## What is a Simple Harmonic Oscillator and Its Applications

- Frank
- May 25, 2022
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In our day-to-day life, we observe different kinds of motions such as the Linear motion of a car, Vibratory motion of a string, circular motion of a clock, etc… One of the most interesting and essential types of motion is Periodic motion. A body is said to be moving in a periodic motion when it repeats its path after each time interval. An example of periodic motion is the motion of clock hands, rotation of the earth, the motion of a pendulum, etc..When this periodic motion is about a fixed reference point it is called an Oscillatory motion. Simple Harmonic Oscillator is a special case of the oscillatory motion.

### What is a Simple Harmonic Oscillator?

An oscillator that performs the simple harmonic motion is called the Simple Harmonic Oscillator. The periodic to and fro motion of particles towards a fixed mean point is called the oscillatory motion. It is denoted by the formula F =-kx^{n}, where n is an odd number which denotes the number of oscillations. When the value of n = 1, the oscillatory motion is called the simple harmonic motion.

Simple Harmonic Oscillator consists of a horizontally placed spring whose one end is attached to a fixed point and the other end is attached to a moving object of mass m. The position of the mass when in equilibrium is called the mean position. When the mass is pulled parallel to the axis of the spring, it starts moving to and fro about the mean position. A restoring force, opposite to the direction of displacement, acts upon the mass pulling it towards the mean position. This device is now known as a simple harmonic oscillator.

### Simple Harmonic Oscillator Equation

In simple harmonic motion, the restoring force is directly proportional to the displacement of the mass and acts in the direction opposite to the displacement direction, pulling the particles towards the mean position.

According to Newton’s law, the force acting on the mass m is given by F =-kx^{n}. Here, k is the constant and x denotes the displacement of the object from the mean position. Displacement is proportional to the acceleration of the mass about the mean position. In simple harmonic motion, the value of n = 1.

As the acceleration is proportional to displacement, **a =d ^{2} x/ dt**

^{2}. Substitute the values in newton’s equation.

Thus, **F = ma**, **F = -kx.**

Therefore, **-kx = ma—-(1)**

**-kx = m(d ^{2} x/ dt^{2})**

By rearranging,** -kx/m = (d ^{2} x/ dt^{2}).—-(2)**

The function whose second derivative is itself with a negative sign will be the **simple harmonic oscillator solution** for the above equation. Sine and Cosine functions satisfy this requirement.

**f(x) = sin x, (d ^{2} x/ dt^{2})(f(x)) = -sin x**

**f(x) = cos x, (d ^{2} x/ dt^{2})(f(x)) = -cos x**

For simplicity sin (Φ) is chosen. The phase angle describes the displacement positions of the mass from the mean point. At the mean position, Φ = 0. When the mass moves in the forward direction and reaches the maximum point, Φ = π/2. When the mass returns to mean motion after maximum forward position, Φ = π. When the mass moves in a backward position and reaches a maximum point, Φ = 3π/2 and now when it moves to the mean position, Φ = 2π.

The taken by the mass to complete one complete to and fro cycle is called the Period denoted by T. The number of such oscillation occurring per unit time is called the frequency of oscillation, f. A denotes the extream positions of the object and also called as amplitude. Thus, the displacement of the simple harmonic motion is an algebraic sinusoidal function given as

**x= A sin ωt—-(3)**

Where ω is the angular frequency derived as Φ/t. From Eqn (2)

**-kx/m = (d ^{2} x/ dt^{2}). ω =2πf, T=1/f**

x= A sin(2πft+Φ), substitute in (2)

**-k( A sin(2πft+Φ)/m = -4π ^{2}f^{2}Asin(2πft+Φ)**

By solving, **f = (1/2π)√(k/m)**

**ω = √(k/m)**

Thus, x = Asin√(k/m)t is the equation of a simple harmonic oscillator.

### Simple Harmonic Motion Graphs

In a simple harmonic oscillator, restoring force acting on the spring is always directed in the opposite direction to the displacement of the mass. When the mass is moving towards the positive extream position +A, the acceleration and force are negative and are maximum. When the object moves towards the mean position from the +A position, the velocity increases whereas acceleration is zero at the mean position.

The velocity and speed of the simple harmonic oscillator can be derived from the above **simple harmonic oscillator waveform**. The displacement of the object is given by x = Asinωt=Asin√(k/m)t. Velocity is given as V = ωA cos ωt. Acceleration is given as a = -ω^{2} x. The period is given as T = 1/f where f is the frequency given as ω/2π, where ω = √(k/m).

Force acting on the mass at mean position is 0 and its acceleration is also 0. In a simple harmonic oscillator, acceleration is proportional to displacement. The sign of force depends on the displacement direction of the object from the mean position.

### Simple Harmonic Oscillator Applications

Simple Harmonic Oscillator is a spring-mass system. It is applied in Clocks as an oscillator, in guitar, violin. It is also seen in the Car-shock absorber where springs are attached to the car wheel to ensure the smoother ride. Metronome is also a simple harmonic oscillator that generates continuous ticks which helps the musician to play a piece with constant speed.

A simple harmonic motion comes under the oscillatory motion category of periodic motion. All oscillatory motions are periodic in nature but not all periodic motions are oscillatory. The restoring force in a simple harmonic oscillator obeys Hooke’s Law.

Simple harmonic motion depends on the stiffness of the restoring force and the mass of the object. A simple harmonic oscillator with large mass oscillates with less frequency. The oscillator with high restoring force oscillates with high frequency. The displacement, velocity, amplitude and force parameters of the simple harmonic oscillator are always calculated from the mean position of the spring. Frequency and period of the oscillations are not affected by the amplitude. What are the velocity and acceleration of the object when the spring is in its mean position?