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  Graphs Describing Simple Harmonic Oscillations
Graph of displacement against time
Recall that the relation between displacement and time for a body oscillating with s.h.m. is
So, a graph of x against t has the form shown below
 
Graph of velocity against time
Similarly, it has been shown that the relation between velocity and time is
So, a graph of v against t has the same form but is "shifted" to the left
 
Graph of acceleration against time
In order to find the relation between acceleration and time we simply take the basic equation which "defines" s.h.m.
and substitute for x, giving
So a graph showing how the acceleration of the body varies with time looks like this
 
N.B.
If we remember that the slope of an x against t graph represents velocity and the slope of a v against t graph represents acceleration, then we can more or less predict the shapes of the second two graphs by looking at the slopes of various points on the previous one.
 
For example
 
These observations tell us that the v against t graph must start out at a positive maximum, then curve down to zero, then reach a negative maximum (of the same magnitude as the positive one) etc
 
Of course, if you prefer a more mathematical sounding explanation... the second equation is the derivative of the first and the third equation is the derivative of the second.
 
Graphs of Energy against Time
If an oscillation is s.h.m., then the total energy possessed by the oscillating body does not change with time.
(Remember that an oscillation is a periodic conversion of energy between kinetic and potential energy.)
 
In practice, the total energy of most mechanical oscillations decreases with time, usually because of air resistance or something similar.
An oscillation in which the total energy (and therefore the amplitude) decreases with time is called a damped oscillation.
 
Graph of Total Energy against Time
This one is fairly easy to predict!
 
 
Graph of Kinetic Energy against Time
The form of this graph can be deduced by starting from the velocity against time graph and remembering that, in general, kinetic energy is given by
So the KE possessed by a body oscillating with s.h.m. is
 
Graph of Potential Energy against Time
Knowing that the total energy is constant, allows us to deduce that the potential energy graph must be have the same form as the kinetic energy graph but inverted.
 
   
  Mechanics Index Page