The graphs at the right describe the variation with time of distance, velocity and acceleration for simple harmonic motion when the where magnitude of both the amplitude and the angular frequency are equal to 1.
The graphs have been described using sin and cos functions on previous pages.
However, the
graphs have also been described as
sinusoidal because they have the
same shape, but they
start at a
different point in the wave cycle.
The
distance graph starts at the point of
maximum amplitude.
This was described using a cos function: x = Xcos ωt
The
velocity graph starts at the point where the
amplitude is 0 and then decreases.
This was described using a sin function: x = –ωXsin ωt
The
acceleration graph starts at the point where the
amplitude is a
minimum.
This was described using a cos function: x = –ωX2cos ωt
The
phase of a wave is the
fraction of the wave cycle which has elapsed before measurement begins and time is equal to zero. The phase for these three waves is different. Recall that the entire cycle is 2π.
The phase of the
velocity graph and the
distance graph differ by a
quarter of a cycle (±π/2).
If one is a sin function, the other must be a cos function with a sign difference.
The phase of the
velocity graph and the
acceleration graph differ by a
quarter of a cycle (±π/2)
If one of these is a sin function, the other is a cos function with no sign difference.
This means that the phase of the
acceleration graph and the
distance graph differ by a
half a cycle (±π).
These are the same function (cos or sin), but one is positive and one is negative.