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.
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4936730/494510_98233.jpeg)
Hence, when the displacement is positive the acceleration is negative(and vice versa). This can be described by the equation:
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4936730/749099_87220.jpeg)
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').
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4936730/682020_993846.jpeg)
separating the variable and integrating,
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4936730/303767_180517.jpeg)
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,
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4936730/692416_816286.jpeg)
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.
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4936730/84461_850278.jpeg)
SHM and Circular Motion
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4936730/366113_459109.jpeg)
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.
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4936730/409171_61741.jpeg)
That's the end of the topic in A2/A-Level Mathematics.