13.1 INTRODUCTION

In our daily life we come across various kinds of motions. You have already learnt about some of them, e.g., rectilinear motion and motion of a projectile. Both these motions are non-repetitive. We have also learnt about uniform circular motion and orbital motion of planets in the solar system. In these cases, the motion is repeated after a certain interval of time, that is, it is periodic. In your childhood, you must have enjoyed rocking in a cradle or swinging on a swing. Both these motions are repetitive in nature but different from the periodic motion of a planet. Here, the object moves to and fro about a mean position. The pendulum of a wall clock executes a similar motion. Examples of such periodic to and fro motion abound: a boat tossing up and down in a river, the piston in a steam engine going back and forth, etc. Such a motion is termed as oscillatory motion. In this chapter we study this motion.

The study of oscillatory motion is basic to physics; its concepts are required for the understanding of many physical phenomena. In musical instruments, like the sitar, the guitar or the violin, we come across vibrating strings that produce pleasing sounds. The membranes in drums and diaphragms in telephone and speaker systems vibrate to and fro about their mean positions. The vibrations of air molecules make the propagation of sound possible. In a solid, the atoms vibrate about their equilibrium positions, the average energy of vibrations being proportional to temperature. AC power supply give voltage that oscillates alternately going positive and negative about the mean value (zero).

The description of a periodic motion, in general, and oscillatory motion, in particular, requires some fundamental concepts, like period, frequency, displacement, amplitude and phase. These concepts are developed in the next section.

13.2 PERIODIC AND OSCILLATORY MOTIONS

Fig. 13.1 shows some periodic motions. Suppose an insect climbs up a ramp and falls down, it comes back to the initial point and repeats the process identically. If you draw a graph of its height above the ground versus time, it would look something like Fig. 13.1 (a). If a child climbs up a step, comes down, and repeats the process identically, its height above the ground would look like that in Fig. 13.1 (b). When you play the game of bouncing a ball off the ground, between your palm and the ground, its height versus time graph would look like the one in Fig. 13.1 (c). Note that both the curved parts in Fig. 13.1 (c) are sections of a parabola given by the Newton’s equation of motion (see section 2.6),

$h=u t+\frac{1}{2} g t^{2}$ for downward motion, and

$h=u t-\frac{1}{2} g t^{2}$ for upward motion,

with different values of $u$ in each case. These are examples of periodic motion. Thus, a motion that repeats itself at regular intervals of time is called periodic motion.

Very often, the body undergoing periodic motion has an equilibrium position somewhere inside its path. When the body is at this position no net external force acts on it. Therefore, if it is left there at rest, it remains there forever. If the body is given a small displacement from the position, a force comes into play which tries to bring the body back to the equilibrium point, giving rise to oscillations or vibrations. For example, a ball placed in a bowl will be in equilibrium at the bottom. If displaced a little from the point, it will perform oscillations in the bowl. Every oscillatory motion is periodic, but every periodic motion need not be oscillatory. Circular motion is a periodic motion, but it is not oscillatory.

There is no significant difference between oscillations and vibrations. It seems that when the frequency is small, we call it oscillation (like, the oscillation of a branch of a tree), while when the frequency is high, we call it vibration (like, the vibration of a string of a musical instrument).

Simple harmonic motion is the simplest form of oscillatory motion. This motion arises when the force on the oscillating body is directly proportional to its displacement from the mean position, which is also the equilibrium position. Further, at any point in its oscillation, this force is directed towards the mean position.

In practice, oscillating bodies eventually come to rest at their equilibrium positions because of the damping due to friction and other dissipative causes. However, they can be forced to remain oscillating by means of some external periodic agency. We discuss the phenomena of damped and forced oscillations later in the chapter.

Any material medium can be pictured as a collection of a large number of coupled oscillators. The collective oscillations of the constituents of a medium manifest themselves as waves. Examples of waves include water waves, seismic waves, electromagnetic waves. We shall study the wave phenomenon in the next chapter.

