__What is called a “Simple Harmonic Motion”?__

*A periodic oscillatory motion in which a particle moves to and fro about a mean position under the action of a restoring force which is directly proportional to its displacement from the mean position and is always directed towards the mean position, is called Simple Harmonic Motion.*

A simple harmonic motion possess following characteristics –

- It should be a “to and fro” type of vibratory motion.
*Restoring force*must be proportional to the displacement of particle from mean position.- Restoring force must be acting in a direction opposite to the direction of
*displacement*.

### General conditions of Simple Harmonic Motion

A simple harmonic motion is an oscillatory motion. General condition of an oscillatory motion is given by –

F = - k x ^ n

Where, ( n ) is any odd number ( 1, 3, 5, ....... ) etc. and ( x ) is the displacement of particle from mean position.

*The simplest form of oscillatory motion is a simple harmonic motion for which ( n = 1 ) . *

Therefore, for a simple harmonic motion, we have –

F = - k x^{1} = - k x

In this equation, following points should be noted –

- Minus ( – ) sign indicates that, restoring force ( F ) always acts in a direction opposite to the direction of undergoing displacement ( x ) .
- Quantity ( k ) is a positive constant called
or*force constant**spring factor.*

Therefore, \quad \text {Restoring force in simple harmonic motion} \ \propto \ \text {Displacement of particle}

If, ( m ) and ( a ) are the *mass* and *acceleration* of a particle in linear motion, then from *Newton’s second law of motion*, we have –

F = m a

Therefore, \quad m a = - k x

Or, \quad a = - x \left ( \frac { k }{ m } \right )

But mass ( m ) and force constant ( k ) are constant for a given oscillating system.

Therefore, \quad a \propto x

Thus, a simple harmonic motion may also be defined in a different way as follows –

*If a particle moves to and fro about a mean position under an acceleration which is directly proportional to displacement of particle from the mean position and is always directed towards that mean position, then its motion is called a simple harmonic motion.*

__EXAMPLES –__

- Oscillations of loaded
*springs*. - Vibrations of tuning forks.
- Oscillations of a freely suspended magnet in an uniform
*magnetic field.*

__Oscillatory Motion__

*If a particle moves to and fro repeatedly about its mean position then its motion is called oscillatory motion or vibratory motion or harmonic motion.*

An oscillatory motion has following characteristics –

- Oscillatory motion repeats again and again about a mean position i.e. at ( x = 0 ) .
- Motion remains confined within extreme positions ( x = \pm A ) .
- A force ( F ) is always acting on the particle which tries to bring the particle in its mean position. This force is called
*restoring force.*

If, *displacement* of the particle is reckoned from its mean position, then –

F = - k x ^ n

Where, ( x ) is the displacement of particle from mean position and ( n ) is any odd number 1, 3, 5, ......... etc.

**EXAMPLE –**

- Swinging motion of
*pendulum*of a wall clock. - Motion of a piston of automobile engine.
- Vibrations of string of a guitar.
*Oscillations*of a mass body suspended from a coil spring.

Oscillatory motion is of two types –

- Periodic motion – which repeats at regular intervals of time.
- Non-periodic motion – which repeats at irregular intervals of time.

__Periodic Motion__

*A motion which repeats again and again at regular intervals of time is called a periodic or harmonic motion.*

In nature, all types of oscillatory motions are periodic motion. But due to the presence of *damping forces* offered by medium particles, the amplitude of oscillations dies out with time and thus it becomes a non-periodic motion.

A periodic motion can be of the following types –

- To and fro vibratory motion in a straight line.
- A uniform circular motion.
- A uniform elliptical motion.

__EXAMPLE – __

- The motion of planets around the sun.
- Motion of moon around the earth.
- Motion of hands of a clock.
- Heart beat of human.

**TO BE NOTED –**

All oscillatory motions are periodic motions because they repeat their cycle after fixed time intervals. But it is not necessary that, all periodic motions will also be oscillatory motions.

**Reason –** Oscillatory motion is a **“to and fro”** type of vibratory motion undergoing in a straight line. But a periodic motion can be a “to and fro” type vibratory motion or a circulatory motion. A circular motion is a periodic motion but it is not an oscillatory motion.

__Uniform Circular Motion is a Simple Ha____rmonic Motion__

*When a particle executes an uniform circular motion, the foot of its projection on a fixed diameter executes a simple harmonic motion.*

Consider about the figure shown. A particle is moving anti-clockwise direction with uniform *angular velocity* ( \omega ) in a circle of radius ( A ) .

When the particle or point ( P ) moves round the circumference of circle, its projection ( P' ) will execute a “to and fro” motion along a fixed diameter ( XX' ) .

**PROOF –**

The particle has a *centripetal acceleration* ( a _{ c } ) acting along the radius towards the centre O which keeps the particle moving in circular path.

Magnitude of this *centripetal acceleration* will be –

a _ { c } = \omega ^2 A

Let, PQ represents the centripetal acceleration ( a _ { c } ) . Perpendicular PP' and QQ' are drawn from points P \ \text {and} \ Q respectively on diameter XX' *. *Then P'Q' will represent the projection of acceleration ( a _ { c } ) on fixed diameter XX' .

From geometry of the figure, we have –

- OP' = OP \cos ( \omega t + \phi _ { 0 } )
- OQ' = OQ \cos ( \omega t + \phi _ { 0 } )

### Instantaneous Acceleration

Therefore, *instantaneous acceleration* of point P' moving on straight path XX' will be ( a _ { t } )

Acceleration of particle P' at any instant ( t ) will be the projection of the acceleration ( a _ { c } ) on XX' .

In the figure, P'Q' represents the instantaneous acceleration ( a _ { t } ) of point P' .

Therefore, \quad ( a _ { t } ) = P'Q' = ( OP' - OQ' )

= [ OP \cos ( \omega t + \phi _ { 0 } ) - OQ \cos ( \omega t + \phi _ { 0 } ) ]

= [ ( O P - O Q ) \cos ( \omega t + \phi _ { 0 } ) ]

= - a _ { c } \cos ( \omega t + \phi _ { 0 } )

= - \omega ^ 2 A \cos ( \omega t + \phi _ { 0 } ) = - \omega ^ 2 x

Therefore \quad ( a _ { t } ) = - ( \omega ^2 ) x

From above equations, it is clear that the motion of point P' has following characteristics –

- It is moving “to & fro” along fixed diameter XX' .
- Acceleration ( a _ { t } ) is proportional to the displacement ( x ) .
- Acceleration ( a _ { t } ) is acting in a direction opposite to the direction of displacement ( x ) .

*These characteristics are similar to the characteristics of a Simple Harmonic Motion. Hence, the motion of point P' is a simple harmonic motion.*

See numerical problems based on this article.