Physics 101- Introduction to Mechanics
April 26- Simple Harmonic Oscillations






  Phase Transitions and How Energy related to SHO

How can you model the transition from liquids to gases?  What is the difference between gases and liquids (at the molecular level)?

Let's look at the potential energy stored in the system of two atoms.  One is taken to be fixed at the origin and the other oscillates about it.


  • For 1-D systems, one can usually use the analogy that the potential energy curve causes behavior just like a roller coaster with the same shape.  (This works because gravitational potential energy is proportional to height.)
  • To determine the motion, graphically add the total energy to the potential energy graph.
  • If E and U cross then you have a point where K is 0- a "turning point".
    • If there are no turning points between where the particle is at any given moment and infinity, then the particle is free to travel to infinity, in other words it is "unbound".
    • If a particle is trapped between turning points (it simply oscillates between these points), then the particle is "bound".
    • Any place where the particle does oscillate you can approximate the potential by a parabola.  This means that, for small oscillations about the equilibrium point, the motion can be treated as simple harmonic oscillations.  (Essentially this is what we've already done when we looked at ideal springs.  We said we were going to approximate their complicated behavior, by a relatively simple potential/ force.)
  • If E is below U then this tells you that K must be less than zero, which is impossible so this is an area that is forbidden.  (Well,... this is no longer true when working with small, quantum systems.  There it is possible for particles to exist in classically "forbidden regions"; this is what's known as tunneling.)
  • Notice that the slope of the potential curve gives you the force on the particle.  A steeper slope means a greater force on the particle.  (This about how a car behaves on a roller coaster track.)  This force points "down hill"- if the slope is positive, then the force is in the negative direction.   (F= - dU/ dx)


Notice how we can approximate the valley of this complicated curve by a parabola.  This is what we try to do with other complicated systems.  Often if we only have small oscillations about an equilibrium point, we are able to describe that portion of the potential energy curve by a parabola.  We know the solution to this problem; therefore we approximate the complicated system with the simpler simple harmonic oscillator.

This is the beauty/ power of simple harmonic oscillations.  Any oscillation can be approximated by this solvable model.


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Jeff Phillips
Loyola Marymount University
Spring 2002