# Periodic Function

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## What is a Periodic Function?

One can understand periodic function meaning as the motion that occurs repetitively over the course of fixed time intervals. Periodic function examples include rocking a chair, which is a circular motion. In other words, one can also define a periodic function as the motion that returns to its initial position after a fixed duration of time.

### Understanding the Difference Between Periodic and Oscillatory Motion

After going through the periodic function definition, one can easily get confused with oscillatory motion at first glance. But, not all periodic functions are oscillatory at the same time. One of the biggest differences between the two is that, while periodic motions can be repetitive at times, oscillatory motion is only constrained at an equilibrium point.

For better understanding, one can take the example of a bob of a pendulum. It oscillates along its equilibrium position in a periodic manner. During its movement, the displacement takes place from zero to positive to negative passing through its initial position. Such a motion is periodic and oscillatory at the same time. Another aspect of the oscillating motion is Simple Harmonic Motion (SHM), where the restoring force of the periodic motion is directly proportional to that of its displacement.

The Formula for Periodic Function

One can define the periodic function f, along with a non-zero constant in the same case:

f (x+P) = f (x)

The function is applicable for all the values of x in the same domain. While the constant P is termed as the period of a function.

Derivation of Periodic Function Equation

For an oscillating object, its periodic function can be defined as:

f(t) = Acosωt

With the cosine part repeating itself after a certain point of time, it can defined as:

cosθ = cos(θ+2π)

cos(ωt) = cos(ωt+2π) ——(1)

Considering the time period to be T:

f(t) = f(t+T)

Acosωt = Acosω(t+T)

Acosωt = Acos(ωt + ωT) ——(2)

So from equation 1 and 2, we can derive:

ωT = 2π

Thus, T = 2πω

### Time Period of Periodic Function

The time period of the periodic function is given by;

T = 2πω, ω is the angular frequency of the oscillating object.

### Frequency of Periodic Function

We all know that the frequency is given by the total number of oscillations per unit time. For periodic motion, frequency is given by;

F = 1/T;

F = 1/ (2πω);

### Solved Questions

Example 1

State whether a motion can be periodic but not oscillatory or not?

a) True

b) False

Solution

The answer is option A. You can often find motions that are periodic but not oscillatory. For example, a uniform circular motion is a periodic motion, but there is no restoring force being applied on it. So, it is not an oscillatory motion.

Example 2

For a given pendulum, if l is the length of the bob, while its mass is m, and it is moving along the circular arc with angle θ. So if a spherical mass M is placed at the end of the circle, what is the momentum of the sphere gained by the moving bob?

a) Infinity

b) Zero

c) Constant

d) Unity

Solution

The answer is B. The sphere will not attain any momentum through the bob at the end of the circle. This is because, at the end of the circle, the velocity of the bob becomes zero.

Example 3

Let us assume that a 2 kg body is suspended from a stretchable spring. So, if someone pulls down the spring, it is released with an oscillating motion vertically. What is the name of the force that is applied to the body, when the spring passes through its mean position?

a) Force equal to the pull

b) Force equal to the weight of the body

c) Force equal to gravity

d) Conservative force

Solution

The answer is B. It is imperative to understand that at the mean position, the total acceleration of the body is zero. So, the resultant force that is applied by the spring is the same as that of the weight of the body.

1. Define SHM (Simple Harmonic Motion) with reference to motion.

Motion can be defined as the type of phenomenon in which a body can change its position within a given time frame. Based on the characteristic of the motion, they can be further divided into different types. Some of the most prominent types of motion include:

• Reciprocating motion

• Linear motion

• Rotary motion

• Oscillating motion

Simple Harmonic Motion or SHM, on the other hand, is a type of oscillating motion. In this type of motion, the total force exerted on the object is actually restoring force. In other words, SHM can be portrayed as periodic motion. In this motion, the object tends to move to and fro along a fixed line. The pendulum of a clock is one of the best examples of simple harmonic motion.

2. Define angular frequency with context to periodic motion.

Angular frequency, mainly represented using the Greek alphabet, ω (Omega), is used for periodic motion. Angular frequency can be defined as the angular displacement that occurs per unit time. In other words, the rate of change of sinusoidal waveform (waveform describing a smooth periodic oscillation) can be defined as angular velocity too. It is mainly expressed in the form of an argument of a sine function.

For example,

y(t) = sin(θ(t)) = sin(ωt) = sin(2πft)

or, ω=2πf

The SI unit of angular frequency is radians per second.