Simple Harmonic Motion

Simple harmonic motion can be used to describe the behavior of simple oscillating systems. Classic examples include pendulums and springs (basically anything that goes up and down or left and right periodically).

The defining condition for SHM is the following:

Acceleration is proportional to displacement from a point of equilibrium and always directed towards that point

Displayed mathematically:

a = -kx

a = force
x = displacement
k = some positive constant

A mass on a spring

Disclaimer: I'm using the convention for springs where tension is expressed in terms of the natural length and a modulus of elasticity. The same logic applies if you use the simpler relation T= kd. Just substitute k for λ/l.

Take for example, a mass on a light spring:

m = mass
g = gravitational field strength
l = natural length of the spring
λ = modulus of elasticity of the spring
e = extension of the spring
In this position, the mass is in equilibrium. The tension in the spring, λe/l, equals the weight of the mass, mg. Now what happens if we stretch the mass downwards? The extension of the spring will increase slightly, so the force of tension on the mass will also increase. Now the mass is no longer in equilibrium: there is a resultant force on it acting upwards. So it will accelerate upwards.

The mass will accelerate upwards until it has returned to the equilibrium position. It is now back where it started, with one important difference: it has some upwards velocity. At this point the mass will move upwards and we get a new picture:

Now the has traveled mass above the point of equilibrium. This means that the extension of the spring has gone down, so the force of tension on the mass has also gone down. There is now a resultant force downwards as the weight of the mass is greater than the tension from the spring. The mass now accelerates downwards until it returns to the equilibrium position with some downward velocity. The same thing happens again. And again. And again. In theory. In practice, energy almost always leaks away from the system.

This is simple harmonic motion: the mass moves down, is accelerated upwards, moves up, is accelerated downwards, moves down, and so on. The maximum value of the displacement is called the amplitude of the motion and is usually denoted by A.

Disclaimer: the rest of this section is devoted to a derivation of the defining SHM equation. There's nothing beyond GCSE maths and I think that it's useful to be able to get your head around it, but you don't need to in order to get full marks on the AQA Physics A2. Feel free to skip to the next section, entitled 'Important Equations', if you don't care about the maths.

It is possible to derive the defining SHM equation by considering a point in the motion of the mass-spring system. If we take downwards to be the positive direction then for a point in this picture:

Defining x as the extension of the spring, we get the following:

With the important result:

Where x* is an adjusted form of the extension which gives displacement from the equilibrium position:

x*=x-(glm/λ)

This is the defining equation for SHM, with x* as the displacement.

It is worth noting that when x*=0:

Which is the result you'd expect: when our displacement is zero, we are in the equilibrium position and there is no resultant force.

Another way of thinking about this is to say that the factor by which we adjusted x, glm/λ, is exactly the value which x would need to take for the system to be in equilibrium. I.e. when the spring has extended by glm/λ, the tension in the spring exactly equals the weight of the mass.

Important Equations

The defining SHM equation is this:

a = -kx

The value of k will depend on the exact nature of the SHM system concerned. We found earlier that for a mass on a spring, k=λ/lm. You should always be able to find a value of k by considering the forces at work. A general result for a mass on a pendulum is k=g/l where l is the length of the pendulum.

The maximum value of the acceleration occurs when displacement is at it's maximum value. I.e. when x=A

a(max) = -kA

Another important equation which gives the frequency of a SHM system is this:

2πf=√k

f=(√k)/2π

f = frequency in Hz or s^-1

Which means that you can always replace k with (2πf)^2 in any equation.

To find the displacement of the system after a certain length of time t use the following equation:

x = Acos(2πft)

t = time in s

Similarly to find the velocity for a given displacement:

v = 2πf√(A^2-x^2)

The maximum velocity is achieved when the system is at it's equilibrium point. This is fairly intuitive: think at what point on a swing are you travelling fastest? Plugging x=0 into the above equation yields:

v(max) = 2πfA

Finally, to find the period of the system (how long it takes to move through a single complete oscillation) simply recall that:

T = 1/f

And so,

T = 2π/(√k)

All of these equations can be derived from the defining SHM equation, but it's quite difficult and I won't be doing it here (FP3 differential equations methods required).

Sine and Cosine Waves

Cosine

The formula for displacement as a function of time is:

x = Acos(2πft) = Acos(2πt/T)

The cosine function varies between 1 and -1, so the maximum and minimum values of x as time varies are A and -A respectively. 

