# What is Harmonic Oscillator : Equation of Motion

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## What is a harmonic oscillator?

A harmonic oscillator is a type of dynamic system that exhibits periodic motion or oscillation around an equilibrium position. The motion of a harmonic oscillator is characterized by a restoring force that is proportional to the displacement from the equilibrium position and is directed opposite to the displacement. The most common example of a harmonic oscillator is a mass-spring system.

Mass

Spring constant

### The equation of motion for a simple harmonic oscillator is given by Hooke's Law:

if we use the math language the force is opposite and proportional to the displacement x: Harmonic oscillator equation: F = -kx

where:
• F is the restoring force,
• k is the spring constant (a measure of the stiffness of the spring),
• x is the displacement from the equilibrium position.
This equation implies that the force exerted by the spring is directly proportional to the displacement from the equilibrium position and acts in the opposite direction. This relationship results in simple harmonic motion.

The equation of motion for a simple harmonic oscillator can be expressed in terms of acceleration (a) and displacement (x):

### a = −m/k(x)

Where,
• m is the mass of the object.
The solution to this differential equation yields a sinusoidal or cosine function, representing the oscillatory behaviour of the harmonic oscillator:

### x(t)=A cos(Ï‰t+Ï•)

Where,
• A is the amplitude (maximum displacement from equilibrium),
• Ï‰ is the angular frequency (Ï‰ = √k/m)
• t is time,
• Ï• is the phase angle.

Harmonic oscillators are prevalent in physics and engineering and can be found in various systems, including mass-spring systems, pendulums, vibrating strings, and electrical circuits with capacitors and inductors. The concept of harmonic oscillation is crucial in understanding and analysing vibrational phenomena in nature and technology.