Periodic Motion
I.
Introduction to Oscillations
In this chapter, we describe and analyze periodic motion, also called oscillatory motion. An object which
exhibits periodic motion is, simply put, an object which repeats the same motion over and over again.
We first describe some of the definitions needed to understand oscillations in general. Then we find the
kinematic equations (position, velocity and acceleration functions) for oscillations and subsequently
analyze the conditions under which oscillatory motion occurs. This will lead us to a better understanding
of two specific oscillatory systems, the mass-spring and pendulum. Finally, we will analyze the mass-
spring using energy conservation to learn even more about this important system.
II.
Displacement, Amplitude and Period
In all oscillating systems, there is a location called equilibrium, where the net force acting on the object
is zero.
The displacement, generically labeled
𝒙(𝒕)
, is the position of the object relative to the equilibrium
location. The displacement changes with time and therefore is a function of time.
The amplitude
𝑨
is defined as the magnitude of the maximum displacement from equilibrium that
the object attains.
The period
𝑻
is defined as the time it takes for the object to complete one full cycle of the motion.
One full cycle means that the object must return to the same position with the same velocity.
The period has SI units of seconds.
We also define two other quantities which are related to the period, the frequency and the angular
frequency.
The frequency is defined as the inverse of the period,
? ≡
1
𝑇
.
It has units of inverse seconds, also known as Hertz;
1 Hz = 1 s
−1
. Conceptually, the frequency is the
number of cycles that the object undergoes in a given time.
The angular frequency
𝜔
(not to be confused with angular velocity) is defined as
𝜔 ≡ 2𝜋?
.
This quantity is defined to take advantage of the connection between circular motion and oscillatory
motion. In an analogy with circular motion, one cycle can be thought of as 2
𝜋
radians. Thus the angular
frequency is the number of radians that the object traverses in a given time.