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Harmonic Oscillation (2013)

Page history last edited by Joe Redish 7 years, 11 months ago

Working Content > Oscillations and Waves

 

Pre-requisites:

 

There are three core situations that lead to a system oscillating.

  1. The system has a stable point where all the forces on the system are balanced (net force = 0).
  2. If the system deviates away from that stable point (for whatever reason) it experiences a force that tends to push it back to where it started.

That force tends to accelerate the system back towards its stable point. If the motion of the system is governed by anything like Newton's second law, the force makes it accelerate towards the stable point. As it approaches the stable point, the net force weakens since it vanishes at the stable point. But if the object is moving, it will continue to move through the stable point, not stopping -- since there is no force there to stop it. This gives us our third core concept necessary for oscillation:

  1. The system overshoots the stable point. The force will then build up to slow it down, but not before it has gone the other half of the oscillation. The situation will then repeat.

 

We'll go through this in detail in specific cases so it might make more sense then.

 

If there is a stable point, then the system's potential energy should look like a well with a minimum point. The (classical) system will sit at the minimum point when it is stable. (A quantum system never can sit exactly at rest at a minimum point, which is why we draw our molecular bound states not exactly at the bottom of the potential well.) If the restoring force can be treated as linear -- or equivalently, if the potential energy can be treated as a parabola -- then the motion is called harmonic. It's called "harmonic" because the solution of Newton's second law (a second order differential equation that determines the motion of the object) are sines and cosines of time with a particular frequency -- just like the result produced by a pure musical tone heard at a particular point in space. The simplest example of this is a mass on a spring, so we will treat that example in great and gory detail.

 

But harmonic oscillation has value far beyond the case of a mass on a spring. For almost any stable situation, the energy for small enough deviations around the stable point can be approximated by a quadratic (parabola) and this is equivalent to a mass on a spring. What we learn in mathematical modeling is

 

If the equations for two different systems are the same, then each one serves as a good analogy for the other.

 

Since the equations that describe the mass on a spring appear in so many other cases, it is a useful analogy, and a valuable conceptual tool to have in your toolkit.

 

Follow-ons:

 

Joe Redish 3/11/12

 

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