Force Oscillations: Approximating Amplitude & Energy Decreases

[zeromaxxx]

Homework Statement

A lightly-damped oscillator is set in motion and observed as the oscillations decrease in amplitude. Derive approximate values for the factors by which i) the amplitude, and ii) the energy will decrease after Q cycles, where Q is the “quality parameter” defined in the notes. Assume Q is large enough that the period does not differ significantly from the period without damping.

Homework Equations

Q= $$\omega$$₀/2$$B$$

The Attempt at a Solution

 

[jbsteele]

Since the period does not differ significantly from the period without damping, then the Q value is large enough. Therefore, Q = \omega_ 0/2B > 1The factor by which the amplitude decreases after Q cycles is given by A_{Q}/A_0 = e^{-2BQ}where A_Q is the amplitude after Q cycles and A_0 is the initial amplitude. Therefore, the factor by which the amplitude decreases after Q cycles is A_{Q}/A_0 = e^{-2BQ}The factor by which the energy decreases after Q cycles is E_{Q}/E_0 = e^{-4BQ}where E_Q is the energy after Q cycles and E_0 is the initial energy. Therefore, the factor by which the energy decreases after Q cycles is E_{Q}/E_0 = e^{-4BQ}