13.2.1 Period and frequency

We have seen that any motion that repeats itself at regular intervals of time is called periodic motion. The smallest interval of time after which the motion is repeated is called its period. Let us denote the period by the symbol $T$. Its SI unit is second. For periodic motions, which are either too fast or too slow on the scale of seconds, other convenient units of time are used. The period of vibrations of a quartz crystal is expressed in units of microseconds $\left(10^{-6} \mathrm{~s}\right)$ abbreviated as $\mu \mathrm{s}$. On the other hand, the orbital period of the planet Mercury is 88 earth days. The Halley’s comet appears after every 76 years.

The reciprocal of $T$ gives the number of repetitions that occur per unit time. This quantity is called the frequency of the periodic motion. It is represented by the symbol $v$. The relation between $v$ and $T$ is

$$ \begin{equation*} v=1 / T \tag{13.1} \end{equation*} $$

The unit of $v$ 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 $\mathrm{Hz}$ ). Thus,

1 hertz $=1 \mathrm{~Hz}=1$ oscillation per second $=1 \mathrm{~s}^{-1}$

Note, that the frequency, $v$, is not necessarily an integer.

13.2.2 Displacement

In section 3.2, we defined displacement of a particle as the change in its position vector. In this chapter, we use the term displacement in a more general sense. It refers to change with time of any physical property under consideration. For example, in case of rectilinear motion of a steel ball on a surface, the distance from the starting point as a function of time is its position displacement. The choice of origin is a matter of convenience. Consider a block attached to a spring, the other end of the spring is fixed to a rigid wall [see Fig. 13.2(a)]. Generally, it is convenient to measure 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 [see Fig. 13.2(b)]. The term displacement is not always to be referred in the context of position only. There can be many other kinds of displacement variables. The voltage across a capacitor, changing with time in an $\mathrm{AC}$ circuit, is also a displacement variable. In the same way, pressure variations in time in the propagation of sound wave, the changing electric and magnetic fields in a light wave are examples of displacement in different contexts. 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 case of periodic motion, this function is periodic in time. One of the simplest periodic functions is given by

$$ \begin{equation*} f(t)=A \cos \omega t \tag{13.3a} \end{equation*} $$

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

$$ \begin{equation*} T=\frac{2 \pi}{\omega} \tag{13.3b} \end{equation*} $$

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,

$$ \begin{equation*} f(t)=A \sin \omega t+B \cos \omega t \tag{13.3c} \end{equation*} $$

is also a periodic function with the same period $T$. Taking,

$$ A=D \cos \phi \text { and } B=D \sin \phi $$

Eq. (13.3c) can be written as,

$$ \begin{equation*} f(t)=D \sin (\omega t+\phi), \tag{13.3d} \end{equation*} $$

Here $D$ and $\phi$ are constant given by

$$ D=\sqrt{A^{2}+B^{2}} \text { and } \varphi=\tan ^{-1} \frac{B}{A} $$

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.

13.3 SIMPLE HARMONIC MOTION

Consider a particle oscillating back and forth about the origin of an $x$-axis between the limits $+A$ and $-A$ as shown in Fig. 13.3. This oscillatory motion is said to be simple harmonic if the displacement $x$ of the particle from the origin varies with time as :

$$ \begin{equation*} x(t)=A \cos (\omega t+\phi) \tag{13.4} \end{equation*} $$

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. Fig. 13.4 shows 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. Fig. 13.5 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. 13.6. Let us understand these quantities.

The amplitutde $A$ 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 same $\omega$ and $\phi$ but different amplitudes $A$ and $B$, as shown in Fig. 13.7 (a).

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 plase 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. 13.7 (b).

