What is Periodic Motion?

A motion that repeats itself at regular intervals of time is called periodic motion. The waveforms shown below are examples of periodic waveforms that exhibit periodic motion as they repeat after a regular interval of \(T\) known as the period.

What is Oscillatory Motion?

The motion of a body is said to be oscillatory motion if it moves to and fro about a fixed point after regular intervals of time. The fixed point about which the body oscillates is called the mean position or equilibrium position. The motion of a pendulum is the most common example of oscillatory motion. The bob of a pendulum is always in a to-and-fro motion about a fixed point as shown below.

What is Period?

The smallest interval of time after which the motion is repeated is called it’s period. Let us denote the period by the symbol \(T\). Its SI unit is second.

We can also say the time taken to complete one wave (or to complete one cycle) is known as the period of the waveform. For the waveform shown below \(y(t)=\sin (2 t)\) the time period is \(T=\pi\).

As \(\omega=2\), in the function \(y(t)=\sin (2 t)=\sin (\omega t)\)

\(
\text { Period: } \quad T=\frac{2 \pi}{\omega}, \quad T=\frac{2 \pi}{2}=\pi
\)

What is Frequency?

The reciprocal of \(T\) gives the number of repetitions that occur per unit time. This quantity is called the frequency of the periodic motion (we can say the number of oscillations in one second is called frequency). It is represented by the symbol \(f\). The relation between \(f\) and \(T\) is
\(
f=1 / T \dots(14.1)
\)
The unit of \(f\) is thus \(\mathrm{s}^{-1}\). After the discoverer of radio waves, Heinrich Rudolph Hertz (1857-1894), a special name has been given to the unit of frequency. It is called hertz (abbreviated as Hz). Thus,
1 hertz \(=1 \mathrm{~Hz}=1\) oscillation per second \(=1 \mathrm{~s}^{-1} \dots(14.2)\)

In terms of the angular frequency \(\omega=\frac{2 \pi}{T}=2 \pi f\)

Example 1: On an average, a human heart is found to beat 75 times in a minute. Calculate its frequency and period.

Solution:

\(
\begin{aligned}
\text { The beat frequency of heart } & =75 /(1 \mathrm{~min}) \\
& =75 /(60 \mathrm{~s}) \\
& =1.25 \mathrm{~s}^{-1} \\
& =1.25 \mathrm{~Hz} \\
\text { The time period } T & =1 /\left(1.25 \mathrm{~s}^{-1}\right) \\
& =0.8 \mathrm{~s}
\end{aligned}
\)

Displacement

we defined the displacement of an object as the change in its position vector. It refers to the change with time of any physical property under consideration. Consider a block attached to a spring, the other end of the spring is fixed to a rigid wall as shown in the figure below(Fig. 14.2(a)). 

Generally, it is convenient to measure the displacement of the body from its equilibrium position. For an oscillating simple pendulum, the angle from the vertical as a function of time may be regarded as a displacement variable (Fig. 14.2(b)). 

The displacement variable may take both positive and negative values. In experiments on oscillations, the displacement is measured for different times. The displacement can be represented by a mathematical function of time. In the case of periodic motion, this function is periodic in time. One of the simplest periodic functions is given by

\(f(t)=A \cos \omega t \dots(14.3a)\)
If the argument of this function, \(\omega t\), is increased by an integral multiple of \(2 \pi\) radians, the value of the function remains the same.The function \(f(t)\) is then periodic and its period, \(T\), is given by
\(
T=\frac{2 \pi}{\omega} \dots(14.3b)
\)
Thus, the function \(f(t)\) is periodic with period \(T\),
\(
f(t)=f(t+T)
\)
The same result is obviously correct if we consider a sine function, \(f(t)=A \sin \omega t\). Further, a linear combination of sine and cosine functions like,
\(
f(t)=A \sin \omega t+B \cos \omega t \dots(14.3c)
\)
is also a periodic function with the same period T. Taking,
\(
A=D \cos \phi \text { and } B=D \sin \phi
\)
Eq. (14.3c) can be written as,
\(
f(t)=D \sin (\omega t+\phi) \dots(14.3d)
\)
Here \(D\) and \(\phi\) are constant given by
\(
D=\sqrt{A^2+B^2} \text { and } \phi=\tan ^{-1}\left(\frac{B}{A}\right)
\)
The great importance of periodic sine and cosine functions is due to a remarkable result proved by the French mathematician, Jean Baptiste Joseph Fourier (1768-1830): Any periodic function can be expressed as a superposition of sine and cosine functions of different time periods with suitable coefficients.

Example 2: Which of the following functions of time represent (a) periodic and (b) non-periodic motion? Give the period for each case of periodic motion \([\omega\) is any positive constant].
(i) \(\sin \omega t+\cos \omega t\)
(ii) \(\sin \omega t+\cos 2 \omega t+\sin 4 \omega t\)
(iii) \(\mathrm{e}^{-\omega t}\)
(iv) \(\log (\omega t)\)

Solution:

(1) \(\sin \omega t+\cos \omega t\) is a periodic function, it can also be written as \(\sqrt{2} \sin (\omega t+\pi / 4)\).

Now \(\sqrt{2} \sin (\omega t+\pi / 4)=\sqrt{2} \sin (\omega t+\pi / 4+2 \pi)\)
\(
=\sqrt{2} \sin [\omega(\mathrm{t}+2 \pi / \omega)+\pi / 4]
\)
The periodic time of the function is \(2 \pi / \omega\).

(ii) This is an example of a periodic motion. It can be noted that each term represents a periodic function with a different angular frequency. Since the period is the least interval of time after which a function repeats its value, \(\sin \omega\) thas a period \(T_0=2 \pi / \omega ; \cos 2 \omega t\) has a period \(\pi / \omega=T_0 / 2\); and \(\sin 4 \omega t\) has a period \(2 \pi / 4 \omega=T_0 / 4\). The period of the first term is a multiple of the periods of the last two terms. Therefore, the smallest interval of time after which the sum of the three terms repeats is \(T_0\), and thus, the sum is a periodic function with a period \(2 \pi / \omega\).

(iii) The function \(e^{-\omega t}\) is not periodic, it decreases monotonically with increasing time and tends to zero as \(t \rightarrow \infty\) and thus, never repeats its value.

(iv) The function \(\log (\omega t)\) increases monotonically with time \(t\). It, therefore, never repeats its value and is a nonperiodic function. It may be noted that as \(t \rightarrow \infty, \log (\omega t)\) diverges to \(\infty\). It, therefore, cannot represent any kind of physical displacement.