Over the next few weeks, Paul Gluck, co-author of Physics Project Lab, will be describing how to conduct various Physics experiments. In this first post, Paul explains how to investigate motion on a cycloid, the path described by a point on the circumference of a vertical circle rolling on a horizontal plane.
If you are a student or an instructor, whether in a high school or at university, you may want to depart from the routine of lectures, tutorials, and short lab sessions. An extended experimental investigation of some physical phenomenon will provide an exciting channel for that wish. The payoff for the student is a taste of how physics research is done. This holds also for the instructor guiding a project if the guide’s time is completely taken up with teaching. For researchers it seems natural to initiate interested students into research early on in their studies.
You could find something interesting to study about any mundane effect. If students come up with a problem connected with their interests, be it a hobby, some sport, a musical instrument, or a toy, so much the better. The guide can then discuss the project’s feasibility, or suggest an alternative. Unlike in a regular physics lab where all the apparatus is already there, there is an added bonus if the student constructs all or parts of the apparatus needed to explore the physics: a self-planned and built apparatus is one that is well understood.
Here is an example of what can be done with simple instrumentation, requiring no more than some photogates, found in all labs, but needing plenty of building initiative and elbow grease. It has the ingredients of a good project: learning some advanced theory, devising methods of measurements, and planning and building the experimental apparatus. It also provides an opportunity to learn some history of physics.
The challenge is to investigate motion on a cycloid, the path described by a point on the circumference of a vertical circle rolling on a horizontal plane.
This path is relevant to two famous problems. The first is the one posed by Johann Bernoulli: along what path between two points at different heights is the travel time of a particle a minimum? The answer is the brachistochrone, part of a cycloid. Secondly, you can learn about the pendulum clock of Christian Huygens, in which the bob and its suspension were constrained to move along cycloid, so that the period of its swing was constant.
Here is what you have to construct: build a cycloidal track and for comparison purposes also a straight, variable-angle inclined track. To do this, proceed as follows. Mark a point on the circumference of a hoop, lid, or other circular object, whose radius you have measured. Roll it in a vertical plane and trace the locus of the point on a piece of cardboard placed behind the rolling object. Transfer the trace to a 2 cm-thick board and cut out very carefully with a jigsaw along the green-yellow border in the picture. Lay along the profile line a flexible plastic track with a groove, of the same width as the thickness of the board, obtainable from household or electrical supplies stores. Lay the plastic strip also along the inclined plane.
Your cycloid track is ready.
Measure the time taken for a small steel ball to roll along the groove from various release points on the brachistochrone to the bottom of the track. Compare with theory, which predicts that the time is independent of the release height, the tautochrone property. Compare also the times taken to descend the same height on the brachistochrone and on the straight track.
Design a pendulum whose bob is constrained to move along a cycloid, and whose suspension is confined by cycloids on either side of its swing from the equilibrium position. To do this, cut the green part in the above picture exactly into two halves, place them side by side to form a cusp, and suspend the pendulum from the apex of the cusp, as in the second picture. The pendulum string will then be confined along cycloids, and the swing period will be independent of the initial release position of the bob – the isochronous property. Measure its period for various amplitudes and show that it is a constant.
Have you tried this experiment at home? Tell us how it went to get the chance to win a free copy of the Physics Project Lab book. We’ll pick our favourite descriptions on 9th January. Good luck to all entries!
Featured image credit: Advanced Theoretical Physics blackboard, by Marvin PA. CC-BY-NC-2.0 via Flickr.