6.1.4

# Simple Harmonic Motion

Test yourself

## 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 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:
• $a_{max}=\omega^2A$
• 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 $x=A\cos({\omega}t)$ where A is the amplitude
• The defining equation of SHM is $a=-{\omega}^2x$

### Maximum points

• The maximum displacement can be found when $\cos({\omega}t)=1$
• The maximum displacement is called the amplitude A
• The maximum speed is given by $v={\omega}A$
• The maximum acceleration is given by $a={\omega}^2A$