Changing reference frames
Your eye and brain can easily make the jump into a new reference frame.
Build the Snack Relative Motion Pendulum, or go to the exhibit.
Notice that there are two pendulums, one with a pendulum bob that oscillates right and left, and a second one which is a table that goes forward and back. Make each pendulum oscillate by itself. Notice that both pendulums have the same period.
Now start the table pendulum oscillating and also the pendulum bob. Notice how the pendulum bob seems to be moving forward and backward even though it is only moving side to side. Hold a hand up near the pendulum and watch its pure side to side motion relative to the hand.
Your eye and brain jump into the reference frame of the oscillating table. In this frame, the pendulum bob moves both right and left and front to back. Your eye and brain add the two motions creating one apparent motion. This apparent motion is the sum of two sinusoidal motions at right angles.
Experiment by starting the pendulum bob oscillating, then pulling the table toward you and releasing it. Notice that depending on the timing of your release you will see the pendulum move along a diagonal line, in a circle, or in an ellipse. The direction of the diagonal line and the direction of rotation around the circle or ellipse (clockwise versus counterclockwise.) depends on the position and motion of the pendulum bob when you release the pendulum table.
If the pendulum bob is at the extreme end of its swing when you release the table you will see a straight line motion. (The sum of two sine waves at right angles which are in phase, or 180 degrees out of phase is a straight line.)
If you release the table as the bob passes through the center of its motion it will seem to move in a circle, the result of two sinusoidal motions out of phase by 90 or 270 degrees.
Any other release position produces an elliptical motion.
An Inquiry Challenge.
See if you can determine how to release the table after it has been pulled toward you to produce a clockwise circular motion, then a counterclockwise motion. (Pulling the table toward you and then releasing it is the safest way to avoid being hit by the swinging table.)
Continue your exploration so that you can create each of the two straight line motions.
Now start with the table at its central rest position and with the pendulum moving. See if you can push the table away from you at the exact right time to create motion in a straight line or in a circle. Be careful that you don't get hit by the table on its return swing.
A model for polarized light
The relative motion pendulum can be used to model polarization of light. Light is a transverse wave, this wave can be modeled as the result of two oscillations at right angles to each other and to the direction of propagation of the light. Depending on the phase of these oscillations the light can be linearly polarized, or circularly polarized or elliptically polarized.
If when going through a material one direction of oscillation is delayed relative to another, linear polarization can be turned into circular polarization and vice versa. To show this, set up a linear oscillation with the relative motion pendulums then hold the pendulum bob briefly (for about 1/4 of a period) notice how the linear polarization becomes circular. This can be done in a material where the speed of light is different along two different polarization directions. Just the right thickness of such a material (a quarter of a wavelength of light) will turn linear polarization into circular polarization. Such a device is known as a quarter wave plate.
A piece of stretched cellophane tape has a speed of light that is slower for night polarized alone the direction of its stretch than perpendicular to the stretch. Thus by shining linearly polarized light into cellophane tape each color will turn to circular polarization, then to linear polarization at right angles to the initial polarization, then to circular again and so on. Since the difference in speeds is itself dependent on the color of the light, white polarized light will become colored when it is passed through a piece of cellophane tape and then through a polarizing filter. See the Snack Polarized Light.
It really pays to break the polarization into two parts at right angles when the light is traveling through a medium that has two different speeds of light which change for light polarized in two directions at right angles to each other.
Adding two equal amplitude, equal frequency sinusoidal oscillations at right angles can produce a linear motion, elliptical, or circular.
In the same way any constant frequency linear, circular, or elliptical motion can be analyzed as two perpendicular sinusoidal oscillations.
The coupled equations for x and y motion as a function of time, t, at angular frequency w radians per second with one oscillation at a phase difference of q radians are:
y = cos wt, x = sin (wt + q)
If q = 0 this is a description of circular motion. If q = 90 this describes linear motion.
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Scientific Explorations with Paul Doherty
2 October 2002