Consider a particle oscillating back and forth about the origin of an \(x\)-axis between the limits \(+A\) and \(-A\) as shown in Fig. 14.3.

This oscillatory motion is said to be simple harmonic if the displacement \(x\) of the particle from the origin varies with time as:
\(
x(t)=A \cos (\omega t+\phi) \dots(14.4)
\)
where \(A, \omega\) and \(\phi\) are constants.

Thus, simple harmonic motion (SHM) is not any periodic motion but one in which displacement is a sinusoidal function of time.  We can calculate the positions of a particle executing SHM at discrete value of time, each interval of time being \(T / 4\), where \(T\) is the period of motion. This is shown in Fig. 14.4 below.

Displacement as a continuous function of time for simple harmonic motion

Fig. 14.5 plots the graph of \(x\) versus \(t\), which gives the values of displacement as a continuous function of time. The quantities \(A\), \(\omega\), and \(\phi\) which characterize a given SHM have standard names, as summarised in Fig. 14.6. Let us understand these quantities.

Displacement as a function of time plot with \(\phi\)

The amplitude \(A\) of SHM is the magnitude of maximum displacement of the particle. [Note, \(A\) can be taken to be positive without any loss of generality]. As the cosine function of time varies from +1 to -1, the displacement varies between the extremes \(A\) and \(-A\). Two simple harmonic motions may have the same \(\omega\) and \(\phi\) but different amplitudes \(A\) and \(B\), as shown in Fig. 14.7 (a).

Phase \(\phi\)

While the amplitude \(A\) is fixed for a given SHM, the state of motion (position and velocity) of the particle at any time \(t\) is determined by the argument \((\omega t+\phi)\) in the cosine function. This time-dependent quantity, \((\omega t+\phi)\) is called the phase of the motion. The value of phase at \(t=0\) is \(\phi\) and is called the phase constant (or phase angle). If the amplitude is known, \(\phi\) can be determined from the displacement at \(t=0\). Two simple harmonic motions may have the same \(A\) and \(\omega\) but different phase angle \(\phi\), as shown in Fig. 14.7 (b).

Finally, the quantity \(\omega\) can be seen to be related to the period of motion \(T\). Taking, for simplicity, \(\phi=0\) in Eq. (14.4), we have
\(
x(t)=A \cos \omega t \dots(14.5)
\)
Since the motion has a period \(T, x(t)\) is equal to \(x(t+T)\). That is,
\(
A \cos \omega t=A \cos \omega(t+T) \dots(14.6)
\)
Now the cosine function is periodic with period \(2 \pi\), 1.e., it first repeats itself when the argument changes by \(2 \pi\). Therefore,
\(
\omega(t+T)=\omega t+2 \pi
\)
that is \(\omega=2 \pi / T \dots(14.7)\)
\(\omega\) is called the angular frequency of SHM. Its S.I. unit is radians per second. Since the frequency of oscillations is simply \(1 / \mathrm{T}\), \(\omega\) is \(2 \pi\) times the frequency of oscillation. Two simple harmonic motions may have the same \(A\) and \(\phi\), but different \(\omega\), as seen in Fig. 14.8. In this plot the curve (b) has half the period and twice the frequency of the curve (a).

Example 1: Which of the following functions of time represent (a) simple harmonic motion and (b) periodic but not simple harmonic? Give the period for each case.
(1) \(\sin \omega t-\cos \omega t\)
(2) \(\sin ^2 \omega t\)

Solution:

(a)
\(
\begin{aligned}
\sin \omega t & -\cos \omega t \\
& =\sin \omega t-\sin (\pi / 2-\omega t) \\
& =2 \cos (\pi / 4) \sin (\omega t-\pi / 4) \\
& =\sqrt{ } 2 \sin (\omega t-\pi / 4)
\end{aligned}
\)
This function represents a simple harmonic motion having a period \(T=2 \pi / \omega\) and a phase angle \((-\pi / 4)\) or \((7 \pi / 4)\)
(b)
\(
\begin{aligned}
& \sin ^2 \omega t \\
& \quad=1 / 2-1 / 2 \cos 2 \omega t
\end{aligned}
\)
The function is periodic having a period \(T=\pi / \omega\). It also represents a harmonic motion with the point of equilibrium occurring at \(1 / 2\) instead of zero.