In A2/A-Level Mathematics, we will discuss simple harmonic motion.
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 w2 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 'non-uniform 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 mid-position 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 SHM-circle 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(x-coord.) of a particle A is made on the diameter along the x-axis. This projection, as the particle moves around the circle, is the SHM displacement about O.
That's the end of the topic in A2/A-Level Mathematics.