Physics 101- Introduction to Mechanics
April 24- Simple Harmonic Oscillators

 

 

 

 
 

 

  What do the following have in common?
 
 

Each can (at least in some way) be described by the same basic physics/ math- simple harmonic oscillators.
 
 

Consider an ideal spring
F= -kx  A spring pulls (pushes) with a force that is proportional to the displacement from equilibrium. (As always this is simply an approximation to reality.)
 

For static situations (a hanging mass/ spring), you have a balance of forces so,  kx= mg.  (yawn)

Letís look at something new and different- dynamic/ oscillating situations.  (For now let's consider a horizontal spring, where the mass is on a frictionless tabletop.)
 
 

Plug our force expression into F=ma where you use the definition of acceleration to get a differential equation:
 
 






What is the solution?  What is x(t)?  Well, the answer is given as 
x(t)= A sin (wt +f).  You don't need to know how to solve the equation, you can verify that the solution is correct by substituting it into the equation and seeing that it works.

where A is the amplitude (meter), w is the angular frequency (radians/ sec) (notice that w is NOT angular velocity, this is a new variable), and f is the phase constant (radians).

The angular frequency describes how fast (or how frequently) an object is oscillating.  The phase constant is important since it helps us to describe where the object is at t= 0, and consequentially at all times.  Without f, you wouldn't know if your object started at the top or bottom of its motion (or somewhere else).
 
 

A glider on an air track is connected by a spring to the end of the air track.  If it is pulled 3.5cm in the +x direction away from its equilibrium point and then released from rest at t=0, what is the phase constant f?

  • 0
  • p/4
  • p/2
  • p
  • 3 p/2
  • other

 
 

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Jeff Phillips
phillips@lmu.edu
Loyola Marymount University
Spring 2002