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# Periodic and Oscillatory Motion: Definition, Examples & Difference

Have you seen an ant trying to climb on a wall and fall down it try a number of times repeatedly and have you seen the motion of a pendulum of a clock? What type of motion is in both situations? Ant is repeating its motion in a definite time interval so it is periodic motion and pendulum of a clock is moving to and fro of a definite point so motion of pendulum is oscillatory motion.

Table of content

• Periodic motion
• Oscillatory motion
• Types of Oscillatory Motion
• Period and frequency
• Practice problem
• FAQs

## Periodic motion

If a body or a particle repeats its motion along a definite path after a regular interval of time, its motion is said to be periodic motion.

When a particle or a body performs motion repeatedly along a definite path after a regular interval of time, this motion is said to be periodic motion. That regular interval is termed as the time period.

• The path of periodic motion can be linear, circular, elliptical in shape or in the shape of any other curve.
• Examples: The Earth's motion around the Sun, the motion of the hands of a clock

• A body performing uniform circular motion comes under the category of periodic motion.
• The fixed time interval after which the motion is repeated is known as the time period (T) of periodic motion.

## Oscillatory motion

The periodic motion of a body which is to and fro in nature about a fixed position is known as oscillatory motion.

• The term oscillation refers to the to and fro motion of a body about a fixed position along a fixed path.
• Examples: The motion of a simple pendulum, the motion of a vibrating string, etc.

In case of a simple pendulum, the bob of the pendulum performs to and fro motion about the equilibrium position. The bob reaches an extreme position (where it stops momentarily, i.e., = 0) on both the sides of the lowest position.

• All oscillatory motions are periodic motions but the reverse is not true. In order to qualify for oscillatory motion, the body must be moving to and fro about a fixed point.

Example : Motion of a particle moving on a circular path is periodic and circular motion of projection of particle on diameter of circle is oscillatory.

Now, lLet us examine the case when a particle is being acted upon by a force , where n is an integer.

Case 1:→ Even integer (0, 2, 4……….)

For negative as well as positive displacements of the body, the force will always be directed along the-ve x-axis. This clearly indicates that the body cannot perform oscillatory motion (to and fro motion) about its mean position, = 0 (F = 0 at x = 0).

The motion of the particle is not oscillatory for even integral values of n.

Case 2:→ Odd integer (1, 3, 5……….)

In this case for the negative displacement (-x), the force will be directed towards the +ve x-axis.

Or

Similarly, for the positive displacement (+x), the force will be directed towards the -ve x-axis.

Or

Hence, the force will always act opposite to the displacement and it will tend to bring the body back to its mean position (= 0).The body will perform to and fro motion about = 0

The motion of the particle will be oscillatory for the odd integral values of n.

## Types of Oscillatory Motion\

1. Harmonic oscillation
• The term harmonic oscillation means periodic oscillation.
• Harmonic oscillations are those whose equations of motion can be expressed as simple trigonometric functions.

Or

Or

• The equations involving simple trigonometric functions become periodic in nature due to the periodicity of and . We know that sin and cos have a period of , and the values oscillate between +1 and -1.

2. Non-harmonic oscillations

• Non-harmonic oscillations are those whose equations of motion cannot be expressed as simple trigonometric functions.

Example:

Or

Or

## Period and frequency

The smallest interval of time after which the motion is repeated repeats itself is called its time period or period. In general period is denoted by the symbol . Its SI unit of period is second.

Example: If a ball is thrown in an upward direction and it goes to the highest point and returns to ground then collides and repeats the same then time taken by the ball upto return is called as period of motion of ball.

• As we know the trigonometric function repeats its value after a specific interval so,The periodic motion represented by trigonometric function has the same period as that of trigonometric function.

For example periodic motion represented by the has the period .

The reciprocal of the time period gives signifies the number of repetitions that occur per unit time. This is called the frequency of the periodic motion. It is The frequency is represented by . The unit of frequency of oscillation is . It is also called hertz, abbreviated as Hz.

The period and frequency is related as

## Practice problem

Q. Human heart beats 72 times in a minute. Find its frequency and period.

A. Frequency is heart beat ()= number of beat in 1 sec

Time period

Q. A particle is thrown upward with a velocity of . andIt collides elastically with the ground. What is the frequency of thise periodic motion? (Take acceleration due to gravity ).

a) 1 Hz

b) 2 Hz

c) 3 Hz

d) 0.5 Hz

A.As motion will repeat itself after as the ball hits the ground every time.

So, the time taken for the ball to reach the topmost highest position is.

The same time ball will take to return to the ground, so the total time

Hence the time period of oscillation is 1 s .

As we know the frequency . Ans.

Q. In which of the following functions of time represent (a) periodic and (b) non-periodic motion? Find the period for each case of periodic motion [ is any positive constant].

(i) sin t + cos t

(ii) sin t + cos 2 t + sin 4 t

(iii)

(iv) log (t)

A.

(i) sin t + cos t is a periodic function, it can also be written as 2 sin (t + ).

Now 2 sin (t + ) = 2 sin (t + +2) = 2 sin [ (t + ) + ]

The periodic time of the function is .

(ii) This is also an example of a periodic motion. It can be noted that in this motion each term represents a periodic function with a different angular frequency. As period is the least interval of time after which a function repeats its value, sin t has a period ; cos 2 t has a period ; and sin 4 t has a period . In this the time period of the first term is a multiple of the time periods of the last two terms. Therefore, the The smallest interval of time after which the sum summation of the three terms repeats is hence the sum is a periodic function with a time period .

(iii) This function is not periodic; it decreases with increasing time and tends to zero and never repeats its value.

(iv) The function log(t) increases with time t and never repeats its value so it is a nonperiodic function.

Q. A Force on a particle in motion is given by as, . What is the value of n for which the motion will be oscillatory?

a) 3

b) 4

c) Any integer

d) Any real value

A. If motion is oscillatory then force on the body will be toward the mean position.If the n is even the force will act in only one direction and motion will be rectilinear.

If n is odd the direction of motion will be opposite to the direction of x.Hence correct option will be (a) 3 .

## FAQs

Q. Is the force on the extreme position is Zero for an oscillation motion?

A. No, it has a maximum value of extreme position.

Q. Circular motion an example of a) Periodic b) Oscillatory c) Simple Harmonic d) Rectilinear motion ?

A. Circular motion is an example of periodic motion

Q. Can an oscillatory motion be non-periodic?

A. No, every oscillatory motion needs to be periodic.

Q. What is the characteristic of oscillatory motion?

A. In oscillatory motion force always acts in a direction opposite to the displacement towards the mean potion

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