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.

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 tb, and one associated with the restoring force (k), that we'll call tk. 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 tb >> tk. 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.