
Problems for
Intermediate Methods in Theoretical Physics
Edward F. Redish 
Dimensions of damping
The equation of motion for a damped simple harmonic oscillator is
In some cases, the damping (bv) dominates, in others the restoring force (kx). We cannot compare b and k directly (Why not?) to decide. A dimensional analysis will help.
 From the parameters of the problem, m, b, and k, we can construct two different natural times: one associated with the damping (b), that we'll call t_{b}, and one associated with the restoring force (k), that we'll call t_{k}. Find these two "natural times" and prove that we cannot construct a natural length in this case.
 Suppose we are asking which term dominates for small times. Suppose t_{b} >> t_{k}. Which term do you think would dominate for small times? the damping term or the restoring term? Explain your reasoning.
 Construct a (nearly) dimensionless equation by replacing the time by a dimenionless time
From this equation, find a dimensionless combination of the problem's constants. Show why if this constant is >> 1 you expect one of the terms to dominate and if this constant is << 1, you expect the other term to dominate. Explain your reasoning and specify which terms dominate in which case.
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Edward F. Redish
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University of Maryland
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Last revision 15. September, 2004.