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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
| Table of Content |
Key Terms: Periodic motion, oscillatory motion, Simple Harmonic motion, Angular frequency, Time period, Harmonic oscillations, Amplitude, Phase.
Periodic Motion
[Click Here for Sample Questions]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
Types of Periodic Motion
[Click Here for Sample Questions]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
[Click Here for Sample Questions]Some Important Formulae of SHM
- 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
- 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 vmax = aω
- At the extreme positions, displacement is the same as magnitude. So, y = a and v is zero.
- 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, Amax = – aω2
- The Time period in SHM is expressed as
T = 2π √Displacement / Acceleration
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Things to Remember
[Click Here for Sample Questions]- 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.
Sample Questions
Ques: Is the motion of a simple pendulum simple harmonic without any assumption? (2 marks)
Ans: The motion of the simple pendulum is not strictly harmonic because here we assume that the angle is small. So, sin∅ = ∅.
Ques: Give examples of SHM. (2 Marks)
Ans. Some practical examples of S. H. M. are:
- Motion of piston in a filled gas cylinder.
- Atoms vibrate in a crystal lattice.
- The motion of a spring.
Ques: What is the effect in the time period of periodic motion if the amplitude of the simple pendulum is increased? (1 mark)
Ans: If the amplitude of the simple pendulum is increased there is no effect on the time period of periodic motion.
Ques: Is there any difference between forced oscillation and resonance? (2 marks)
Ans: External frequency is not matched with the oscillation of the object. In the case of resonance external frequency is matched with the frequency of the object.
Ques: Is only acceleration and displacement sufficient for simple harmonic motion? (2 marks)
Ans: Acceleration and displacement are not sufficient, because the direction of acceleration is not mentioned. In SHM, the direction of acceleration is always opposite to that of displacement.
Ques: Water in a U-tube executes S.H.M. What will be the time period for a mercury-filled U-tube at the same height? (2 marks)
Ans: The time period of the liquid in a U-tube of simple harmonic motion does not depend upon the density of the liquid. So, the time period will be the same, for the mercury is filled in the U-tube.
Ques: How is the length of the second's pendulum related to the acceleration due to gravity? (1 mark)
Ans: The length of the second's pendulum is directly proportional to acceleration due to gravity.
Ques: Does the direction of acceleration at various points during the oscillation remain towards the mean position? (2 marks)
Ans: No, it is not always true. The resultant of Tension in the string and weight of the bob is not always towards the mean position.
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