When a body moves periodically in a straight line on either side of a point, the motion of body is called the simple harmonic motion.
Thus, the simple harmonic motion is a special case of the periodic motion, obviously, a simple harmonic motion is definitely a periodic motion, but all the periodic motions are not the simple harmonic motions.
For example, the motion of earth around the sun is a periodic motion, but it is not the simple harmonic motion. On the other hand, the motion of a simple pendulum is simple harmonic motion as well as the periodic.
In a simple harmonic motion, the body moves periodically in a straight line on either side of its an position such that its acceleration is proportional to the displacement of the particle and the direction of acceleration is always towards the mean position. In other words, the motion of a body under a restoring force is a simple harmonic motion.
The force, which is directly proportional to the displacement of the body from its mean position and is directed towards the mean position is called the restoring force i.e., the restoring force tends to bring the body back to its mean position.
1.2.1 Conditions or Characteristics of Simple Harmonic Motion
Following are the conditions (or characteristics) of simple harmonic motion:
(i) The motion must be in straight line on either side of a definite point (mean position).
(ii) The moving body must pass from its mean position repeatedly after a definite time i.e., motion must be periodic.
(iii) The acceleration of the moving body must always be proportional to the displacement of the body from its mean position and the direction of acceleration must always be towards the mean position, i.e.,
Acceleration α displacement and in the direction opposite to displacement.
If at any instant, the displacement of body from its mean position is x, the acceleration of the body is
But by Newton’s law of motion, force = mass x acceleration
Force α – x
Thus, the force acting on the body must be proportional to the displacement of the body from its mean position and its direction must be towards the mean position (i.e., the force must be the restoring force).
A system, whose motion is simple harmonic, is known as the simple harmonic oscillator.
For example, motion of simple pendulum for small amplitude, motion of the compound pendulum, motion of the mass attached at the free end of a spring rigidly fixed at its other end, motion of torsional pendulum etc.