1. 1.2.12 Graphical analysis of SHM
Objectives
Plot graphs showing how displacement and acceleration vary with time during simple harmonic motion.
Select and apply the equation for the maximum speed of a simple harmonic oscillator

2. As we saw in the results from the Pendulum sensor.
250
200
150
100
Angle 50
Velocity
acceleration
100
200
300
400
500
600
-50
-100
-150
You need to note the phase relationship between the displacement, velocity and acceleration and also that the frequency is not affected by the amplitude of the swing!

3. Looking at them separately we get:
Graph of displacement x against time t
Graph of acceleration a against time t
Graph of velocity against time t

4. x = xocos(t) = xocos(2f)t
Observations are:
Displacement and Acceleration are in antiphase
Velocity and Displacement are out of phase by /2 rads or quarter of a cycle
So we know the motion known as SHM can be described by the equation:
x = xocos(t) = xocos(2f)t
Gives the shape of the curve
=2f or
=2/T
Maximum value - AMPLITUDE

5. v = xo = 2fxo a = 2 xo a = - 2x Also as described before:
a = -(2f)2x
a = - 2x
It is also useful to be aware of the following facts:
The maximum velocity is given by
v = xo = 2fxo
The maximum acceleration is given by
a = 2 xo

6. The graph for the acceleration against displacement is shown below.
Since a = - 2x or a  -x

7. The graph for the velocity against displacement is shown below.

8. Combining these two graphs we can draw the graph below
Combining these two graphs we can draw the graph below. The values for each is dependent on the Amplitude and therefore limits the graph to the outer edge which is the value of the amplitude.