The special form of simple oscillatory motion which is most simple is known as Simple Harmonic Motion. In this motion the body moves in a simple line on both the sides of its mean position. The definition of simple harmonic motion, “Simple harmonic motion is the projection on the diameter of a circle of the same motion as on the circumference of that circle.”
Suppose, as shown in a particle of mass m is moving with angular speed ω0 on the circumference of a circle of radius ‘a’ and center O. At any instant, when the particle is at point P, then the perpendicular drawn from the position of the particle to the diameter YY’ has its foot at point M. When the particle doing circular motion is at point X, then the foot point M of the perpendicular is moving along the diameter YY’ and is at point O. When the particle is at point Y then the foot point M also reaches at Y point. When particle is at X’ point then the foot point of perpendicular again comes back to O.
The particle reaching Y’ point the foot point of the perpendicular is also at Y’ and after completing one cycle when the particle is at point X then the foot point of the perpendicular also moves from Y’ to O. It is clear that when any particle moving on a circular path completes one cycle then foot point M of the perpendicular, moves in a simple line to and fro of the center O of the circle. This is the simple harmonic motion of point M. Therefore, projection of the foot point of perpendicular in uniform circular motion is simple harmonic motion.
Fig: Motion on the circumference of circle as a projection on diameter of the circle is a S.H.M.
Point M moving from O to Y, Y to Y’ and Y’ to O is called one complete vibration.
We know that in circular motion the centrifugal force on the particle (mω20α in magnitude) is as always towards the center of the circle. Hence, the force acting on the point P of the particle will be in the direction of OP. The component of force in YY’ direction will be (mω20 α sinω0t)
It is clear from fig. that mass of the particle m, angular velocity ω0 are constant then as a result force is directly proportional to displacement y, and its direction is opposite to displacement y. Hence, force F is called Restoring Force.
Hence, Simple ha monic motion is that imaginary or oscillating motion of an object to and fro its mean position in which the restoring force is;
(i) directly proportional to the displacement of the object from its mean position.
(ii) always points (moving) towards the mean position.
