This high-school physics demonstration will take students from a
simple
pendulum swinging in a plane to complex patterns called Lissajous figures when a pendulum is
allowed
to swing in two directions.
The aim of this demonstration is to make physics concepts more
tangible
to students while also applying principles outside of a textbook context. Oh yes, and to make
some pretty sand art ;-)
physical phenomena first hand.
principles beyond the scope of the textbook.
interactions with physics.
The position of the bob of a pendulum swinging back and forth in a plane as illustrated in Figure 1 can be expressed as) where x is the horizontal position of the bob, t is time, , is the angular frequency, is the phase and A is the amplitude. However, if you imagine the pendulum dropping sand as it swings, you will see nothing more than a single line of sand on the floor under the plane in which the bob goes back and forth. In order to see the sinusoidal behavior, you would need to be able to distinguish sand that fell at one time from sand that fell at a different time when the bob was in the same place. That could be achieved by putting a paper on the floor and slowly pulling it at a constant speed. But another way of seeing a time dependent trace is if the pendulum is not limited to swinging in a plane, but instead, is allowed to swing in both directions. In this case, the motion in each orthogonal direction must be sinusoidal as the resulting more complex motion can be seen as a superposition of two perpendicular planar swings. The equations of motion of this 2D pendulum are as follows:
.
These parametric equations form a 2-dimensional curve which the sand will trance out as the pendulum swings. The period, T, for each planar swing is, , where g is the gravitational constant, and l is the radius of rotation of the swing (the string length for the pendulum in Figure 1). The angular frequency is then , and so we see that for a simple pendulum hanging from a single string (as in Figure 1) the period/frequency must be the same for motion in both the x and y directions. This restricts the shapes the sand can trace out to being ellipses.
Now, the pendulum in this demonstration is the pendulum in Figure 2, which does not have the same radius of rotation for swings in the x plan as in the y plane. For swings in the x-direction the radius of rotation, l_{x }in Figure 2, is the length of the bottom string, from the bob to the intersection of the Y shaped structure. On the other hand, when the pendulum is swung in the y-direction, the radius of rotation, l_{y} in Figure 2, is the vertical distance from the bob to the top of the Y, or the bar from which the pendulum is hung. Thus, this pendulum can have different periods and angular frequencies in the two directions and so, depending on the exact period in each direction, it can make a whole set of looped figures called Lissajous figures after the 19^{th} century French physicist Jules Lissajous. The Lissajous table (Figure 3) shows how the shapes change based on the ratio of and , the angular frequencies in the x and y directions respectively.
Lesson Overview
This activity is designed for a Physics 11 or 12 classroom in British Columbia, Canada.
All the documents used for the presentation can be downloaded here:
It is also useful to have a bob on a string other than the one built into the 3D pendulum to use for explanations (a 2^{nd} bowling ball was used in this demonstration but even a simple rock tied to a string is fine).
The flow of the lesson can be broken into 5 parts which are outlined below along with some of the approaches used in this demo.
Introductory remarks:
Students will be encouraged to contribute as much as possible to this so it will serve as a warmup exercise. It will also serve as a brief assessment of where the class is at for the person running the activity.
Assign directions for x and y axes relative to the classroom walls, then swing the bob on a string used above for measuring the period, in the y=0 plane. This will give the sinusoidal trajectory x(t) discussed in part 1. Then, rotate, and swing the bob in the x=0 plane. Ask what the trajectory is now. Since the choice of coordinate system doesn’t affect the motion, one concludes that it must be the same function but with y in the place of x. Finally swing the pendulum in a circle or other 2d figure. Ask again about the trajectory. You’ll likely hear “complicated” or the equivalent in blank stares. At this point it may be helpful to do a little Socratic questioning along the lines of “Yes, but… Does motion in one direction affect the motion in the other direction? What is the x component of the trajectory? What is the y component?”.
This part of the lesson starts with Part 1 of the worksheet which has students connect to Geogebra’s online parametric grapher and plot some trajectories. The first questions lead students to discover the importance of the frequency ratio to the shape of the trajectory, and the 2^{nd} page has students fill in a Lissajous table, a table of different trajectories based on frequency ratio and phase (see worksheet solution).
At the end of the time for the first part of the worksheet have a brief full class discussion/sharing of results to make sure the entire class has developed an understanding of the impact of frequency on trajectory shape.
It’s time to start swinging the actual 3d demo pendulum! The slides suggest starting with a 3/2 frequency ratio. This choice is mostly based on it giving a figure that is complex enough to be interesting without being too messy (if it has a lot of lines of sand a more precise swing is required).
The first swing done; it will naturally bring up the question of why it doesn’t look exactly like the figures from the Lissajous table. This is an opportunity to have students practice thinking like a scientist!
This section mostly consists of students playing with/experimenting with the 3d pendulum, but it is also where students will calculate the string lengths from periods (part 2 of the worksheet). A first calculation (the worksheet uses a ¾ ratio) and swing can be done together as a class and subsequent ones can follow as time allows.
Some concluding remarks can touch on:
There may be a possibility to borrow some of the material from the physics department if you want to run this activity at school near Vancouver.
Feel free to contact me with any questions.
Email:88butterflys@gmail.com