Periodic Motion

Any object that moves back and forth, in and out, or to and fro, is oscillating. When the oscillations take a constant amount of time to repeat, the object's motion is called periodic. 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 = 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 can similarly see that the x-part of the velocity is given by

vx = - v sin wt

or to put it in terms of , since v = r,

vx = - wAo sin wt

Finally we know that centripital acceleration is given by

ac = v2/R = w2R

so that gives us

ax = - w2Ao cos wt

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

and to the regular frequency, f, by

w = 2pf, f = 1/T

Using what we know about trigonometry, we can find the maximum values for these quantities:

• xmax = Ao
• vmax = 2pfAo
• amax = 4p2f2Ao

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.