Periodic Motion

Any behavior that repeats itself regularly is called periodic. Periodic motion is also called oscillatory motion, that is objects experiencing this kind of motion undergo oscillations. Examples of this are children swinging on swings, guitar strings being strummed, and the vibrations of quartz crystals in watches.

There are two necessary conditions for periodic motion:

1. The moving object must have inertia/mass
2. There must be some restoring force moving the object back towards its equilibrium position (lowest energy state)

Many cases can be considered using periodic motion. The complex rhythms of the heart are just compositions of the simple rhythms of pendulums.

Simple Harmonic Motion

We want to find a general description of motion for which acceleration is proportional to negative displacement (i.e., in which a restoring force is present). This is called simple harmonic motion.

In order to find this we will use a common example of periodic motion, the Earth moving around the sun. We know that it undergoes periodic motion, i.e., that it takes 365.25 days to make a complete revolution. If we watch the Earth's motion from a point on the solar plane, we see that its motion is just like that of a mass attached to a spring: In the figure to the right, the arrow on the lower half of the y-axis represents the point from which we are looking. We are really only seeing the x-projection of the motion. We can easily find an equation for the position at any time. We know that :

wt = q

From the figure, we see that the x component of the position is given by:

rx = R cos q

So we have that:

rx = R cos wt

R, the radius, is the maximum point reached. We'll call it the maximum amplitude, Ao, so our x displacement, x, is given by:

x = Ao cos wt

We should also take into account that the motion doesn't need to begin in any certain place. So we define a phase constant, f (phi), that describes the position shifted away from the origin. We are finally left with:

x = Ao cos (wt - f)

From calculus, we know that the derivative of cosine is negative sine so:

dx/dt = v = -wAo sin (wt - f)

and

dv/dt = a = -w2Ao cos (wt - f)

So now we have equations for the position, velocity, and acceleration of an oscillating body at any time. As you can see from a comparison to what you learned in trigonometry, x represents the amplitude of the motion,with Ao being its maximum, and w represents the angular frequency of the body. This is related to the period, T, by

w = 2p /T

For practice with these ideas, try the following problems:

Page 401, Problems 13.10, 13.12, 13.15, 13.16

Next stop, masses, springs and pendulums.