Oscillations

Periodic Function

A periodic function is a repetitive motion that occurs in fixed time intervals. So the function comes to its initial point after a fixed amount of time. A common example of a periodic function is the motion of a rocking chair, swing set, etc. Anything that is in a circular motion is an ideal example of the periodic function too.

periodic function

Introduction to a Periodic function

It is the motion that returns to the same value at regular intervals. It is important to know that all periodic motions are periodic functions. This gives a perception that this function and oscillatory motion are the same. However, not all of these functions are oscillatory. A periodic function can be any motion that is repetitive. Oscillatory motion can reach a point of equilibrium.

To understand this, the periodic function displacement of an object. Let us consider a pendulum oscillating in equilibrium. The displacement will start from zero and reach a positive point. It then goes back to zero and then to a negative point.

In a graphical representation, the interval length between two identical points is termed as the Period. It is generally the horizontal distance along the x-axis that is considered. Function travels through this specific distance in a repeat cycle creating the Periodic function.

The mos consistently used Periodic functions includes sine (sin), cosine (cos), tangent (tan), cotangent (cotan), secant (sec), and cosecant (cosec).

In addition to the usable trigonometrical periodic variations, other periodic functions are the light and sound waves.

Formula to calculate Periodic Function 

The formula to calculate this function is as follows.

f(x+P) = f(x)

Here f is said to be a periodic function if that is the case of a non-zero constant P for all values of x.

If we extend the function h to all of R by the equation, then h(t+2)=h(t)

The value of a Period in a periodic function depends on certain aspects.

  • If the function is repeating in the presence of a constant period.
  • If the time interval between two waves is a constant
  • When fx= f(x+p), P represents the real number

Periodic Function Equation and its Derivation 

The equation for it is given for an oscillating object as follows:

  • The cosine function will repeat itself with respect to trigonometry. This will further give the time period for the particular periodic motion. Here omega is the angular frequency. This is the angular displacement that takes place per unit time.
  • Similarly, the frequency for the function is derived from the time period. This is because the total number of oscillations at a given time will be the frequency.

Hence, we can state that f = 1/T

SHM and Periodic Function

SHM is a simple harmonic motion. A pendulum is the best example of this type of motion. In this type of motion, the object will to and fro creating an at a given amount of time. When the given motion can be represented in the form of a sine curve, it is a simple harmonic motion. In this case, the restoring force is opposite in direction to the displacement.

The force that is being exerted on the object during the movement is called the restoring force. Also, this force is directly proportional to the displacement. This kind of motion is periodic motion as well as an oscillating motion. We can refer to this as a special case of a periodic function.

The ideal example of such a case of a periodic function is the movement of a pendulum. In everyday life, the pendulum clock will be a good example.

FAQ on Periodic Function

Question 1: What is a period in a periodic function?

Answer 1: When the function has a repetitive pattern, we define it as a Periodic Function. The pattern is a consistent graph representation of periods that has the same interval length between each cycle.

Question 2: How does one recognize if a function is periodic?

Answer 2: The presence of a positive numerical value represented as T determines the functional periodic. As we know that f(x+T) = f(x), hence the minimum value of T is determined as the period of a specific function.

Question 3: What are the different parts of a periodic function?

Answer 3: It comprises two primary components. They are a) Period b)Function

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sam

very helpful

Sandaras Edirisinghe
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Sandaras Edirisinghe

It was so much helping. Thank u for that. 👍

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