Finally, the quantity $\omega$ can be seen to be related to the period of motion $T$. Taking, for simplicity, $\phi=0$ in Eq. (13.4), we have

$$ \begin{equation*} x(t)=A \cos \omega t \tag{13.5} \end{equation*} $$

Since the motion has a period $T, x(t)$ is equal to $x(t+T)$. That is,

$$ \begin{equation*} A \cos \omega t=A \cos \omega(t+T) \tag{13.6} \end{equation*} $$

Now the cosine function is periodic with period $2 \pi$, i.e., it first repeats itself when the argument changes by $2 \pi$. Therefore,

$$ \omega(t+T)=\omega t+2 \pi $$

$$ \text{that is } \quad \omega=2 \pi / T \tag{13.7}$$

$\omega$ 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 $\mathrm{A}$ and $\phi$, but different $\omega$, as seen in Fig. 13.8. In this plot the curve (b) has half the period and twice the frequency of the curve (a)

13.4 SIMPLE HARMONIC MOTION AND UNIFORM CIRCULAR MOTION

In this section, we show that the projection of uniform circular motion on a diameter of the circle follows simple harmonic motion. A simple experiment (Fig. 13.9) helps us visualise this connection. Tie a ball to the end of a string and make it move in a horizontal plane about a fixed point with a constant angular speed. The ball would then perform a uniform circular motion in the horizontal plane. Observe the ball sideways or from the front, fixing your attention in the plane of motion. The ball will appear to execute to and fro motion along a horizontal line with the point of rotation as the midpoint. You could alternatively observe the shadow of the ball on a wall which is perpendicular to the plane of the circle. In this process what we are observing is the motion of the ball on a diameter of the circle normal to the direction of viewing.

Fig. 13.10 describes the same situation mathematically. Suppose a particle $\mathrm{P}$ is moving uniformly on a circle of radius $A$ with angular speed $\omega$. The sense of rotation is anticlockwise. The initial position vector of the particle, i.e., the vector $\overline{\mathbf{O P}}$ at $t=0$ makes an angle of $\phi$ with the positive direction of $x$-axis. In time $t$, it will cover a further angle $\omega t$ and its position vector

will make an angle of $\omega t+\phi$ with the + ve $x$-axis. Next, consider the projection of the position vector OP on the $x$-axis. This will be $\mathrm{OP}^{\prime}$. The position of $\mathrm{P}^{\prime}$ on the $x$-axis, as the particle $\mathrm{P}$ moves on the circle, is given by

$$ x(t)=A \cos (\omega t+\phi) $$

which is the defining equation of SHM. This shows that if $\mathrm{P}$ moves uniformly on a circle, its projection $\mathrm{P}^{\prime}$ on a diameter of the circle executes SHM. The particle P and the circle on which it moves are sometimes referred to as the reference particle and the reference circle, respectively.

We can take projection of the motion of $\mathrm{P}$ on any diameter, say the $y$-axis. In that case, the displacement $y(t)$ of $\mathrm{P}^{\prime}$ on the $y$-axis is given by

$$ y=A \sin (\omega t+\phi) $$

which is also an SHM of the same amplitude as that of the projection on $x$-axis, but differing by a phase of $\pi / 2$.

In spite of this connection between circular motion and SHM, the force acting on a particle in linear simple harmonic motion is very different from the centripetal force needed to keep a particle in uniform circular motion.

13.5 VELOCITY AND ACCELERATION IN SIMPLE HARMONIC MOTION

The speed of a particle $v$ in uniform circular motion is its angular speed $\omega$ times the radius of the circle $A$.

$$ \begin{equation*} V=\omega A \tag{13.8} \end{equation*} $$

The direction of velocity $\overline{\mathbf{v}}$ at a time $t$ is along the tangent to the circle at the point where the particle is located at that instant. From the geometry of Fig. 13.11, it is clear that the velocity of the projection particle $\mathrm{P}^{\prime}$ at time $t$ is

$$ \begin{equation*} v(t)=-\omega A \sin (\omega t+\phi) \tag{13.9} \end{equation*} $$

where the negative sign shows that $v(\mathrm{t})$ has a direction opposite to the positive direction of $x$-axis. Eq. (13.9) gives the instantaneous velocity of a particle executing SHM, where displacement is given by Eq. (13.4). We can, of course, obtain this equation without using geometrical argument, directly by differentiating (Eq. 13.4) with respect of $t$ :

$$ \begin{equation*} v(t)=\frac{\mathrm{d}}{\mathrm{d} t} x(t) \tag{13.10} \end{equation*} $$