The maximum values of the cos function are when it's domain (2πt/T), = any integer multiple of 2π (0, 2π, 4π etc). This will occur when:

t = 0, corresponding to the system being released at maximum displacement
t = T, corresponding to the system having completed a single full oscillation and having returned to it's starting position
t = nT where n is a positive integer, corresponding to the system having completed n cycles

The same occurs for the minimum values of the cos function, just shifted by T/2. This should make sense as T/2 is the time it takes for the system to move from being at amplitude on one side to at amplitude on the other side (i.e. half one complete cycle).

So this is how the displacement equation works: the cos function dictates where in the oscillation the system is and then you multiply by A to get the actual displacement in space.

Sine

You can also use a sine function instead.

x = Asin(2πft) = Asin(2πt/T)

In this case the graph will look more like this:

This function behaves in exactly the same way, except it is shifted so that it passes through the origin. This means that when t = 0, x also = 0, so the system is at the equilibrium position at the start of the motion.

Use the cosine version when the system is moved to amplitude and then released, like us extending the spring and then letting go. Use the sine version when the system is initially at equilibrium and given some impulse at t = 0.

Angular Frequency

SHM is closely related to circular motion. If you look at the diagram below you will see that the object on the right is undergoing circular motion with constant angular speed. If you look only at it's vertical displacement (the object on the left) then you get something undergoing SHM.

If you look at the SHM this way, then one complete oscillation is analogous to a complete turn around the circle, i.e. the object has turned through an angle of 2π. It is for this reason that we talk about SHM systems as having an 'angular frequency'. It's your angular speed around the hypothetical circle in 'phase space'.

The angular frequency is this:

ω = 2πf = 2π/T

The expression 2πf crops up a lot in the SHM equations, so it is often replaced with ω.

Finally, memorize this equation. It will probably save your life in the exam.

k = ω^2 = (2πf)^2

Acceleration and Velocity

We saw earlier how to calculate the acceleration and velocity of the system as functions of displacement. I.e. we can calculate how fast the system is moving and how much it is accelerating given how far it is from the equilibrium position. But what if we want to look at acceleration and velocity as functions of time?

Recall the equation for displacement as a function of time (using the angular frequency substitution explained just above):

x = Acos(ωt)

If we differentiate this with respect to time we get an equation for velocity as a function of time. This requires C3 maths (differentiating trig and the chain rule).

v = -Aωsin(ωt)

Differentiate again and we get acceleration as a function of time.

a = -A(ω^2)cos(ωt)

If you draw out these graphs you will see that whenever x is positive, a is negative. This is exactly what you'd expect given the earlier equation a=-kx.

Similarly you will see that v = 0 whenever x is at a maximum or minimum. This corresponds to the fact that when the system is at amplitude it will be instantaneously at rest. Also, when x = 0, v will be at a maximum or minimum. This also corroborates with the reasoning we used earlier to find the formula for maximum speed: on a swing you are at your fastest at the bottom of the arc, i.e. when your displacement is zero!

Resonance

For any SHM system we found that there is a frequency it will tend to oscillate at. Given the defining SHM equation:

a = -kx

We saw that this frequency, f, is given by:

f=(√k)/2π

This also called the 'natural frequency' of the SHM system. Left to it's own devices the system will always try to oscillate at this frequency. Now think about a child on a swing. If you push the swing out then it will tend to oscillate at it's natural frequency. But how can you get it to swing even further out?

Pushing in one direction won't help; you'll just end up pinning the swing on one side with it not moving at all. So you have to push periodically, sometimes pushing hard and sometimes not pushing at all (or if you're really committed, sometimes pulling).

Consider what would happen if you applied your force at any old frequency. Sometimes you'll be pushing the swing in the same direction as it is moving and you'll speed it up. But sometimes you'll be pushing against it, slowing it down.

So what's the right frequency to push at? The natural frequency of the swing! If you push at this frequency then you'll always be pushing with the swing, never against it. The fancy way of saying this is that you'll be 'in phase' with the swing.

This is the effect called resonance: a force applied to a SHM system at it's natural frequency which causes the amplitude of the motion to constantly increase. This can be a very bad thing: if we did it to the mass on the spring from earlier we may end up stretching the spring so much that it breaks. If you apply a periodic force in the form of a sound compression wave at the the right frequency then you can cause glass to oscillate so violently that it breaks.