
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.
This page prepared by
Edward F. Redish
Department of Physics
University of Maryland
College Park, MD 20742
Phone: (301) 4056120
Email: redish@umd.edu
Last revision 15. September, 2004.