Oscillations: Class 11 Physics · Chapter 13

13.1 Introduction

Some motions happen once and are over. A ball thrown to a fielder follows its arc and lands; a train runs from Pune to Mumbai and stops. Most of the motions in the earlier chapters of this course were of this one-off kind. Other motions come back to where they started, again and again: the Moon circling the Earth, a planet going round the Sun, the tip of a ceiling fan blade. Such motions are periodic.

This chapter looks at a special family of repeating motions in which a body goes back and forth on either side of a middle position. Drive over a speed breaker and the car keeps bobbing on its springs for a second or two. A diver leaves the end of a springboard and the board keeps quivering. Pluck a tanpura string and it blurs as it vibrates. Each of these is an oscillation (Fig. 13.1).

Oscillations sit underneath a surprising amount of physics. Your eardrum moves to and fro whenever a sound reaches it. The quartz crystal that keeps time in a wristwatch vibrates tens of thousands of times a second. The voltage at a wall socket in India rises and falls through $50$ complete cycles every second. And inside a solid that looks perfectly still, every atom is jiggling about its place in the crystal, more vigorously the hotter the solid is.

To describe any of these we need a small, exact vocabulary: period, frequency, displacement, amplitude and phase. The next section builds it, and the rest of the chapter uses it to study the simplest and most important oscillation of all, simple harmonic motion.

13.2 Periodic and oscillatory motions

Watch a car's turn indicator: on, off, on, off, with the same timing every time. Or follow the second hand of a wall clock, whose angle grows steadily for a minute and then starts again from twelve. If you plot what these do against time, the graph is made of one piece copied over and over (Fig. 13.2). That is the test for periodic motion: after a fixed interval, the motion repeats itself exactly.

Why do some repeating motions go back and forth? Drop a marble into a valley-shaped track (Fig. 13.3). Placed gently at the lowest point, it stays put: gravity and the push of the track cancel there, so this is its equilibrium position. Roll it a little way up either slope and the balance breaks; the leftover force always aims back down towards the bottom. That restoring force speeds the marble up on the way down, it overshoots, climbs the other side, is turned back again, and an oscillation is under way.

Every oscillation is periodic, but the reverse is not true. The tip of a fan blade repeats its path every turn, yet it never reverses direction along a line, so it is periodic without being oscillatory. The same is true of a planet in its orbit.

Oscillation and vibration mean the same thing in physics. Habit picks the word: a tall building swaying in a storm, a few seconds per sway, is said to oscillate, while the prongs of a tuning fork, flicking back and forth hundreds of times a second, are said to vibrate.

Leave a swing alone and every arc is a little lower than the one before, until it hangs still. Friction at the hooks and the drag of the air drain its energy; this fading is called damping. A parent giving a well-timed push each cycle replaces the lost energy and keeps the swing going. For most of this chapter we ignore damping and study the cleanest case of all, simple harmonic motion, in which the restoring force grows in exact proportion to the distance from equilibrium. A real, slowly damped oscillator behaves like this only over stretches of time much shorter than the time it takes to die away.

Oscillators rarely live alone. Tie a row of beads along a stretched string and a jiggle given to the first bead is handed on to the next, and the next. A disturbance passed along a chain of linked oscillators in this way is a wave. Sound in air, the shaking of the ground after an earthquake and light are all waves, and they are the subject of the next chapter.

13.2.1 Period and frequency

The shortest time after which a periodic motion repeats is its period, written $T$. Its SI unit is the second. A motion that repeats after $T$ also repeats after $2T$, $3T$ and every whole-number multiple of $T$, but only the least such time is called the period.

Periods in nature span an enormous range. A tuning fork marked $512\mathrm{H}\mathrm{z}$ completes each vibration in about $2$ milliseconds, while sunspots on the Sun wax and wane over a cycle of roughly $11$ years and Saturn needs about $29$ years to go once round the Sun. So it is often handier to say how many repeats happen each second. This number is the frequency, written $\nu$ (the Greek letter nu):

Frequency is measured in ${\mathrm{s}}^{-1},$ repeats per second. This unit has its own name, the hertz (Hz):

A frequency need not be a whole number. A pendulum that completes $7$ swings in $10\mathrm{s}$ has $\nu =0.7\mathrm{H}\mathrm{z};$ nothing requires a complete swing to finish exactly when a second ends.

13.2.2 Displacement

Until now, displacement has meant a change of position. Oscillations need a wider meaning. Here the displacement is the amount by which whatever is repeating differs from its equilibrium value, at a given moment. Measuring from equilibrium is a choice, but almost always the most convenient one.

So the displacement of a block on a spring is its distance $x$ from where the spring is relaxed, and that of a pendulum is most naturally the angle $\theta$ between the string and the vertical. It need not be a length or an angle at all. For the mains supply it is the voltage between the two pins of a socket, which swings positive and negative; for a sound reaching your ear it is how far the air pressure has risen above, or fallen below, its normal value (Fig. 13.4). Displacements carry a sign, and studying an oscillation means recording how the displacement changes with time.

Mathematically, a periodic displacement is a periodic function of time. The simplest ones are the sine and the cosine. Take

The cosine repeats whenever its argument grows by $2\pi$. So $f$ returns to the same value when $\omega t$ grows by $2\pi$, that is, when $t$ grows by

and $f(t+T)=f(t)$ for every $t$. The same holds for $A\sin \omega t$. A mixture of the two with the same $\omega$ is periodic too, and it turns out to be a single shifted sine curve.