The method of reference circle can be similarly used for obtaining instantaneous acceleration of a particle undergoing SHM. We know that the centripetal acceleration of a particle $\mathrm{P}$ in uniform circular motion has a magnitude $v^{2} / \mathrm{A}$ or $\omega^{2} \mathrm{~A}$, and it is directed towards the centre i.e., the direction is along PO. The instantaneous acceleration of the projection particle $\mathrm{P}^{\prime}$ is then (See Fig. 13.12)

$$ \begin{align*} a(t) & =-\omega^{2} A \cos (\omega t+\phi) \\ & =-\omega^{2} x(t) \tag{13.11} \end{align*} $$

Eq. (13.11) gives the acceleration of a particle in SHM. The same equation can again be obtained directly by differentiating velocity $v(t)$ given by Eq. (13.9) with respect to time:

$$ \begin{equation*} a(t)=\frac{\mathrm{d}}{\mathrm{d} t} v(t) \tag{13.12} \end{equation*} $$

We note from Eq. (13.11) the important property that acceleration of a particle in SHM is proportional to displacement. For $\mathrm{x}(t)>0$, $a(t)<0$ and for $x(t)<0, a(t)>0$. Thus, whatever the value of $x$ between $-A$ and $A$, the acceleration $a(t)$ is always directed towards the centre.

For simplicity, let us put $\phi=0$ and write the expression for $x(t), v(t)$ and $a(t)$

$x(t)=A \cos \omega t, v(t)=-\omega A \sin \omega t, a(t)=-\omega^{2} A \cos \omega t$ The corresponding plots are shown in Fig. 13.13. All quantities vary sinusoidally with time; only their maxima differ and the different plots differ in phase. $x$ varies between $-A$ to $A ; v(t)$ varies from $-\omega A$ to $\omega A$ and $a(t)$ from $-\omega^{2} A$ to $\omega^{2}$. With respect to displacement plot, velocity plot has a phase difference of $\pi / 2$ and acceleration plot has a phase difference of $\pi$.

13.6 FORCE LAW FOR SIMPLE HARMONIC MOTION

Using Newton’s second law of motion, and the expression for acceleration of a particle undergoing SHM (Eq. 13.11), the force acting on a particle of mass $m$ in SHM is

$$ \begin{align*} F(t) & =m a \\ & =-m \omega^{2} x(t) \end{align*} $$

$$ \text{i.e.,} \quad F(t)=-k x(t) \tag{13.13}$$

$$ \text{where} \quad k=m \omega^{2} \tag{13.14a}$$

$$ \text{or} \quad \omega=\sqrt{\frac{k}{m}} \tag{13.14b}$$

Like acceleration, force is always directed towards the mean position-hence it is sometimes called the restoring force in SHM. To summarise the discussion so far, simple harmonic motion can be defined in two equivalent ways, either by Eq. (13.4) for displacement or by Eq. (13.13) that gives its force law. Going from Eq. (13.4) to Eq. (13.13) required us to differentiate two times. Likewise, by integrating the force law Eq. (13.13) two times, we can get back Eq. (13.4).

Note that the force in Eq. (13.13) is linearly proportional to $x(t)$. A particle oscillating under such a force is, therefore, calling a linear harmonic oscillator. In the real world, the force may contain small additional terms proportional to $x^{2}, x^{3}$, etc. These then are called non-linear oscillators.

13.7 ENERGY IN SIMPLE HARMONIC MOTION

Both kinetic and potential energies of a particle in SHM vary between zero and their maximum values.

