5.3.1 Simple harmonic oscillations

Examples of oscillating motion include a simple pendulum, end of a ruler hanging over the edge of a desk, a volume of water in a U-shaped tube. In each example the object always starts in an equilibrium position and a force must be applied to displace it (it now begins to oscillate). When an object is displaced it travels back toward the equilibrium position at increasing speed (it is accelerating). It begins to slow down once it has reached the equilibrium position and eventually reaches maximum displacement. The maximum displacement is known as amplitude (the same as in waves). Again, it speeds up whilst returning to equilibrium position and then slows down to it's maximum negative displacement (etc, etc). There are a few key words we should cover before we delve into this topic:

• Displacement - the distance from the equilibrium position
• Amplitude - maximum displacement/maximum distance from the equilibrium position
• Period - the time taken for to complete one full oscillation
• Frequency - the number of complete oscillations occurring per unit time

NOTE: like in waves, we can also get phase difference with oscillating objects.


Angular frequency is a term used to describe the motion of an oscillating object and is closely related to the angular velocity of an object in circular motion. We can find it using the following equations:

ω = 2π / T

ω = 2πf

(T = the period of the oscillator, f is its frequency).

Simple harmonic motion
SHM is a kind of oscillating motion for which the acceleration is directly proportional to the displacement and acts in the opposite direction. In other words:

a = -ω² x

NOTE: the minus sign indicates that the acceleration of the object acts in the direction opposite to the displacement.

A graph of acceleration against displacement for an object moving in SHM has a gradient equal to -ω² (-angular frequency²). the gradient is constant implying that the frequency and period of an oscillating object in SHM is constant. It is also important to note that objects moving in SHM are isochronous, that is, the period T is independent of the amplitude A. This is because as amplitude increases speed increases, so the period does not change.

We need to know a quick experiment to determine the period and frequency of objects moving with SHM:
Using a pendulum:

• Attach a mass to a string on a clamp stand (ie, make a pendulum)
• Time the time taken to complete 10 oscillations on a stopwatch - divide this by 10 to obtain the period (T)/time taken to complete one oscillation
• Repeat with different amplitudes to show a consistent T. This proves that the period is independent of the amplitude.

Using a spring:

• Place a fiducial marker at the equilibrium position (this provides a clear point from which to start/stop time measurements)
• Repeat as above (take time for 10 oscillations etc).

Displacement-time graphs
The graph of displacement against time for an oscillator moving in SHM creates a sinusoidal shape. If no energy is transferred to the surroundings (ie damping) the amplitude will remain constant.

Lets take a pendulum as an example. At zero displacement the pendulum is at/moving through equilibrium position and the maximum/minimum displacement it is at the top/bottom of it's swing, respectively. The gradient is equal to the velocity (v) of the oscillator. At maximum displacement (amplitude) the velocity is zero as the mass is effectively turning around (it is momentarily stopped/turning and going back in the opposite direction, not sure if that made sense sorry). Velocity is maximum as the pendulum moves through its equilibrium position.

It is important to note that the velocity-time graph is similar to the displacement-time graph, only it looks like it's beed shifted along a bit (by T/4). The acceleration-time graph looks as though it has been shifted along by T/4 from the velocity-time graph, so by T/2 from the displacement-time graph. This makes it look the same as the displacement-time graph, only inverted (hence a = -ω² x ∴so a ∝ -x).

Photo credit: Kerboodle OCR A physics textbook

So, what do all these graphs actually mean? Well, we know that a mathematical definition of SHM is a = -ω² x. We know that displacement of a SHM oscillator varies with time in a sinusoidal manner (meaning that the x-t graphs can be a sine or cosine graph). The common solutions to the equation a = -ω² x are:

x = Acosωt

x = Asinωt

Now what's the difference I hear you shout. You use either one or the other, and which one you use depends on where the object begins oscillating from. If an object begins oscillating from it's amplitude use x = Acosωt. If it begins oscillating from it's equilibrium position, use x = Asinωt.

So, in SHM velocity varies with time. When angular frequency (ω) is increased with no change in amplitude the oscillator will be travelling the same distance in a shorter time interval. So Vmax increases (meaning the x-t graph gradient will increase).

Now consider increasing the amplitude. We know SHM is isochronous so it will travel the same distance in the same time period, so again velocity will increase. The velocity of a SH oscillator at displacement x is given as follows:

v = ± ω √(A²-x²)

We are aware that velocity has a direction as it is a vector so a velocity at any particular displacement will have a positive or negative value (depending on the direction in which the oscillator is moving). At the equilibrium position velocity is a maximum and x is zero:

v = ω A