Periodic Function

What is Periodic Function?

A body is said to be in periodic motion if the motion it’s executing is repeated after equal intervals of time, like a rocking chair, a swing in motion. A periodic function can be defined as:

A function returning to the same value at regular intervals.

Though periodic motion and oscillatory motion sound the same, not all periodic motions will be oscillatory motion. The major difference between a periodic motion and oscillatory motion is that periodic motion is relevant to any motion that repeats over time, but the oscillatory motion is unique to those motions that execute about an equilibrium point or between two states. A periodic function can define all periodic motions.

Periodic function of a pendulum bob

Periodic function of a pendulum bob

To understand the concept of periodic function let’s consider a pendulum bob, oscillating along its equilibrium position, the trajectory of the bob is shown below, now if the bob is oscillating then, its displacement will also vary from zero to positive and back to zero and negative, this can be easily illustrated with a graph,

Periodic Function (Displacement/Time Graph)

Periodic Function (Displacement/Time Graph)

Periodic Function Formula

A function f is said to be periodic if, for some nonzero constant P, it is the case that:

 f (x+P) = f (x)

for all values of x in the domain. A nonzero constant P for which this is the case is called a period of the function.

Periodic Function Equation

Let’s take a case of an oscillating object, its displacement in periodic motion is represented by a function which is periodic in time;

\(f(t)\) = \(Acosωt\)

We have to concentrate on the elements of this function, the cosine function repeats itself in time from trigonometry we know the following;

\(cosθ\) = \(cos(θ + 2π)\)

\(cos(ωt)\) = \(cos(ωt + 2π)\) ——(1)

Suppose time period is T;

\(f(t)\) = \(f(t + T)\)

\(Acosωt\) = \(Acosω(t + T)\)

⇒ Acosωt = Acos(ωt + ωT) ——(2)

From equation (1) & (2) we can say that;

\(ωT\) = \(2π\)

Therefore,

\(T\) = \(\frac{2π}{ω}\)

Therefore, the time period of periodic motion is given by the above expression,
The frequency of this periodic function can be given by the time period, the frequency is the number of oscillations per unit time, so if we know the time for one oscillation then we can find the frequency by:

\(f\) = \(\frac{1}{T}\)

The movement of planets around the sun, the motion of a yo-yo are all examples of periodic functions. Though the example of a pendulum is a special case of periodic function because it is executing simple harmonic motion, the difference lies in how the motion is expressed mathematically, if the periodic function can be represented by a sine curve then the motion is said to be simple harmonic motion, like a weight on spring oscillating, a swing, etc. Simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.

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

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