Sample Physics Modeling Cycle

The Constant Force Particle Model

The Paradigm Lab Investigation

This modeling cycle begins, as always, with a paradigm lab.  In the pre-lab discussion, students use their background knowledge about forces to describe the force diagram for a modified Atwood's Machine set-up.  The students are shown a low friction lab cart attached to a light string that in turn passes over a low friction pulley to a second mass suspended below.  When released, the system accelerates.  From the previous unit on forces, the students know that the net force on either mass or the system cannot equal zero.  The students identify the measurable variables of net force, mass, and acceleration of the system.   Agreement is then reached that since only two variables should be related at any time, the third must be kept constant.  Since it is unlikely that one would be able to keep the acceleration constant over many trials, acceleration is better suited as the dependent variable.  Thus, two scenarios are agreed upon:  First, the net force will be kept constant, the mass changed and the acceleration will be measured.  Second, the total mass of the system will be kept constant, the net force changed and the acceleration will be measured.  This will result in two graphs, one of acceleration vs. mass, the other of acceleration vs. net force.  The tools for measuring acceleration are provided.  Groups are challenged to carefully select a measuring device that will easily give reliable data for the acceleration of the system.  Photogates and picket fences, ultrasonic motion detectors, and Smart Pulley set-ups are provided.  The pre-lab discussion concludes with an agreed-upon purpose:  To graphically and mathematically model the relationship between the net force, mass and acceleration of a system of objects.

Students collect the data and immediately analyze their results to determine the acceleration of their system.  Using their group's computers, acceleration values are determined.   After collecting all appropriate data, new graphs of acceleration vs. net force and acceleration vs. system mass are generated.  (See photo #1 below.)  From these graphs, mathematical models are generated.  Students prepare whiteboards that summarize their group's results.  (See photo #2 below.)  The summaries show sketches of the graphs as well as any related equations.  The students then present their results to the class where comparisons with other groups results are done.

In the post-lab discussions, students are guided to analyze the slopes of their linear graphs.  The acceleration vs. system mass graph yields an inverse relationship.  Using the graph straightening techniques from a previous unit, students replot the acceleration v. the inverse of the system mass.  The straight line graph that results yields a slope that is equal to the constant net force applied to the system.  The acceleration vs. net force graph yields a slope that should approximate the inverse of the system mass.  Both graphs then yield the hoped for net F = ma mathematical model.

Model Deployment

Following the post-lab discussion, the model is deployed through several sets of problems.  These problems are carefully chosen to cut the contextual strings to the paradigm lab.  The results of the model development from the lab are used in a variety of situations that become further and further removed from the lab situation.  Ultimately, the model is seen as the tool for solving a wide variety of problem situations.  Student presentations, again using whiteboards (See photo #3 below.) to summarize solutions, initiate vigorous discussion that often return to the paradigm lab for justification.

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