Ubiquity of Vibrations and Waves
Degrees of Freedom
Simple Harmonic Motion (and relation to 1D systems near equilibrium)
Waves are everywhere – everything waves (Howard Georgi)
Waves and vibration are ubiquitous – nearly all systems under some conditions undergo oscillation of some form. The electric guitar demo in this lecture showed a chain of events that involved wave and oscillation phenomenon at many levels:
- neuro-electronic pulses
- vibration of guitar string
- vibration of electromagnetic field
- pulses sent through electronics of guitar down to amplifier
- amplifier vibrates speaker panel using electromagnetic coils
- sound waves created by speaker travel through air
- your eardrums vibrate
- new electronic pulse sent through your brain
Many forms of oscillation are present in this “simple” demonstration!
We introduced the “period” (the time which a repeating phenomenon takes to repeat itself). This is often referred to as “T”
Degrees of freedom – it is crucial in systems to identify the moving parts, and the ways that they can move. The number of degrees of freedom is the number of coordinates necessary to specify what a system looks like in a stationary picture. For a point particle in a normal room, there are 3 degrees of freedom (the x,y and z coordinates that locate the particle). For a extended rigid body (like a bowling ball), there are 6 degrees of freedom: The x,y, and z coordinates along with 3 angles describing the orientation of the object. A mass on a spring that can only move in 1 direction counts as a single degree of freedom.
Simple Harmonic Motion – Simple harmonic motion is pure sinusoidal motion. This kind of motion is what a mass on an ideal spring undergoes when you displace it from equilibrium and let it go.
Mathematica Demo – go to the following site to get Mathematica:
http://its.syr.edu/mobile/licenses/Mathematica.html
The mathematica file we used in class is here (use shift-click to download rather than opening in the browser):
https://jhubisz.expressions.syr.edu/phy360/wp-content/uploads/sites/5/2016/08/Week-1.nb
SHM is ubiquitous – As we discussed, all systems sufficiently near an equilibrium point undergo simple harmonic motion when they are displaced from equilibrium. The reason for this is that all functions (like the potential energy) are quadratic in the displacement from equilibrium, and so the force, which is the negative of the derivative of the potential energy, is linear. See the full lecture notes for details.
EXAMPLE DEMO:
Pendulum – potential energy is sinusoidal, but near minimum is is quadratic, and force is linear. Thus for small oscillations, the pendulum is just like a mass on a spring, and undergoes simple harmonic motion. This is not the case for large oscillations! The motion is then periodic, but NOT sinusoidal.