Simple Harmonic Motion

Physics: Cambridge International A Level, topic 7.1.1

Simple Harmonic Motion (SHM)

SHM is any motion in which the acceleration is directed towards a fixed point (or equilibrium position for 1-D examples) AND is directly proportional to the negative of the displacement.

Formulae

Mathematically, the definition of SHM can be expressed in a formula:
$a=-\omega^2 x$ $a = - ω^{2} x$
- a is the acceleration.
- $\omega = 2\pi f$ , where f is the frequency of oscillation
- x is the displacement.

Acceleration-displacement graph

The acceleration-displacement graph is a straight-line graph that passes through the origin.
The angular frequency can be found from the gradient:
gradient $= -\omega^2$
$\omega = \sqrt{-gradient}$

Calculating maximum acceleration

If we are given the time period, T (or the frequency f), ω² can be calculated.
The maximum acceleration is given by:
$a_{max}=\omega^2A$ $a_{ma x} = ω^{2} A$
- where A is the maximum displacement.

Simple Harmonic Motion (SHM) Graphs

SHM can be represented graphically.

Acceleration and displacement

The acceleration can be found graphically, and so the displacement or vice versa from a graph.
E.g. if you have a displacement-time graph, you can calculate the period and angular frequency.
The displacement at any point can be read off the graph. This means you can calculate the acceleration (and then even the net force) because $a=-\omega^2 x$ .

Velocity and displacement

The velocity at any given time is found from the gradient of a displacement-time graph.
By finding the gradient at each point in time, we can produce a velocity-time graph from a displacement-time graph.

Velocity and acceleration

In a similar way, you can use a velocity-time graph to get the corresponding acceleration-time graph. This is because the gradient at a point on any velocity-time graph gives the acceleration at that moment in time.
You can then see that the acceleration-time graph is exactly the same shape as the displacement-time graph, reflected in the x-axis.

Simple Harmonic Equations

The condition for simple harmonic motion is that the acceleration is directed towards a fixed point and that the magnitude of the acceleration is proportional to the negative of the displacement.

Oscillations

For SHM the acceleration must be proportional to the negative of the displacement. $a{\alpha}-x$
If an object is under SHM it's displacement is found by $x=x_0\cos({\omega}t)$ where x_0 is the maximum displacement.
The defining equation of SHM is $a=-{\omega}^2x$

Velocity

The velocity of an object in simple harmonic motion is a sine function:
$v=v_0\sin(\omega t)$
Where $v_0$ is the maximum velocity.
We can also calculate the velocity of an object in SHM by using the equation:
$v = \pm\omega(\sqrt{x_0^2-x^2})$

Maximum points

The maximum displacement can be found when $\cos({\omega}t)=1$
The maximum displacement $x_0$ is the amplitude of the cosine wave.
The maximum speed is given by $v_0={\omega}x_0$
The maximum acceleration is given by $a={\omega}^2x_0$

Test yourself on Simple Harmonic Motion

Which statements about simple harmonic motion are true?
What is the correct equation for the maximum acceleration?
What can be found using the gradient of an acceleration-displacement graph?
Which statements about SHM graphs are true?
How do an acceleration-time and a displacement-time graph differ in simple harmonic motion?

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