# Oscillatory Motion - Definition, Examples, Derivation and Simple Harmonic Motion

Oscillation, in general, is defined as the repetition of a specific action or motion. A body is said to be in an oscillatory motion if it travels continuously from its initial point to an endpoint repeatedly in a given time frame. Oscillation can be observed in both mechanical and dynamic systems. Once initiated in an oscillatory motion, the desired situation of a body is if it keeps going about that “vibrating motion” indefinitely without stopping, except on the interruption using an external force. However, this is not feasible due to friction being an influencing factor and owing to friction; the object will eventually slow down its motion to come to a stop.

In the above explanation, oscillatory motion is referred to as a vibrating motion because it describes the former precisely. However, generally, if the frequency of the motion is very high, the motion is said to be a vibrating motion, and if it has a low frequency, the motion is said to be an oscillating motion. An object is said to satisfy the condition of oscillatory motion only if the object passes a specific point during each of its oscillations, which is its starting point and eventual endpoint. Oscillating motion is often confused with periodic motion. Periodic motion is when an object repeats a specific motion in equal intervals of time (e.g., the revolution of planets around the sun), whereas oscillating motion is when the object goes to and fro from the position of equilibrium (e.g., when a child plays on a swing in a park). Based on the direction of motion, oscillation can be classified into two as linear oscillatory motion and circular oscillatory motion.

## Derivation of the Three Equations

• Linear oscillatory motion: Here, the object moves in either an up and down motion or left and right motion. E.g., strings in stringed musical instruments like guitar, violin etc., ships and boats moving along the waves in oceans.
• Circular oscillatory motion: In this type of oscillatory motion, the object moves in either a clockwise or an anti-clockwise motion. E.g., the motion of the pendulum found in the watch, when a spinning top is spun.

## Simple Harmonic Motion

A very common example of an oscillating motion is a simple harmonic motion. It is observed when an object with a finite mass moves up and down, linearly in a straight line, with an acceleration of magnitude directly proportional to the object's length from its equilibrium position. This can be seen when a spring that is fixed at one end is loaded with a weight on the other end. Due to the mass of the object and the effect of gravitation acting on it, the spring moves down from its position of rest. As the spring goes down, a restoring force due to the tension present in the string acts on it and tries to restore it to its original position. Now, the weight makes the spring go down and crosses the position of equilibrium due to its momentum, and this motion is repeated until the spring eventually comes to a rest. The time taken for one such oscillation to occur is called the oscillatory period. An expression for this motion, in terms of the restoring force, as given by Hooke’s law, can be shown as follows.

F = - kx

where

• F = the restoring force acting on the weight by the spring (in newtons N)
• K = the spring constant (in N/ m)
• x = the measured displacement of the weight from its position of equilibrium (in meter m)
• the negative sign is to signify the motion of the weight downwards

## Example of Oscillation

• Mechanical oscillation: Double pendulum, tuning fork, vibrating string, Foucault pendulum, Helmholtz resonator.
• Electrical oscillation: Alternating current (AC), RLC circuit, astable multivibrator, Butler oscillator, Wien bridge oscillator.
• Electro-mechanical oscillation: Crystal oscillator
• Optical oscillations: Laser, Toda Oscillator, quantum oscillator.