Table of Contents

**Introduction:**

The meaning of a periodic function can be understood as a movement that repeats over a fixed time interval. An example of a periodic function is a chair shake, a circular motion. In other words, a periodic function can also be defined as a movement that returns to its original position after a period of time. After defining the periodic function, it is easy to confuse it with oscillatory motion at first glance. However, not all periodic functions oscillate at the same time. One of the main differences between the two is that periodic motion can be repeated from time to time, whereas oscillatory motion is limited only to the point of equilibrium. The most commonly used periodic functions are sine (sin), cosine (cos), tangent (tan), cotangent (cotan), secant (sec), and cosecant (cosec). Besides the periodic change of trigonometric functions, other periodic functions are light and sound waves.

**Overview:**

A periodic function is a repetitive action that occurs at fixed intervals. So, the function will reach the starting point after a set amount of time. A movement that returns to the same value at regular intervals. It is important to know that all periodic motion is a periodic function. This gives the idea that this function and oscillatory motion are the same. However, not all of these features vibrate. A periodic function can be any repeating action. An oscillatory motion can reach an equilibrium point. Typical examples of periodic functions include movements such as rocking chairs and swings. Anything that has circular motion is an ideal example of a periodic function.

Consider a pendulum swinging in equilibrium. Offset starts at 0 and reaches a positive point. Then it goes back to zero and then back to negative points.

The length of the gap between two identical points in a graphical representation is called a period. Usually, the horizontal distance along the x-axis is taken into account. A function creates a periodic function by moving a defined distance in a repeating period.

Although periodic and oscillatory motion sounds the same, not all periodic motion is oscillating. The main difference between periodic and oscillatory motion is that cyclic motion refers to any motion that is repeated over time whereas oscillatory motion is inherent to motion that occurs near an equilibrium point or between two states. A periodic function can define any periodic behaviour.

**Formula to calculate Periodic Function:**

The formula to calculate the periodic function is:

**f(x+P)=f(x)**

Here, f is considered as a periodic function if that is the case of a non-zero constant P for all values of x.

When 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 basically depends on certain aspects such as:

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

**Periodic Function Equation and its Derivation**

#### The equation for a periodic function given for an oscillating object:

- The cosine function repeats for trigonometry. This will also give you time for certain periodic movements. where omega is each frequency. This is the angular displacement per unit time.
- In the same way, 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.

Thus, we can conclude that,

**f=1/T**

The motion of the planets around the sun and the motion of the yo-yo are all examples of periodic functions. The example of a pendulum is a special case of a periodic function because it has simple harmonic motion, but there is a difference in the way the motion is expressed mathematically. If the periodic function can be expressed as a sine wave, then the motion is said to be a simple harmonic motion, such as oscillation or swing of a load applied to a spring. SHM (Simple Harmonic Motion) is a type of periodic motion in which the restoring force acts directly. It is proportional to the displacement and acts in the opposite direction of the offset.

Remember that the motion of a simple pendulum approaches simple harmonic motion only if the angle is small.

**SHM and Periodic Function:**

It is known that SHM is a simple harmonic motion. A pendulum is the best example of this type of movement. In this type of motion, the object moves back and forth, creating a period of time. When a given motion can be expressed as a sinusoid, it is a simple harmonic motion. In this case, the restoring force is in the opposite direction to the displacement.

The force acting on an object while it is in motion is called the restoring force and it is directly proportional to the displacement. These types of motion are periodic motion and oscillatory motion. We can call this a special case of periodic functions.

An ideal example of such a periodic function is the motion of a pendulum. A pendulum clock is a good example in everyday life.

Also read: **Periodic Motion**

**Frequently Asked Question (FAQs):**

**Question 1: What do you mean by a period in a periodic function?**

**Answer: **If a function has a repeating pattern, define it as a periodic function. A pattern is a consistent graphical representation of a period with the same length of intervals between each period.

**Question 2: What are the different components of a periodic function?**

**Answer: **Periodic function comprises two primary components such as Period and Function.

**Question 3: What is simple harmonic motion?**

**Answer: **Simple harmonic motion or SHM is a type of oscillatory motion. The total force acting on an object in this type of motion is actually the restoring force. That is, the SGM can be expressed as a periodic motion. In this motion, the object tends to move back and forth along a fixed-line. The clock pendulum is one of the best examples of simple harmonic motion.