Any object that moves back and forth, in and out, or to and fro, is oscillating. When theoscillationstake a constant amount of time to repeat, the object's motion is calledperiodic. Things such as children on swings, guitar strings, and vibrations of quartz crystals are all examples of periodic motion.In the last chapter, we discussed the periodic motion of things moving in circles, such as the planets. It turns out that if this motion is viewed from the side it can be considered an oscillation. Take a look:

This is the same kind of motion undergone by a mass attached to a spring, moving back and forth. The motion when an object is acted upon by a "restoring" force that pulls it backward toward its equilibrium point is called,

simple harmonic motion. The equilibrium point is the "natural" state of the system, the state where the least energy needs to be expended to keep it there.We can find general equations for simple harmonic motion by consider the planet moving in circular motion. If the place we are looking from is at the bottom of the y-axis, the displacement we're talking about is that along the x-axis.

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 can similarly see that the x-part of the velocity is given by

v_{x}= - v sin wtor to put it in terms of , since v = r,

v_{x}= - wAo sin wtFinally we know that centripital acceleration is given by

a_{c}= v^{2}/R = w^{2}Rso that gives us

a_{x}= - w^{2}Ao cos wtSo 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 /Tand to the regular frequency, f, by

w = 2pf, f = 1/TUsing what we know about trigonometry, we can find the maximum values for these quantities:

x_{max}= Aov_{max}=2pfAoa_{max}=4pAo^{2}f^{2}Now we can find out about real physical situations.

For homework in this chapter, do Page 306, Multiple Choice questions 1 - 26.Next stop, masses, springs and pendulums.