**Simple Harmonic Motion** (SHM)

In simple language to and fro motion of a particle about a fixed point (called equilibrium position) along with a straight line then its motion is called linear Simple Harmonic Motion (SHM).

Simple Harmonic Motion (SHM) is the motion in which the restoring force is directly proportional to the displacement of the body from its equilibrium position. The restoring force directs towards the mean position.

Also we can say that SHM is a special case of oscillatory motion. Because simple harmonic motion is an oscillatory motion in which the acceleration of the particle at any position is directly proportional to the displacement from the equilibrium position.

It is very important to note that all the Simple Harmonic Motions are oscillatory as well as periodic. However all oscillatory motions are not SHM.

In this type of oscillatory motion displacement, velocity, acceleration and force vary w.r.t time. It varies in such a way that it can be described by either sine (or) the cosine functions collectively called sinusoids.

**For Example:** spring-mass system

**Conditions for Linear SHM**

The condition of Linear SHM is when restoring force ‘F’ or acceleration ‘a’ acting on the particle should always be directly proportional to the displacement of the particle. And also directs towards the equilibrium position.

F∝−x

a∝−x

Where x is the displacement of particle from equilibrium position, F is Restoring force and A is acceleration.

**Mechanical System **

To make it more clear let us take an example of a mechanical system executing simple harmonic motion. The figure shows a simple mechanical oscillator having mass attached to the spring.

When a body attached to a spring is displaced from its equilibrium or mean position. The spring starts exerting a restoring force on the body. Thus it tends to restore the body to the equilibrium or mean position. This restoring force is responsible for the oscillation of the system.

As we know that Simple Harmonic Motion of SHM is the motion in which the restoring force is directly proportional to the displacement of the body from its equilibrium position.

Two things are responsible for mechanical oscillation i.e. Inertia & Restoring force.

For mass-spring system, the restoring force for small oscillations obeys Hooke’s law:

F=−kx ………….(1)

where k is the stiffness of the spring. Here the coordinate x=0 corresponds to the equilibrium position. At which the force of gravity is balanced by the initial tension of the spring.

Then, according to Newton’s second law

………….(2)

From equation (1) and (2)

………….(3)

Equation (3) can

………….(4)

or

⇒x′′+ ω^{2}x=0

On comparing equation (3) and (4)

………….(5)

Solution of differential equation is

x= Acosωt ………….(6)

Thus, the mass on the spring will perform undamped oscillations with the circular **frequency**

The **time period** of oscillation, respectively, will be equal to

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