Table of Contents

## Simple Harmonic Motion:

Simple Harmonic Motion is a fundamental kind of oscillatory motion. It results because of a typical acceleration that acts on the oscillating body.

A particle is said to execute simple harmonic motion if it moves to and fro about a fixed point periodically, under the action of a force F which is directly proportional to its displacement y from the fixed point and the direction of the force is opposite to that of the displacement. |

If the particle is disturbed from its equilibrium position so that its displacement is “y” then for the particle to perform the simple harmonic motion, the restoring force F must be such that

F ∝ -y F = -ky ……….(i) i.e., Restoring Force = – k (Displacement) |

Where “k” is a constant known as the ** force constant**. The force constant is defined as the restoring force per unit displacement. The minus sign in the above equation shows that “F” and “y” are oppositely directed.

If the mass of the particle is “m”, then the particle will experience an acceleration “a” which is given by-

a = F/m = -ky/m Writing, ω^{2} = k/mwe have, a = – ω^{2}yi.e. Acceleration = – ω^{2} (Displacement) |

Since *ω*^{2} is a positive constant, an alternative definition of a simple harmonic motion is as follows:

Simple harmonic motion is that motion in which the instantaneous acceleration is proportional to the displacement measured from the equilibrium position and the acceleration and displacement are oppositely directed. |

### Geometrical Interpretation of Simple Harmonic Motion:

Consider a particle P moving anti-clockwise with uniform angular velocity *ω* along the circumference of a circle of radius a and center O as shown in the figure. Let initially (t = 0) the reference particle be at X. When the particle was at X, the projection of the perpendicular drawn on the diameter YY’ of the circle from X is at O. When the particle revolving along the circumference reaches the point P, at an instant t, the projection is at N and when it reaches at Y the projection is also at Y. When the particle reaches X’ the projection comes back from Y to O. When the particle while going along the circumference reaches from X’ to Y’, the projection moves along the diameter from O to Y’, and when the particle reaches the starting point X, the projection comes back to O. Thus when a particle moves uniformly along a reference circle, the projection of the uniform circular motion moves in a straight line to and fro about the center O.

Thus Simple Harmonic Motion may be defined geometrically as the projection of uniform circular motion upon one of the diameters of the circle. |

### Characteristics of Simple Harmonic Motion:

The important characteristics of a Simple Harmonic Motion are the following-

- The particle or a body moves in a straight line and the motion is periodic with linear restoring force.
- The function of the force developed in the particle is to restore its position of rest.
- In this motion time period is independent of amplitude.
- The amplitude of the motion is the same on both sides of the position of rest.
- In this motion, the particle oscillates indefinitely with constant amplitude if the resistance of the medium is zero.
- Every oscillatory motion is not SHM but every SHM is oscillatory; only those vibratory motions are simple harmonic for which the restoring force is linear.

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