Period Motion: Definition, Examples, Formula

Periodic motion referred to the motion that repeats itself in a fixed interval. In our daily life, we have been faced with some repetitive and non-repetitive motions. Periodic motions are repetitive motions in which the body repeats its motion after a regular interval of time. Some examples of periodic motions are:

• The motion of the planet around the sun
• The motion of the moon around the earth
• The motion of the bob of the simple pendulum

If the motion of the body is non-repetitive, then it is said to be a Non-periodic motion. In Non-periodic motion, the motion of the body does not repeat itself in fixed intervals of time. Some examples of non-periodic motions are:

• Local wind patterns
• Bouncing off the ball 
• The motion of waves in a sea

Read Also: Wave Nature of Electromagnetic Radiation

Key Terms: Periodic motion, oscillatory motion, Simple Harmonic motion, Angular frequency, Time period, Harmonic oscillations, Amplitude, Phase.

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Periodic Motion

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The body's motion is said to be periodic if it repeats itself after a regular interval of time. 

• The orbital motion of the Earth around the sun, the motion of the arms of a clock, the motion of a simple pendulum, etc. are examples of Periodic Motion. 
• The time taken by the body to complete one repetition or cycle is called its time period (T).
• The number of the cycle completed by the body in one second is called its frequency (f).
• The time period and frequency are reciprocal of each other.

Periodic Motion

Read More: Wave Optics

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Types of Periodic Motion

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The various type of Periodic motions are:

• Oscillatory Motion
• Harmonic Oscillation
• Simple Harmonic Motion

Oscillatory Motion

Oscillatory motion is a special type of periodic motion. If a body moves to and fro about a fixed point after regular intervals of time then the motion of the body is said to be oscillatory motion.

• It is also called Vibratory motion.
• The fixed about which the body moves to and fro in an oscillatory motion is known as the equilibrium position or mean position.
• The maximum displacement of the body from its mean position is called amplitude.
• Every oscillatory motion is a periodic motion, but every periodic motion is not oscillatory.
• The motion of the planet around the sun is periodic but not oscillatory.
• To and fro motion of a pendulum is oscillatory as well as periodic.

A periodic motion occurs to and fro or back and forth about a fixed point, which is known as oscillatory motion. Examples of oscillatory motion are the motion of a simple pendulum, the motion of a loaded spring, etc. Every oscillatory motion is periodic motion but every Periodic Motion is not Oscillatory Motion.

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Harmonic Oscillation

The oscillation which is expressed as a single harmonic function, in terms of sine or cosine function, is called harmonic oscillation.

Simple Harmonic Motion

The simplest form of oscillatory motion is known as Simple Harmonic Motion. A harmonic oscillation of constant amplitude and of a single frequency that restores force whose magnitude is proportional to the displacement and always moves towards the mean position is known as Simple Harmonic Motion (SHM).

A simple harmonic oscillation is expressed as a function of sine and cosine i.e

y = a sin ωt

or y = a cos ωt

Here

• a is the maximum displacement or amplitude of oscillation.
• ω is the angular frequency = 2πf = 2π/T
• f is the frequency of the oscillation
• T is the time period of the oscillation

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Some Terms which are Related to SHM are:

• Time Period: It is the time taken by the body to complete one oscillation. It is denoted by T.
• Frequency: Frequency is the number of oscillations completed by the body in one second. It is expressed as v.

Its SI unit of frequency is ‘hertz’ or ‘second-1‘.

Frequency = 1 / Time period

• Angular Frequency: The angular frequency is the product of frequency with factor 2π. It is expressed as ω.

Angular frequency (ω) = 2πv

Its SI unit of frequency is ‘hertz’ or ‘second-1‘.

• Displacement: It changes uniformly with time in a periodic motion. It is expressed as y.
• Amplitude: The maximum displacement in any direction from the mean position is known as amplitude. It is expressed as a.
• Phase: A phrase is a physical quantity that expresses the position and direction of motion of an oscillating particle. It is expressed as φ.

Non-harmonic Oscillation

A non-harmonic oscillation is known as the combination of two or more than two harmonic oscillations.

It is represented as y = a sin ωt + b sin 2ωt

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Important Formulae

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Some Important Formulae of SHM

1. Displacement in SHM at any time is expressed as

y = a sin ωt or y = a cos ωt

Where

• a is the maximum displacement or amplitude of oscillation.
• ω is the angular frequency = 2πf = 2π/T
• f is the frequency of the oscillation
• T is the time period of the oscillation

2. The velocity of a particle in SHM is expressed as

v = ω √(a² – y²)

• At the mean position y = 0 and v is maximum. So, the maximum velocity is expressed as v_max = aω
• At the extreme positions, displacement is the same as magnitude. So, y = a and v is zero.

3. Acceleration of a particle in SHM is expressed as

A or α = – ω² y

• A negative sign means the direction of acceleration is opposite to the direction in which displacement is towards the mean position.
• At mean position y = 0 and acceleration is zero.
• At extreme positions acceleration is maximum. So, A_max = – aω²

4. The Time period in SHM is expressed as

T = 2π √Displacement / Acceleration

Also Check:

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Things to Remember

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• The body's motion is said to be periodic if it repeats itself after a regular interval of time. 
• In Non-periodic motion, the motion of the body does not repeat itself in fixed intervals of time.
• If a body moves to and fro about a fixed point after regular intervals of time then the motion of the body is said to be oscillatory motion.
• The fixed about which the body moves to and fro in an oscillatory motion is known as the equilibrium position or mean position.
• The angular frequency is the product of frequency with factor 2π. It is expressed as ω.
• Every oscillatory motion is a periodic motion, but every periodic motion is not oscillatory.