Periodic motion – period, frequency, displacement

Introduction

Periodic motion is an important topic that helps the students understand other significant topics in physics and mathematics. It is essential to study this topic properly because it will help in better understanding of other important topics. 

Motion can be divided into several types based on how the object moves. Let us take the help of an example to understand this topic more clearly. Suppose a car moves forward in a straight line, then it is said to have linear motion. Let us now discuss this interesting topic a bit more in detail.

Meaning of periodic motion

A motion that repeats itself after a fixed interval is known as periodic motion. The time period is the interval of time after which the motion repeats itself. A bouncing ball, a rocking chair, a moving swing, a water wave, and a revolution of the earth around the sun in its orbit are all examples of periodic motion. Here are some other examples of periodic motion that will make the concept clear in your head. 

Some examples of periodic motion

A pendulum’s motion and a tuning fork are some examples of periodic motion. You must have noticed the motion of a pendulum. It passes through the mean or central position after a definite time interval. It can also be classified as an oscillatory motion where an object moves to and fro while staying in a fixed position.

Therefore, we can say that an oscillatory motion is a type of periodic motion. Now that you have a good idea of what a periodic motion is, let’s look at the periodic motion formula. 

Frequency of the periodic motion

We all know by now that a periodic motion repeats after a fixed time interval. If the motion repeats itself for a long time, it has a frequency. So, the frequency is the number of times the motion repeats itself in unit time. You can say that one complete motion is equal to one frequency; the number of times one complete motion is repeated in a given period is its frequency.

The letter f depicts the frequency of the periodic motion. ‘Hertz’ or Hz is the unit used to measure the frequency of a periodic motion.

Frequency (f) = 1/T. This formula is explained in detail in the next part.

The formula of a periodic motion

The time period, T, is the time required by the motion to repeat itself. The standard unit to measure the time period is seconds. 

Frequency (f) is the number of times a motion repeats itself in unit time. Hertz (Hz) is used to measure the frequency.

There exists an inverse relationship between time period and frequency. Mathematically, it can be depicted as follows:

      f = 1/T

Equation (1) is the formula of periodic motion. 

Periodic function 

A periodic function is the one where the value of the function repeats in fixed periods or intervals. 

     f(a+m) = f(a)

Equation (2) shows that the function f (a) has the same value after the ‘m’ time interval. Thus, the function is a periodic function. Equation (2) is also the periodic function formula.

Periodic function equation

The periodic function of an oscillating object can be mathematically represented as:

f(t) = A cos(ωt)

The cosine repeats itself after a fixed time period which can be mathematically represented as follows:

cosθ = cos(θ+2π)

cos(ωt) = cos(ωt+2π) -(1)

let us take T to be the time period :

F(t) = f(t+T)

Acosωt = Acosω(t+T)

Acosωt = Acos(ωt + ωT) -(2)

From equations (1) and (2), we will get the following results: 

ωT = 2π

T = 2π/ω

The time period of a periodic function can be represented as follows: 

T= 2π/ω

ω here is used to represent the angular frequency of an oscillating object.

Oscillatory motion

When an oscillator is disturbed by the application of force, then it moves from its initial position of equilibrium. And then after some time, it returns back to its equilibrium position because of the restoring force. 

Hooke’s law states that the restoring force is proportional to the displacement, and it also depends on the system’s elasticity.

When the restoring force starts bringing the oscillator to its equilibrium position, the velocity starts changing. This changing velocity is opposed by inertia. Moreover, when the oscillator reaches the equilibrium position, it moves beyond its mean position because of its inertia. This motion continues until the deforming force is developed, which brings the oscillator to rest.

Simple Harmonic Motion or SHM

Simple harmonic motion or simply SHM is a periodic motion that has a relationship between displacement from the center and the restoring force. The simple harmonic motion is a special case of oscillation. In linear SHM, the motion occurs between two points and along a straight line. The position of the mean in SHM is a stable equilibrium. 

Let us assume that in simple harmonic mean,

x = displacement position and,

a = acceleration 

Then we will get,

F = – kx

Note that k here is a constant term. It is not a spring constant or a force. The negative sign in the equation denotes that acceleration is opposite to the motion.

Periodic motion is an easy-to-understand concept. The frequency of the periodic motion is denoted by the letter f, and it is equal to 1/T, where T is the time period. The S.I unit of frequency is hertz or Hz.