5.3.3 Damping

An oscillation is amped when an external force that acts on the oscillator has the effect of reducing the amplitude of its oscillations. E.g a pendulum moving through air experiences air resistance which damps the oscillations until the pendulum comes to rest.

When amping forces are small the amplitude of the oscillator gradually decreases with time but the period (T) is almost unchanged. This is light damping.

For larger damping forces the amplitude decreases significantly and the period also increases slightly. This is heavy damping (e.g is a pendulum was stopped with water). 

Very heavy damping occurs if we were to damp the pendulum with for example treacle/oil. Eventually there would be no oscillatory motion. Instead the oscillator would slowly move towards its equilibrium position.

The x-t graph is as follows:

When a system is displaced and allowed to oscillate without any external forces its motion is referred to as free oscillation. The frequency of the free oscillation is known as the natural frequency of the oscillator. A forced oscillation is one in which a periodic driver force is applied to an oscillator. The object will vibrate at the frequency of the driving force (the driving frequency).

If the driving frequency is equal to the natural frequency of an oscillating object resonance will occur. This causes the amplitude of the oscillations to increase dramatically and, if not damped, the system may break. The greatest possible transfer of energy from the driver to the forced oscillator occurs at the resonant frequency. This is why the amplitude of the forced oscillator is maximum. Resonance examples include:

• Tacoma Narrows Bridge
• Types of tuning circuits (e.g in car radios to select the correct frequency radio wave signal)
• Many clocks keep time using the resonance of a pendulum
• Musical instruments have bodies that resonate to produce louder notes
• MRI enables diagnostic scans of inside our bodies without surgery/the use of harmful X-rays (MRI stands for magnetic resonance imaging)

NOTE: damping a forced oscillation has the effect of reducing the maximum amplitude at resonance. The degree of damping also has an effect on the frequency of the driver when maximum amplitude occurs.

Amplitude-driving frequency graphs:

• Light damping: the maximum amplitude occurs at the natural frequency (fo) of the forced oscillator.
• As the amount of damping increases:
• the amplitude of vibration at any frequency decreases
• the maximum amplitude occurs at a lower frequency that fo
• the peak on the graph becomes flatter and broader

Image sources: Kerboodle OCR A Physics textbook

5.3.2 Energy of a simple harmonic oscillator

Okay so this is nice and short. It's just to do with the energy transfers involved in SHM.

As we know, energy must be conserved. Provided there are no frictional forces (e.g damping) the total energy for an object moving in SHM will remain constant:

• At the amplitude the pendulum has zero kinetic energy (it is very briefly stationary). All its energy is in the form of potential energy (GPE in the case of a pendulum, EPE in the case of a horizontal string)
• As it moves through its equilibrium position it has maximum kinetic energy (as it has maximum velocity) and no potential energy.

This graph nicely illustrates how the total energy remains constant:

Credit: Kerboodle OCR A Physics A level textbook

We also need to be able to interpret graphs. In this case I will use an example of a spring on a horizontal glider:

• EPE is given by the equation EP = 0.5kx² (k is the force constant of the spring. This means that a graph of EP-x will be parabolic
• The EPE is always positive and varies from 0 to 0.5kA²
• When at amplitude the glider will be momentarily stationary (so it has no kinetic energy) meaning the total energy of the oscillator equates to 0.5kA²
• The kinetic energy of the glider is the difference between the total energy and the EPE:
• EK = 0.5kA²- 0.5kx²= 0.5k(A²-x²)

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