In a well-known 1938 book entitled "Demonstration Experiments in Physics", editor Richard Sutton describes a setup for producing periodic motion of a cart using weights instead of springs. With today's technology this experiment can be done using an air disk, and data can be collected with PocketLab Voyager's rangefinder. The data clearly shows that not all periodic motions are simple harmonic. The restoring force when weights are used is constant, while the restoring force with springs is proportional to the displacement. Springs produce simple harmonic motion--the period being independent of the amplitude. But the weights produce a periodic motion in which the period decreases with decreasing amplitude.
Lab Setup for Periodic Motion with Weights
Figure 1 shows the lab setup for producing periodic motion with weights. An air disk placed on a smooth table provides a low friction object. Two pieces of string attached to opposite sides of an air disk pass over pulleys on each end of the table. Masses M are attached to the other ends of the string so that the masses just touch the floor when the air disk is at rest at the center of the table. When the disk is moved toward one side of the table, one mass M is lifted while the other mass M is deposited on the floor. The smaller masses m are at a distance above the floor that is approximately half the height of the table. These masses serve two purposes: (1) they restrict the maximum amplitude of the motion to half the table height, and (2) they keep the strings on the pulleys when the larger mass M is resting on the floor.
Photos of the Lab Setup
Figure 2 shows an overhead view of the lab setup used by the author of this lesson. The air disk at the top of the figure is attached to a pair of springs. The disk at the bottom of the figure is attached to weights as described in Figure 1. PocketLab Voyagers are mounted to each air disk with a small piece of modeling clay. The Voyagers are oriented so that their rangefinders are facing a piece of white cardboard at the far right of the table. The white cardboard serves as a reflection surface for the IR beams from the rangefinders. Pulleys from the PocketLab Turbo Track kit are fastened to each end of the table with 3M damage free poster strips.
Figure 3 shows an enlargement of the two air disks. Screw eye hooks easily screw into the foam cushion around each air disk without causing severe damage to the air disks. The air disks were purchased at Educational Innovations, Inc. The 5 N/m springs were purchased at Vernier Software & Technology.
Figure 4 shows the masses that hang from the pulleys on each side of the table. Pinch sinkers and egg sinkers (used in fishing) are especially nice as their positions can easily be adjusted so that they are hanging on the floor when the air disk is at rest on the center of the table. The total mass of the sinkers at the bottom of the string is about 50 gm. The small mass half way down to the floor is a pinch sinker whose mass is about 4 gm. Warning: These sinkers do contain lead. As an alternative, you can use any other appropriate non-lead masses, such as those from science lab weight sets.
The video below shows a typical run for springs and for weights. Since the string and pulleys cause more friction, the weights portion of the video will dampen much more quickly than the springs portion of the video. But the key difference to note is that the period for the springs is constant, while the period for the weights shortens with decreasing amplitude.
Figures 5 and 6 contain Excel charts of position of the air disk versus time. These graphs were obtained from rangefinder data collected by the PocketLab app. The graphs provide clear evidence that the period is constant for the springs, whereas the period decreases with decreasing amplitude for the weights.
Suggested Student Activities for the Periodic Motion Lab
Describe the experiment setup using Figures 1 through 4.
Ask students to predict any differences they would expect for springs vs. weights.
Ask the students to set up the apparatus and then collect rangefinder data for springs and for weights.
Have the students make graphs of rangefinder data versus time for both cases. Do their graphs agree with their predictions?
What do they conclude regarding the period of oscillation when comparing springs versus weights.
Discuss the physics of the two cases with the class, encouraging the students to explain any differences based upon physical concepts.
Discussion of Physics Concepts
For springs the restoring force is given by the equation F = -kx, where k is the spring constant and x is the displacement of the air disk from its equilibrium position. In other words, the restoring force is proportional to the displacement. This is the defining condition for simple harmonic motion SHM.
For weights, the magnitude of the restoring force is constant and is provided by gravity, i.e., F = Mg, where M is the hanging mass and g is the acceleration of gravity. For constant acceleration, we know that:
Therefore, we would expect that t, in this case the period, is proportional to the square root of the displacement (amplitude of the oscillation) d. The smaller the amplitude, the smaller the period. This is exactly what we found in our investigation! This motion does not meet the definition of simple harmonic motion. We have seen two different examples of periodic motion whose differences are extraordinary!