Simple Harmonic Motion: Part One

In the IB physics syllabus, one of the more difficult topics is wave phenomena, in particular the concept of simple harmonic motion. Let us begin by reminding ourselves of the definition.

Simple Harmonic Motion is defined as periodic motion that takes place when the acceleration of an object is proportional to its displacement from its equilibrium position and is always directed toward its equilibrium position.

(Click the picture to see simple harmonic motion occurring)

The two conditions of SHM:

1. The acceleration is directed in the opposite direction to displacement, or towards the equilibrium position.
2. The size of the acceleration is proportional to the displacement from the equilibrium position.

Firstly, we know that the motion is repeated, from the word periodic. Furthermore,  as the object gets further away from the equilibrium position, the acceleration increases, and it always directs itself to the central equilibrium position.   Acceleration is in the opposite direction from the displacement from the equilibrium position. Considering a pendulum example, when the bob is left of the equilibrium position, the acceleration is directed right, in this sense it is a negative relationship.

In a pendulum, simple harmonic motion is exhibited.  Of course, we are ignoring the air resistance that dampens the motion. (Damping will be discussed at length in a later article).  As the bob of the pendulum moves in the path of an arc,  reaching its maximum positions, the accelerating force acting on it is greatest, and the velocity is zero, and as it passes through the mean position, the midpoint between the two extreme positions, the accelerating force acting on it is zero, but velocity is maximized. Here it is in equilibrium.

Terms used in simple harmonic motion

Now we must introduce some mathematics into the concept. But first we must consider the symbols that show certain quantities in simple harmonic motion:

Displacement: x_, which is a measure of the displacement of an object from its equilibrium position at any particular instant during an oscillation.
Amplitude: x₀, is a measure of the maximum displacement of an object from its equilibrium position during an oscillation
Period: T, is a measure of the time taken for the system to complete one oscillation.
Frequency: f, is a measure of the number of complete oscillations that a system makes in one second
Angular frequency: ω, is a measure of the rate that a simple harmonic motion oscillation covers 2π or 360°.  However, in SHM, we typically use radians as a measure of angles, so 2π is preferred.

Equations used in simple harmonic motion

The first equation we will be looking at is the relationship between the time period of one oscillation and the frequency of the oscillations of a system displaying simple harmonic motion:

or  

This shows that the frequency and the time period of an oscillation are in fact inversely proportional.

This brings us on to our second equation:

  or  

Here we have introduced the symbol, ω, which is a measure of the rotational speed of a system.  By substituting the first equation, which links frequency with time period we can form a new equation.

  or  

The next equation that shall be discussed is that which links acceleration of a simple harmonic oscillator to its displacement, this is the defining equation of SHM:

As previously described the acceleration is in the opposite direction from displacement, and thus there is a negative sign. When relative to the equilibrium position, the displacement is above or leftward, the acceleration would be directed below or rightwards. Furthermore, the angular frequency of the system is purely a constant of the simple harmonic oscillator, and thus we see that acceleration is dependent on the displacement.

Another equation concerning the acceleration is about the restoring force, which causes the acceleration into the equilibrium position.  This is given by the equation:

Once again, it is negative as if acceleration is in the opposite direction from displacement, the force must also be, as we can see this from the equation:

which shows that for a given mass, force is proportional to acceleration. Linking back to the previous equation, k is a constant to be found, but only in SHM systems that involve a spring, is k the spring constant.

This concludes the first article on simple harmonic motion.  The next article will delve more into the mathematics behind this phenomenon, and will also include graphs of displacement, velocity and acceleration of the oscillating object.