In section 13.5 we have seen that the velocity of a particle executing SHM, is a periodic function of time. It is zero at the extreme positions of displacement. Therefore, the kinetic energy (K) of such a particle, which is defined as

$$ \begin{align*} K & =\frac{1}{2} m v^{2} \\ & =\frac{1}{2} m \omega^{2} A^{2} \sin ^{2}(\omega t+\phi) \\ & =\frac{1}{2} k A^{2} \sin ^{2}(\omega t+\phi) \tag{13.15} \end{align*} $$

is also a periodic function of time, being zero when the displacement is maximum and maximum when the particle is at the mean position. Note, since the sign of $v$ is immaterial in $K$, the period of $K$ is $T / 2$.

What is the potential energy $(U)$ of a particle executing simple harmonic motion? In Chapter 6, we have seen that the concept of potential energy is possible only for conservative forces. The spring force $F=-k x$ is a conservative force, with associated potential energy

$$ \begin{equation*} U=\frac{1}{2} k x^{2} \tag{13.16} \end{equation*} $$

Hence the potential energy of a particle executing simple harmonic motion is,

$$ \begin{align*} & U(x)=\frac{1}{2} k x^{2} \\ & =\frac{1}{2} k A^{2} \cos ^{2}(\omega t+\phi) \tag{13.17} \end{align*} $$

Thus, the potential energy of a particle executing simple harmonic motion is also periodic, with period $T / 2$, being zero at the mean position and maximum at the extreme displacements.

It follows from Eqs. (13.15) and (13.17) that the total energy, $E$, of the system is,

$$ \begin{aligned} & E=U+K \\ & =\frac{1}{2} k A^{2} \cos ^{2}(\omega t+\phi)+\frac{1}{2} k A^{2} \sin ^{2}(\omega t+\phi) \\ & =\frac{1}{2} k A^{2}\left[\cos ^{2}(\omega t+\phi)+\sin ^{2}(\omega t+\phi)\right] \end{aligned} $$

Using the familiar trigonometric identity, the value of the expression in the brackets is unity. Thus,

$$ \begin{equation*} E=\frac{1}{2} k A^{2} \tag{13.18} \end{equation*} $$

The total mechanical energy of a harmonic oscillator is thus independent of time as expected for motion under any conservative force. The time and displacement dependence of the potential and kinetic energies of a linear simple harmonic oscillator are shown in Fig. 13.16.

Observe that both kinetic energy and potential energy in SHM are seen to be always positive in Fig. 13.16. Kinetic energy can, of course, be never negative, since it is proportional to the square of speed. Potential energy is positive by choice of the undermined constant in potential energy. Both kinetic energy and potential energy peak twice during each period of SHM. For $x=0$, the energy is kinetic; at the extremes $x= \pm A$, it is all potential energy. In the course of motion between these limits, kinetic energy increases at the expense of potential energy or vice-versa.

13.8 The Simple Pendulum

It is said that Galileo measured the periods of a swinging chandelier in a church by his pulse beats. He observed that the motion of the chandelier was periodic. The system is a kind of pendulum. You can also make your own pendulum by tying a piece of stone to a long unstretchable thread, approximately $100 \mathrm{~cm}$ long. Suspend your pendulum from a suitable support so that it is free to oscillate. Displace the stone to one side by a small distance and let it go. The stone executes a to and fro motion, it is periodic with a period of about two seconds.

We shall show that this periodic motion is simple harmonic for small displacements from the mean position. Consider simple pendulum - a small bob of mass $m$ tied to an inextensible massless string of length $L$. The other end of the string is fixed to a rigid support. The bob oscillates in a plane about the vertical line through the support. Fig. 13.17(a) shows this system. Fig. 13.17(b) is a kind of ‘free-body’ diagram of the simple pendulum showing the forces acting on the bob.

