Theory
A particle is said to move with S.H.M when the acceleration of the particle about a fixed point is proportional to its displacement but opposite in direction.
Hence, when the displacement is positive the acceleration is negative(and vice versa).
This can be described by the equation:
where x is the displacement about a fixed point O(positive to the right, negative to the left), and w^{2} is a positive constant.
An equation for velocity is obtained using the expression for acceleration in terms of velocity and rate of change of velocity with respect to displacement(see 'nonuniform acceleration').
separating the variable and integrating,
NB cos^{1}() is the same as arc cos()
So the displacement against time is a cosine curve. This means that at the end of one completete cycle,
Example
A particle displaying SHM moves in a straight line between extreme positions A & B and passes through a midposition O.
If the distance AB=10 m and the max. speed of the particle is 15 m^{1} find the period of the motion to 1 decimal place.
SHM and Circular Motion
The SHMcircle connection is used to solve problems concerning the time interval between particle positions.
To prove how SHM is derived from circular motion we must first draw a circle of radius 'a'(max. displacement).
Then, the projection(xcoord.) of a particle A is made on the diameter along the xaxis. This projection, as the particle moves around the circle, is the SHM displacement about O.
Example
A particle P moving with SHM about a centre O, has period T and amplitude a .
Q is a point 3a/4 from O. R is a point 2a/3 from O.What is the time interval(in terms of T) for P to move directly from Q to R? Answer to 2 d.p.

