# Simple Harmonic Motion

For

This is called

This has a solution (which we may as well assume is 0 at $t = 0$) of the form

If we plot $x$ against the velocity $\Large\frac {dx}{dt}$ then we get a curve in what is called the

*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*.