Let $\theta$ be the angle made by the string with the vertical. When the bob is at the mean position, $\theta=0$

There are only two forces acting on the bob; the tension $\mathrm{T}$ along the string and the vertical force due to gravity (=mg). The force $m g$ can be resolved into the component $m g \cos \theta$ along the string and $m g \sin \theta$ perpendicular to it. Since the motion of the bob is along a circle of length $L$ and centre at the support point, the bob has a radial acceleration $\left(\omega^{2} L\right)$ and also a tangental acceleration; the latter arises since motion along the arc of the circle is not uniform. The radial acceleration is provided by the net radial force $\mathrm{T}-m g \cos \theta$, while the tangential acceleration is provided by $m g \sin \theta$. It is more convenient to work with torque about the support since the radial force gives zero torque. Torque $\tau$ about the support is entirely provided by the tangental component of force

$$ \begin{equation*} \tau=-L(m g \sin \theta) \tag{13.19} \end{equation*} $$

This is the restoring torque that tends to reduce angular displacement - hence the negative sign. By Newton’s law of rotational motion,

$$ \begin{equation*} \tau=I \alpha \tag{13.20} \end{equation*} $$

where $I$ is the moment of inertia of the system about the support and $\alpha$ is the angular acceleration. Thus,

$$ \begin{equation*} I \alpha=-m g \sin \theta \quad L \tag{13.21} \end{equation*} $$

Or,

$$ \begin{equation*} \alpha=-\frac{m g L}{I} \sin \theta \tag{13.22} \end{equation*} $$

We can simplify Eq. (13.22) if we assume that the displacement $\theta$ is small. We know that $\sin \theta$ can be expressed as,

$$ \begin{equation*} \sin \theta=\theta-\frac{\theta^{3}}{3 !}+\frac{\theta^{5}}{5 !} \pm \ldots \tag{13.23} \end{equation*} $$

where $\theta$ is in radians.

Now if $\theta$ is small, $\sin \theta$ can be approximated by $\theta$ and Eq. (13.22) can then be written as,

$$ \begin{equation*} \alpha=-\frac{m g L}{I} \theta \tag{13.24} \end{equation*} $$

In Table 13.1, we have listed the angle $\theta$ in degrees, its equivalent in radians, and the value of the function $\sin \theta$. From this table it can be seen that for $\theta$ as large as 20 degrees, $\sin \theta$ is nearly the same as $\theta$ expressed in radians. Table $13.1 \sin \theta$ as ma function of angle $\theta$

Equation (13.24) is mathematically, identical to Eq. (13.11) except that the variable is angular displacement. Hence we have proved that for small q, the motion of the bob is simple harmonic. From Eqs. (13.24) and (13.11),

$$ \omega=\sqrt{\frac{m g L}{I}} $$

and

$$ \begin{equation*} T=2 \pi \sqrt{\frac{I}{m g L}} \tag{13.25} \end{equation*} $$

Now since the string of the simple pendulum is massless, the moment of inertia $I$ is simply $\mathrm{mL}^{2}$. Eq. (13.25) then gives the well-known formula for time period of a simple pendulum.

$$ \begin{equation*} T=2 \pi \sqrt{\frac{L}{g}} \tag{13.26} \end{equation*} $$

Summary

1. The motion that repeats itself is called periodlic motion.

2. The period $T$ is the time reequired for one complete oscillation, or cycle. It is related to the frequency $v$ by,

$$ T=\frac{1}{v} $$

The frequency $v$ of periodic or oscillatory motion is the number of oscillations per unit time. In the SI, it is measured in hertz :

$$ 1 \text { hertz }=1 \mathrm{~Hz}=1 \text { oscillation per second }=1 \mathrm{~s}^{-1} $$

3. In simple harmonic motion (SHM), the displacement $x(t)$ of a particle from its equilibrium position is given by,

$$ x(t)=A \cos (\omega t+\phi) \quad \text { (displacement) } $$

in which $A$ is the amplitude of the displacement, the quantity $(\omega t+\phi)$ is the phase of the motion, and $\phi$ is the phase constant. The angular frequency $\omega$ is related to the period and frequency of the motion by,

$$ \omega=\frac{2 \pi}{T}=2 \pi \nu \quad \text { (angular frequency). } $$

4. Simple harmonic motion can also be viewed as the projection of uniform circular motion on the diameter of the circle in which the latter motion occurs.

