Simple Harmonic Motion

For small swings of a pendulum, the displacement xx satisfies the differential equation
d2xdt2=k2x\Large\frac {d^2 x}{dt^2}\normalsize = -k^2 x
where tt 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=0t = 0) of the form
x=Asin(kt)x = A \sin(kt) with AA constant.
So the graph of xx against tt is a sine curve.

If we plot xx against the velocity dxdt\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.