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In this experiment we will obtain the Lissajous figures.
These geometric constructions are are obtained when two harmonic oscillations
in perpendicular directions are combined.
We will use a laser and two oscillators fixed to mirrors.
The laser beam projection, reflected in both mirrors
forms the Lissajous figures on the screen.
The setup is very simple, we just need these two oscillators.
with mirrors attached to the tabs.
They have to be oriented perpendicularly.
This one is horizontal,
and this one is oriented vertically.
When the laser beam impinges on both mirror
Light incident on the screen adquires two coupled movements
both of simple harmonic type.
It is very important to adjust in height the two oscillators and the laser
so that the Lissajous figure is well drawn.
The laser beam must reach first the center of the first mirror
and its reflection shouls also impinge on the center of the second mirror
so the figure projected on the screen has the correct shape.
The first thing we will do is to show that we really have
two harmonic oscillators vibrating in perpendicular directions.
For example, if switch on this oscillator
and assign to it a small vibration frequency
about two hertz
what we see is that the laser beam reflected in both mirrors
draws a simple harmonic motion on the wall
with a period of two hertz.
We see on screen and also
as here in the mirror, vibrating horizontally.
However, if the horizontal oscillator is turned off
and we turn on the vertical oscillator,
also to a small frequency, which our eye can appreciate
we see that the prjection of the laser beam on the wall
shows a simple harmonic motion in the vertical direction.
At these low frequencies, for example 2 Hertz
our eye is still capable of seeing this
as a single point moving on the screen.
However, when the frequency increases to around about 10 hertz,
The image looks like a solid line.
This is much clearer
when the frequency increases
for example to about 20 hertz.
Here it is clear that our eye is not capable of
distinguishing the point motion,
our retina can only observe a continuous line.
We now turn both oscillators simultaneously.
We will assign to both oscillators
the same frequency,
for example 20 hertz.
What we have now are two movements,
coupled simple harmonic movements, in perpendicular directions
with a frequency ratio of 1:1.
Ie 20 hertz in a horizontal direction and 20 Hertz in the vertical.
What we see is generically an ellipse
This is the simplest Lissajous figure
that corresponds to a frequency relationship
1:1 between its components.
For an arbitrary phase difference between both oscillations
we get an ellipse.
If the lag was exactly 90 degrees
we could have a circle,
but we need the amplitudes
of the oscillators to be equal
In the extreme case in which the difference is 0 or 180 degrees
The ellipse tends to a straight line
This is the simple case
wherein the relationship in frequency is 1:1.
When the ratio of frequencies varies
Lissajous figures are more complex.
For example, when the frequency is doubled
on the horizontal axis,
so that the horizontal oscillator
has twice the frequency
of the vertical oscillator
ie about 40 Hertz
in the horizontal oscillator
and 20 Hertz in the vertical oscillator.
What I have is a figure a little more complex
which has the approximate shape of an eight.
In fact the perfect "eight" appears
when I have extreme lags
ie 0 or 180 degrees.
When I have a phase shift of 90 degrees.
What I get is a figure
that looks like an arc.
Again, changing the phase between both oscillators
I am sweeping forms
between the "8" and the bow.
From this scheme I can have Lissajous figures
increasingly more complicated.
To do this what I have to do is
modify the frequency relationship between both oscillators
relationships so that they are more complicated.
For example they ones appearig now.
These are the kind of figures that can be found
in B-movie science fiction.
For example, this will could mean that
Martians are attacking us
If a change to more complicated figures
It is not Martians but the Vulcanians and so on !