5. The particle velocity and acceleration during SHM as functions of time are given by,

$$ \begin{array}{rlr} v(t) & =-\omega A \sin (\omega t+\phi) & \text { (velocity), } \\ \\ a(t) & =-\omega^{2} A \cos (\omega t+\phi) & \\ \\ & =-\omega^{2} x(t) & \text { (acceleration), } \end{array} $$

Thus we see that both velocity and acceleration of a body executing simple harmonic motion are periodic functions, having the velocity amplitude $v_{m}=\omega A$ and acceleration amplitude $a_{m}=\omega^{2} A$, respectively.

6. The force acting in a simple harmonic motion is proportional to the displacement and is always directed towards the centre of motion.

7. A particle executing simple harmonic motion has, at any time, kinetic energy $K=1 / 2 m v^{2}$ and potential energy $U=1 / 2 \mathrm{kx}^{2}$. If no friction is present the mechanical energy of the system, $E=K+U$ always remains constant even though $K$ and $U$ change with time.

8. A particle of mass $m$ oscillating under the influence of Hooke’s law restoring force given by $F=-k x$ exhibits simple harmonic motion with

$$ \begin{array}{ll} \omega=\sqrt{\frac{k}{m}} & \text { (angular frequency) } \\ \\ T=2 \pi \sqrt{\frac{m}{k}} & \text { (period) } \end{array} $$

Such a system is also called a linear oscillator.

9. The motion of a simple pendulum swinging through small angles is approximately simple harmonic. The period of oscillation is given by,

$$ T=2 \pi \sqrt{\frac{L}{g}} $$

POINTS TO PONDER

1. The period $T$ is the least time after which motion repeats itself. Thus, motion repeats itself after $n T$ where $n$ is an integer.

2. Every periodic motion is not simple harmonic motion. Only that periodic motion governed by the force law $F=-k x$ is simple harmonic.

3. Circular motion can arise due to an inverse-square law force (as in planetary motion) as well as due to simple harmonic force in two dimensions equal to: $-m \omega^{2} r$. In the latter case, the phases of motion, in two perpendicular directions ( $x$ and $y$ ) must differ by $\pi / 2$. Thus, for example, a particle subject to a force $-m \omega^{2} r$ with initial position $(0$, $A)$ and velocity $(\omega A, 0)$ will move uniformly in a circle of radius $A$.

4. For linear simple harmonic motion with a given $\omega$, two initial conditions are necessary and sufficient to determine the motion completely. The initial conditions may be (i) initial position and initial velocity or (ii) amplitude and phase or (iii) energy and phase.

5. From point 4 above, given amplitude or energy, phase of motion is determined by the initial position or initial velocity.

6. A combination of two simple harmonic motions with arbitrary amplitudes and phases is not necessarily periodic. It is periodic only if frequency of one motion is an integral multiple of the other’s frequency. However, a periodic motion can always be expressed as a sum of infinite number of harmonic motions with appropriate amplitudes.

7. The period of SHM does not depend on amplitude or energy or the phase constant. Contrast this with the periods of planetary orbits under gravitation (Kepler’s third law).

8. The motion of a simple pendulum is simple harmonic for small angular displacement.

9. For motion of a particle to be simple harmonic, its displacement $x$ must be expressible in either of the following forms :

$$ \begin{aligned} & x=A \cos \omega t+B \sin \omega t \\ & x=A \cos (\omega t+\alpha), x=B \sin (\omega t+\beta) \end{aligned} $$

The three forms are completely equivalent (any one can be expressed in terms of any other two forms).

Thus, damped simple harmonic motion is not strictly simple harmonic. It is approximately so only for time intervals much less than $2 \mathrm{~m} / \mathrm{b}$ where $b$ is the damping constant.