PhysicsLAB Resource Lesson
SHM Equations

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In general, the sinusoidal equations for each property graphed at the right are:
 

 
  • where represents the frequency measured in hertz
  • and ω, or the angular velocity, equals and is measured in rad/sec
 
The magnitude of ymax equals the radius of the circle, r, or the amplitude, A, of the vibrating "spring" traced on the sine graph.

Using the relationships from uniform circular motion, the magnitude of the maximum velocity equals


Once again, pulling from the relationships of uniform circular motion, the magnitude of the maximum acceleration is equal to the magnitude of the mass' centripetal acceleration,

 
Our equations can now be written as:
 
 
The knowledge as to which circular trig function to utilize depends on the object's behavior at t = 0 seconds. For our purposes, it will start at one of the four critical locations (B-E) on the x- or y-axes diagrammed below.
 
From B, the position function would be
From C, the position function would be
From D, the position function would be
From E, the position function would be
 
To determine the velocity and acceleration functions, examine the orientation of the velocity and acceleration vectors or the slopes of the respective position and velocity graphs at each of the four critical locations.
 
 
We will limit our discussion to case B when the object has these properties at t = 0:



 
  • an initial position at equilibrium
  • an initial negative instantaneous velocity, and
  • an initial zero instantaneous acceleration.
 
From B, the position function would be
Starting at B, the velocity function would be
Starting at B, the acceleration function would be
 


Use this worksheet to practice writing and understanding these equations.
 
 
Summary of SHM
 
The following list summarizes the properties of simple harmonic oscillators.
 
  1. The oscillator's motion is periodic; that is, it is repetitive at a constant frequency.
  2. The restoring force within the oscillating system is proportional to the negative of the oscillator's displacement and acts to restore it to equilibrium.
  3. The velocity of the oscillator is maximum as it passes through equilibrium, and zero as it passes through the extreme positions in its oscillation.
  4. The acceleration experienced by the oscillator is proportional to the negative of its displacement from the midpoint of its motion.
 



 
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