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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 same form as a graph of sine of angle against angle.  
 
   
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  
 
   
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  
Remember that an oscillation is a periodic conversion of energy between kinetic and potential.  
If an oscillation is s.h.m., then the total energy possessed by the oscillating body does not change with time.  
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  
As it was clearly stated above that the total energy is constant, 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  
 
Therefore the K.E. against time graph has the same form as a sin2 graph, shown below.  
A copy of the v/t graph is added for comparison.  
   
   
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.  
   
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