Periodic and Oscillatory Motion
A phenomenon, process or motion, which repeats itself after equal intervals of time, is called periodic.If a body moves to and fro repeatedly about a mean position it is called oscillatory motion.
For a body to oscillate or vibrate three conditions must be satisfied.
 The body must have inertia to keep it moving across the midpoint of its path.
 There must be a restoring force (elastic etc.) to accelerate the body towards the midpoint.
 The frictional force acting on the body against its motion must be small.
Examples for Periodic Motion
 Heart beat of a person; period about 0.83s for a normal person.
 Motion of earth around the sun; period is one year.
 Motion of Halley's comet around the sun; period is 76 years.
Examples for Oscillatory Motion

Oscillations of a simple pendulum

Vibrations of a mass attached to a spring

Quivering of the strings of musical instruments etc.

Motion of the earth is periodic but not oscillatory because it is not to and fro.
Period and Frequency
Any motion that repeats itself at regular intervals of time is called periodic motion. The smallest interval of time after which the motion is repeated is called its period. The period can be represented as T.
Examples
 The period of vibrations of a quartz crystal is expressed in units of microseconds (10^{6}s).
 The orbital period of the planet mercury is 88 earth days.
Displacement
A block of mass is attached to a spring, the other end of which is fixed to a rigid wall. The block moves on a frictionless surface. The motion of the block can be described in terms of its distance or displacement x from the wall. Suppose we consider an oscillating simple pendulum, its motion can be described in terms of angular displacement Î¸ from the vertical. So the displacement can be expressed in terms of mathematical function of time. In case of periodic motion, this function is periodic in time.
The simplest periodic function can be written as:
f(t) = A Cos Ï‰ t (or) f(t) = A Sin Ï‰ t
Ï‰ is the angular frequency of the oscillation. The argument Ï‰ t in the above function can be increased by an integral multiple of 2Ï€ radians.
The above function will remain the same for each increment of 2Ï€ radians. So its period can be expressed as:
f(t) = f(t+T)
Furthermore, any periodic function can be expressed as a linear combination of sine and cosine functions.
f(t) = A Sin
Ï‰ t + B Cos Ï‰ t
By choosing A = D Cos
Ï† and B = D Sin Ï† , the above function can be written as,
f(t) = D Sin (
Ï‰ t + Ï† )
Where D and
Ï† are constants and can be found by squaring and adding the equation A = D Cos Ï† and B = D Sin Ï† ,
A^{2} + B^{2} = D^{2} Cos^{2}
Ï† +D^{2} Sin^{2} Ï† = D^{2 }(Cos^{2} Ï† + Sin^{2} Ï† )
Dividing the equation