Damped Harmonic Motion: Find Ratio & Periods for Decay

[superdave]

Homework Statement

A damped harmonic oscillator has mass m , spring constant k , damping force
- cv .

(a) Find the ratio of two successive maxima of the oscillations.

(b) If the oscillator has Q = 100 , how many periods will it take for the amplitude to decay to 1/ e
of it’s initial value? What fraction of the initial energy the oscillator has left by this time?

Homework Equations

The Attempt at a Solution

Okay, so for a)

T_d=2pi/omega_d = 2pi/(omega_0^2 - gamma^2)

The ratio is equal to e^-(gamma * T_d)

so r = e^-(c/2m * 2pi/(k/m - (c/2m)^2)^1/2)

But then for b, I'm lost again.

If Q = 100, omega_d=200*gamma.

So the ratio of successive maxima = e ^ - pi/100

But I'm not sure how that let's me figure out how many periods will it take for the amplitude to decay to 1/ e of it’s initial value.

 

[Jhero]

And then, how do I find the fraction of initial energy the oscillator has left by this time?

Hi there! For part (b), you can use the formula for the amplitude of a damped harmonic oscillator: A(t) = A_0 * e^(-gamma*t/2). Set this equal to 1/e of the initial amplitude (A_0) and solve for t. This will give you the time it takes for the amplitude to decay to 1/e of its initial value.

To find the fraction of initial energy the oscillator has left by this time, you can use the formula for energy in a damped harmonic oscillator: E(t) = 1/2 * m * (A_0 * omega_0)^2 * e^(-gamma*t). Plug in the time you found in the previous step and divide it by the initial energy (E_0 = 1/2 * m * (A_0 * omega_0)^2) to get the fraction of initial energy remaining.

Hope this helps! Let me know if you have any other questions.