Simulation fitting of experimental results: A damped pendulum

January 20th, 2012 by Charles Xie
The National Science Foundation recently awarded us a new grant to explore the concept of mixed-reality lab (MRL), which we proposed to combine the power of simulations and sensors to provide a new level of integrated learning experience that connects invisible science concepts with natural phenomena.

One of the proposed ways to integrate sensors and simulations is a strategy called "simulation fitting." When scientists observe something in the natural world, they typically build mathematical models to explain their observations. It is through this process that scientists make sense of their findings, understand the mechanisms more deeply, and derive new ideas for further investigations.

This is also an essential cycle of scientific inquiry we would like students to learn. The MRL project will explore how this "simulation fitting" strategy can improve science learning.

An example we have looked at is a simple pendulum. A swinging pendulum will eventually stop because of damping, which comes from two different sources: air resistance and bearing friction. Air resistance depends on the velocity of the pendulum whereas bearing friction doesn't.

My colleague Ed Hazzard has done a neat experiment that shows the difference of the two damping effects. His pendulum, under the normal circumstance, loses very little energy and can swing for a long time. To slow it down quickly, he attached a piece of paper to create a "sail," thus dramatically increasing the air drag. The result from a rotatory sensor shows the decaying of the rotational angle of the pendulum. In this case, the envelope of the curve looks like an exponential function (see the first image).

Launch the simulation.
Removing the "sail" and inserting some cotton into the bearing to increase the dry friction, he was able to amplify the effect of the dry friction. This time, the result of the rotational angle shows an envelope of a straight line, instead of an exponential curve (see the second image--it had two runs).

To understand these effects, we have created simulations that fit exactly the behaviors (see the third image). These simple simulations are based on solving Newton's equation of motion, with the only difference in the damping force: In one case it is proportional to the velocity; in the other case it is constant.

Our next step is to explore how to translate what we have done into a learning activity.

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