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 y_max 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.