Introductory
Laboratory Apparatus
2002 AAPT Apparatus Competition, Boise State University
Name: J. C. Amato, R. E. Williams, E. J. Galvez
Address: Department
of Physics and Astronomy, Colgate University, Hamilton NY 13346
Phone: 315-228-7653
Fax: 315-228-7187
E-mail: Jamato@Colgate.edu; Rwilliams@Colgate.edu; Egalvez@Colgate.edu
Black Box
Mass-Spring Laboratory Experiment
We describe a simple, but very effective, laboratory exercise for the introductory calculus-based mechanics lab. A vertical mass-spring system is concealed within a long opaque PVC tube and attached by a string to an Atwood’s apparatus. Students are asked to design and carry out an experiment to determine the values of the mass and spring constant. Because of the design of the apparatus, students must combine a static measurement of the spring constant with a dynamic measurement of the oscillation frequency to solve the problem.
Description
What could be simpler than an experiment to find the mass m and spring constant k of a classical textbook mass-spring oscillator? But if the oscillator is hidden from view, is it still a simple matter to determine m and k? Perhaps. Now suppose that the motion is constrained so that the spring is always extended. Is it still an easy experiment? Apparently not! Once the latter complication is introduced, students can succeed only by synthesizing their understanding of Newton’s second law and their understanding of simple harmonic oscillation. The task is surprisingly challenging for beginning students, and it illustrates the process, pain and pleasure of real research. It seems to be a very satisfying experience for our students.
The mass-spring system is concealed within a vertical 4 foot length of 4 inch diameter white PVC pipe. Standard PVC caps are placed on either end, and a 0.5 inch diameter hole is drilled through the center of the top cap. The unknown mass is a short cylinder of brass (or steel) weighing about 200 g. It is glued to a 6 inch length of 2 inch diameter PVC tubing, and attached to a string which passes through the top cap. A short spring (k » 3 N/m) is stretched from a stationary bottom plate (not shown in the figure) through the smaller tubing to the underside of the mass m. The unstretched length of the spring is less than 6 inches, so that it is never relaxed. The string passes over two low friction pulleys (Pasco “Super Pulleys” work well) and is attached to a hanger accommodating calibrated cylindrical masses. The vertical position of the hanger is determined with a meter stick, and the period of oscillation of the system is measured with a stopwatch or other timer. The PVC pipe, pulleys, and meter stick are rigidly attached to a lab bench using standard clamps and rods.
Assuming that the rods, clamps, calibrated mass sets and timers are already available, each apparatus costs about $50 to build, including the cost of the two pulleys. Construction is very quick and simple if there is access to a lathe and drill press. No two unknown masses are the same. Computer interfaced hardware (e.g. photogates or “smart pulleys”) can be used to study the oscillations of the system, but will not make the experimental task easier or more tractable for students.
We introduce the experiment as a “modeling” exercise, that is, we challenge students to measure m and k, but do not prescribe step-by-step instructions. (The lab writeup, distributed to students at the beginning of the lab period, is attached to this document.) The contents of the large pipe are carefully described to students, but they are not allowed to look inside. We insist that students develop a theoretical model and a strategy, and then discuss their ideas with us, before beginning their measurements. This is critical to avoid wasted effort, and to ensure a productive and satisfying experience for the bulk of the class.
The data (dF/dy and the period of oscillation) are reproducible, easily compiled, and easily analyzed to yield best estimates and uncertainty for m and k. We grade students on the accuracy of their results, and, via their lab reports, on the care of their work and completeness of their analyses. Because the goal of the exercise is so specific, the reports tend to be succinct and easy to grade objectively.
We have run this experiment for two years, and it has consistently generated great enthusiasm among our students. We award extra credit to the first team to “publish” correct results, and conversely, we penalize students for incorrect answers. Students seem to like the challenge and the clearly defined “rules.” Surprisingly, groups (two students) tend to work independently, and the successful procedure does not propagate by word of mouth from group to group, or from lab section to succeeding lab section. In our opinion, “black box” labs such as this one contain some of the richness of real experimental research, and help inspire our students to continue their scientific studies.
Measured period for hanging
mass M = .255 kg: t = 2.31 ± .03 s
Physics
121 Laboratory
Black Box
1
The long white tube conceals a mass-spring system attached to a
string. The string loops over two low
friction pulleys and is tied to a hanging mass which can be varied. Your task is to determine (a) the value
of the concealed mass and (b) the value of the spring
constant. You must devise a successful strategy for
doing this, and carry out the measurements with care.
For a lab report, you must describe your procedure, summarize
your data in tables and/or graphs, and describe your calculations leading to
your answers. Your answers must include
error estimates for m and k.
You will be graded on the accuracy of your answers, and also on
how convincing and complete your report is.
You must convince the reader that you’ve done a flawless job, and that
your answer can be trusted to be correct.
Neatness and organization count!
You will also be graded on “originality.” The first team to finish with correct answers gets extra credit.
