Derive equation of SHM

Harmonic motion is defined as a function of single  sine or cosine function.

When an object or particle is under a to and fro motion due to a restoring force that is directly proportional to the displacement from the mean position and also the force is directed towards the mean position. This motion is known as Simple Harmonic motion.

Some Examples of Simple Harmonic motion (SHM) are as follows-:

• Vibration of tuning fork
• Oscillations from a freely suspended magnet .
• Vibration of balance wheel of watch
• Oscillation of a loaded spring.

Equations related to Simple Harmonic motion

Let us consider a body oscillating from the mean position . The displacement by the body occurs and let us consider the displacement is small and hence we can say –

The force under action is directly proportional to the displacement.

Which further can be written as

Restoring force is directly proportional to Displacement

F x. Or

F = -kx ————————————–(1)

Where F = Restoring force

x = Displacement

K=Spring factor and force constant, it is a positive constant. It is given by restoring force per unit displacement with SI unit is Nm-1.

Equation (1) defines Simple Harmonic motion .The negative sign in this equation represents the fact that the force F acts in direction opposite to the direction of displacement x.

Based on Second law of motion

F= ma—————————————–(2)

F= Force , m = mass of the object , a= Acceleration.

From (1) and (2) we can say

ma = -kx

a = -K/m * x

a   x

Hence in simple harmonic motion acceleration is directly proportional to its Displacement and the acceleration is directed to the mean position.

Differential equation for Simple Harmonic motion

According to the equation for Simple Harmonic motion is as follows -:

F= -kx—————-(1)

Where k is spring factor and x displacement and the negative sign shows that the displacement and the force acts in opposite direction.

Based on Newton’s second law of motion we can say –

F = ma

F= m d2x/dt2 ———————-(2)

As acceleration is the differentiation of velocity.

From equation (1) and (2) we get

m d²x/dt²  =-kx

Or d²x/dt²= -k/m *x———————-(3)

By replacing k/m = ꞷ2( omega square)

Then we can substitute the same in equation (3)

d²x/dt²= – ꞷ2x

Or  d²x/dt²+ꞷ²x = 0

Thus, the equation for SHM

Which can be further written as -:

x= A cos ( ꞷt + Φ) ——————-(3)

Then we can say dx/dt =-ꞷAsin ( ꞷt + Φ)

d²x/dt²= -ꞷ2A cos ( ꞷt + phi)

Or d²x/dt²+ꞷ²x = 0

So, the equation (3) is the solution if equation (1)

In this equation x = A cos( ꞷt +Φ )

The simple Harmonic motion is defined at an instant t. Where A is amplitude

Φ= Φ0 + ꞷt is the phase of oscillating particle at initial phase at t =0

Characteristics of Simple Harmonic motion

• The motion of the particle is periodic
• It is the simplest kind of oscillatory motion
• The particle oscillates about the mean position with fixed amplitude and fixed frequency.
• The restoring force is proportional to the displacement from the mean position.
• The simple harmonic motion is represented by the single function of sine or cosine.

Some of the Important terms related to Simple Harmonic motion

• Displacement – It is defined as the distance covered by the oscillating particle from the mean position at any instant t. It is represented by x.
• Amplitude – It is defined as the maximum displacement of the oscillating particle from the mean position, is denoted by A and xmax = +/- A
• Time period- It is defined as the time taken to complete one oscillation.
• Frequency -It is defined as the number of oscillations completed per unit time. It is denoted by v( nu). Frequency is the reciprocal of the time period.
• Angular Frequency- it is obtained by multiplying 2π with v .It is denoted by

ꞷ( omega ) = 2π/T=2πv

• Phase – The phase of a particle is defined as the state of the particle with regard to the position and direction of motion at a particular instant. For example, in equation x= A cos ( ꞷt +Φ0 ) where, phase Φ = ꞷt + Φ0
• Oscillation- It is defined as the one complete back and forth motion of a particle which starts and ends at the same point.

Conclusion

Simple harmonic motion is a harmonic motion which means it can be represented in the terms of single sine or cosine function further the simple harmonic motion is a periodic function wherein the displacement of the particle takes place to and fro about the mean position with the displacement of the body directly proportional to the restoring force.

The particle oscillates with the fixed frequency and amplitude.