Problems for Intermediate Methods in Theoretical Physics Edward F. Redish

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.

1. 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.
2. 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.
3. 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.

Last revision 15. September, 2004.