Any behavior that repeats itself regularly is called periodic.Periodic motionis also called oscillatory motion, that is objects experiencing this kind of motion undergooscillations. 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:

- The moving object must have inertia/mass
- 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 MotionWe 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 = qFrom the figure, we see that the x component of the position is given by:

r_{x}= R cos qSo we have that:

r_{x}= R cos wtR, 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 wtWe 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 = -w^{2}Ao 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,

xrepresents the amplitude of the motion,withAobeing its maximum, andwrepresents the angular frequency of the body. This is related to the period,T, byw = 2p /TFor practice with these ideas, try the following problems:

Page 401, Problems 13.10, 13.12, 13.15, 13.16Next stop, masses, springs and pendulums.