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 AB. 

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. 
