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Simple Harmonic Motion
The Relation between Displacement and Time for a Simple Harmonic Oscillation  
Point p oscillates between points A and B with s.h.m.   
 
Point O is the equilibrium position.   
At time t, the displacement from the equilibrium position is x.   
To find an equation to calculate the displacement at any given time t, it is helpful to consider the auxiliary circular motion, as in the next diagram.   
   
From the diagram, it is clear that   
   
We will assume that, at time t = 0, the point p was at O*.   
If the angular speed of the auxiliary circular motion is ω, then   
   
and therefore the relation we are looking for is   
   
   
The Relation between Velocity and Time for a Simple Harmonic Oscillation   
On the same auxiliary circular motion diagram, we now add a couple of arrows to represent the velocities of p and p'.    
   
The velocity of point p at any instant must be equal to the component of the velocity of point p' in a direction parallel to A-B  
The magnitude of the velocity of point p' is v' = rω  
Therefore the magnitude of the velocity of p is given by   
   
and as  
   
Therefore the relation between v and t is   
   
This reminds us that the maximum speed of p is equal to the speed of p' and occurs when t = 0, that is, at the equilibrium position.   
   
   
*Note that this is not always the choice... some texts assume, not unreasonably, that the oscillation starts from A or B.  
In this case we still end up with sinusoidal variations for x and v (of course) but sin and cos are interchanged.    
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