# Simple Harmonic Motion Definition, Formulas, And Graphs

In this article, I will discuss the definition of simple harmonic motions, formulas, and graphs. Oscillation is one complete movement from the starting or rest position, up, then down and finally back up to the rest position. Examples are pendulum, the beating of the heart, vibration of a guitar string, the motion of a child on a swing e.t.c.

A free oscillation means:

- No energy loss
- No external force acting
- Constant energy
- Constant amplitude

**Definition Of Simple Harmonic Motion**

It is defined as the motion of a particle about a fixed point such that its acceleration a is proportional to its displacement x from the fixed point, and is directed towards the point.

Mathematically,

The negative sign tells us that the acceleration is always in opposite direction to the displacement x

The slope or gradient of this graph = ω^2

__Conditions necessary for a body to execute S.H.M__

__Conditions necessary for a body to execute S.H.M__

- when the body is displaced from equilibrium, there must exist a restoring force
- this restoring force must be proportional to the displacement of the body

(it is always directed to the equilibrium position)

a is acceleration

x is the displacement

**Simple Harmonic Motion Formula**

The major formula in SHM is

the w = 2 pi f where w is angular frequency (unit rad/s) and f is the frequency (unit Hertz)

Also, in S.H.M we can have v = wx, where v is the velocity and x is the displacement.

**Understanding of terms**

Period : it is the time taken for one complete oscillation

Frequency: it is the number of oscillations per unit time

Amplitude : it is the maximum displacement from the rest position.

Displacement : it is the distance from the equilibrium position

F= 1/T unit is Hertz (Hz)

F is frequency, T is period

**Displacement And Velocity in Simple Harmonic Motion**

Displacement and velocity in S.H.M varies with each other. The velocity is always maximum at the equilibrium position. Also, the displacement is maximum when the velocity is zero.

**Simple Harmonic Motion Graphs**

This means when displacement is maximum, velocity is zero and acceleration maximum but in opposite direction.

V is the instantaneous speed

Interchange between Kinetic and potential energy

K. E is maximum when displacemet is zero

P.E is maximum when the displacement is maximum

**Damping**

Damping is an influence within or upon an oscillatory system that has the effect of reducing oscillations

It can now be properly define as reduction in energy of oscillations/ reduction in amplitude due to force opposing motion/ resistive force.

Note

During damping amplitude of oscillation does not decrease linearly also the frequency of the oscillations does not change as the amplitude decreases.

Types of Damping

Light damping: Amplitude decreases gradually as the oscillations continues for a long time

Critical damping: displacement decreases to zero in the shortest time without oscillation

Over damping : displacement decreases to zero in a longer time than for critical damping without any oscillation

Forced oscillation

This occur when an external force is applied to the original frequency causing a change in the frequency of the oscillation

For resonance to occur, there must be a system capable of oscillating freely and also have a way in which the system is forced to oscillate.

some terms

forced frequency: frequency at which object is made to vibrate

natural frequency of vibration: frequency at which object vibrates when free to do so

Resonance

resonance occurs when the natural frequency of vibration of an object is equal to the driving frequency giving a maximum amplitude of vibration

Uses of resonance

oscillation of a child’s swing

microwave to cook food

tuning of radio receiver

Problems of resonance

high-pitched sound waves can shatter fragile object

metal panels on machinery vibrate