We’ve already covered some of the basics of periodic motion
with our discussion of a mass on a spring back in Chapter 5. When
the end of a spring is stretched or compressed, the spring exerts
a force so as to return the mass at its end to its equilibrium
position. The maximum displacement of the mass from its equilibrium
position during each cycle is the amplitude of the
oscillation. One cycle of periodic motion is completed
each time the spring returns to its starting point, and the time
it takes to complete one cycle is the period, T,
of oscillation. The frequency, f,
of the spring’s motion is the number of cycles it completes per second.
A high frequency means each period is relatively short, so frequency
and period are inversely proportional:
Frequency is measured in units of hertz (Hz),
where 1 Hz = 1 cycle/second. The unit
of hertz is technically defined as an inverse second (s–1)
and can be applied to any process that measures how frequently a
certain event recurs.
We can summarize all of these concepts in an equation
describing the position of the mass at the end of a spring, x,
as a function of time, t:
In this equation, A is
the amplitude, f is the frequency,
and T is the period of the oscillation.
It is useful to think of each of these quantities in terms of a
graph plotting the mass’s displacement over time.
The graph shows us an object moving back and forth withina
distance of 1 m from its equilibrium position. It reaches
its equilibrium position of x = 0 at t =
0, t = 2, and t =
Note that one cycle is completed not at t =
2 but at t = 4. Though the
object is at the same position, x = 0,
at t = 2 as it was at t =
0, it is moving in the opposite direction. At the beginning
of a new cycle, both the position and the velocity must be identical
to the position and velocity at the beginning of the previous cycle.