Simple Harmonic Motion (SHM)

Motion of twig of a tree, motion of a cradle, motion of vibrating strings of veena, violin, guitar and sitar, motion of stretched membranes of drums and tabala, motion of prongs of vibrating tuning fork - in all these examples the objects will be making to and fro or up and down motions between two fixed points and are said to be making oscillatory or vibratory motions and these motions are called oscillations or vibrations.

If the time taken for each vibration or oscillation is same, i.e. if the period of oscillation or vibration is constant, then the object is said to be making periodic motion. Even though a body is making uniform circular motion (moving with uniform speed in a circular path, like the hands of a Clock) is not an oscillatory or vibratory motion but it is a periodic motion as the time taken for each revolution is constant.
SHM: It is a special kind of oscillatory or vibratory motion.
Example: simple Pendulum
A bob with negligible mass tied to one end of a torsionless thread (say, ordinary sewing thread) fixed at a point and made to oscillate between two fixed points, with a small amplitude (Not like cradle from one end of the room to the other), but gently which is called simple pendium. (See. Fig-1).

C B or q- Amplitude
A, B - Extreme positions
C - Mean position
When the bob is taken to an extreme position from the mean position and left for oscillation, the bob of the simple pendulam will be oscillating between the two fixed points A and B. From the mean position C, the extent the bob moves to a point is called displacement (x). The maximum displecement is called amplitude which can also be measured in terms of q. After reaching B, the bob falls towards C because of gravity of the earth and possess acceleration (Acceleration due to Gravity).
A body (here the bob is body which is a simple pendulum in motion) is said to be making simple harmonic motion.
a). When it is in to-and-fro motion or up and down motion between two fixed points along a straight line.
b). Its acceleration is directly proportional to its displacement, such that accelaration and displacement are oppositely directed, acceleration towards the mean position and displacement is away from the mean position (Important condition)
acceleration µ - displacement (x)
negative sign (-) indicates that acceleration and displacement are oppositely directed.
Note: SHM is oscillatory or vibratery and period motion. But all oscillatory or vibratory and periodic motions are not simple harmonic motinos. Water Waves: When a stone is dropped into still waters, the particles of the water will experience up and down simple harmonic motion due to which concentric circles of water called waves are formed. Thus, water waves, waves in ropes, sound waves are associated with SHM.

Why this motion is called 'Simple Harmonic Motion' ?
This motion is called "Simple Harmonic Motion" because it is the projection of a particle making uniform circular motion and circular is considered to be the simplest of all the curves. In Fig.2, P is making circular motion with constant speed N, the foot of the perpendicular moving between two fixed points A and B along a straight line is executing SHM. N is the projection of P on the diameter AB.

Harmonic means musical effect. All the musical instruments, the strings of veena, violin, guitar and sitar and streteched membrances of tabla, mridangam and air columns of flute, sehnai make SHM. The atoms in a solid object also constantly experice SHM - so don't you think the most popular motion in nature is SHM ?