Resonance in a driven pendulum system.
Source: Wikipedia
Most of MSU folks are at APS Global Physics Summit
Colloquium, 3:30 pm, 1415 BPS, Guillaume Pignol, University of Grenoble, Ultracold neutrons: a precision tool in fundamental physics
We solved the damped harmonic oscillator equation:
We chose a solution (ansatz) of the form
and computed the roots of the characteristic equation:
We found the roots to be:
We found that when , the roots are complex:
This means that the solution is oscillatory:
The solution is a damped oscillation with frequency .
When , the roots are real:
This means that the solution is not oscillatory:
where and .
The solution is the sum of two exponentials with different decay rates.
When , the roots are real and equal (repeated roots):
This means that the solution is not oscillatory, but also that our ansatz is not sufficient. The correct form of the solution is:
In most cases, we will work with weak damping.
Next week, we can choose what to do in class. What would you like to do?
What do we expect the phase space diagram ( vs ) to look like for a weakly damped harmonic oscillator?
The driven harmonic oscillator equation is:
with and . What is the dimension of the driving force ?
This ODE is a ________ differential equation.
Let , so that the driven harmonic oscillator equation is:
Note:
Note that if the driving follows a sine wave, then we have:
Interesting, , let try to work with
We found that the square amplitude of the driven harmonic oscillator is:
When is the amplitude of the driven oscillator maximized?