot $x$ against the velocity $\Large\frac {dx}{dt}$ then we get a curve in what is called the phase plane. For this case the curve is a closed curve and is in fact an ellipse.