13.3 Simple harmonic motion

Now for the star of the chapter. Put the equilibrium position at the origin of an $x$-axis. A body oscillating along this axis is in simple harmonic motion when a single cosine (or sine) curve describes its displacement at every instant:

The word simple is earned by that shape. Many motions repeat, but only those whose $x$–$t$ graph is a pure sine curve qualify. A strobe photograph shows what such a motion looks like. Hang a weight on a spring, pull it up, let go, and photograph it with flashes at equal time intervals (Fig. 13.6). The images bunch up near the two ends, where the weight is slow, and spread out near the middle, where it is fast. Spread the flashes along a time axis and they trace a cosine curve.

Each constant in Eq. (13.4) has a job. Start with $A$, the amplitude. A cosine never leaves the range from $-1$ to $+1$, so the body never strays further than $A$ from the centre on either side: $A$ is the farthest it gets, and by convention we keep it positive. Change $A$ alone and the motion simply gets wider or narrower, with the turning points still reached at the same instants.

Next, the angle inside the cosine. Think of $\omega t+\varphi$ as a pointer that tells you how far through its cycle the body has got: whether it is at a turning point or crossing the centre, and in which direction. This angle is the phase, and it grows steadily with time. Where the pointer stood when the clock started, $\varphi$, is the phase constant (also called the phase angle). It is fixed by how the motion began: the starting displacement $A\cos \varphi$ leaves two possible angles, and the starting direction picks one. Give two bodies the same $A$ and $\omega$ but different $\varphi$ and their graphs are the same curve slid along the time axis (Fig. 13.7).

The third constant, $\omega$, fixes the period.

So $\omega$, the angular frequency, is the rate at which the phase pointer turns, in radians per second. One full cycle is $2\pi$ radians, which is why $\omega$ is always $2\pi$ times the number of cycles per second. A faster pointer packs more cycles into the same time (step 4 of Fig. 13.7).

13.4 Simple harmonic motion and uniform circular motion

Uniform circular motion and SHM look very different, yet one hides inside the other. Look at a wheel turning steadily edge-on, with your eye in the plane of the wheel, and a point on the rim no longer seems to go round. It slides up and down along a straight line, slowing at the ends and rushing through the middle, exactly like an oscillator. The activity below lets you see this with a bicycle.

To turn this into algebra, three numbers describe the wheel: the radius $A$ of the circle traced by the point $P$, its steady angular speed $\omega$ (anticlockwise, say), and the angle $\varphi$ that the radius $OP$ made with the positive $x$-axis when the clock started. The angle grows at a constant rate, so after a time $t$ it has become $\omega t+\varphi$ (Fig. 13.8).

Drop a perpendicular from $P$ to the $x$-axis and call its foot $P^{\mathrm{\prime }}$. The position of $P^{\mathrm{\prime }}$ is the $x$-component of $OP$:

This is exactly Eq. (13.4). Seen edge-on, so that only one coordinate counts, steady motion round a circle is simple harmonic motion, with the circle's radius as the amplitude, the angular speed as the angular frequency and the starting angle as the phase constant. The particle $P$ is called the reference particle and its circle the reference circle.

Nothing makes the $x$-axis special. The shadow on the $y$-axis follows $y(t)=A\sin (\omega t+\varphi ):$ the same swing and the same timing, but always a quarter of a cycle ($\pi \mathrm{/}2$) away from the $x$-shadow. Turn this round and you get a useful fact: two SHMs along perpendicular lines, with equal amplitudes and a phase difference of $\pi \mathrm{/}2$, combine into uniform circular motion.

Keep the geometry and the physics apart, though. The reference circle is a description of SHM, not its cause. A body in SHM along a line is pushed by a force along that line that grows with its distance from the centre. A body in uniform circular motion needs a centripetal force of constant size that keeps turning to point at the centre. The two forces are quite different.

13.5 Velocity and acceleration in simple harmonic motion

The reference circle gives the velocity and acceleration of an oscillator almost for free. The reference particle $P$ moves round a circle of radius $A$ at angular speed $\omega$, so its speed is

Its velocity points along the tangent to the circle. The oscillator $P^{\mathrm{\prime }}$ always sits directly below or above $P$ on the $x$-axis, so the velocity of $P^{\mathrm{\prime }}$ is simply the $x$-component of the velocity of $P$ (Fig. 13.9). When $OP$ is at angle $\omega t+\varphi ,$ the tangent makes the same angle with the vertical, and the $x$-component is $-\omega A\sin (\omega t+\varphi )\mathrm{.}$ The minus sign says that while $P$ is in the upper half of the circle, $P^{\mathrm{\prime }}$ moves towards negative $x$.

Read Eq. (13.11) slowly, because it is the heart of SHM: the acceleration is the displacement multiplied by the fixed negative number $-{\omega }^2$. Double the distance from the centre and the acceleration doubles; cross to the other side and it flips. So at every point of the swing the body is being urged back towards $x=0$, hardest at the turning points and not at all at the centre.

Eliminating the time between Eqs. (13.4) and (13.9) gives the speed at any position, a result used constantly in problems. Since ${\sin }^2+{\cos }^2=1,$

The speed is greatest, $\omega A$, at the centre and zero at the two ends, just as the strobe photograph showed. With $\varphi =0$ the three quantities are $x=A\cos \omega t,$ $v=-\omega A\sin \omega t$ and $a=-{\omega }^{2}A\cos \omega t\mathrm{.}$ Stack their graphs on one time axis (Fig. 13.10) and slide a cursor along. All three share the period $T$; their peak values are $A$, $\omega A$ and ${\omega }^{2}A$. Each velocity peak arrives a quarter-cycle before the matching displacement peak (a phase lead of $\pi \mathrm{/}2$), while the acceleration is always a mirror image of the displacement (a phase difference of $\pi$).

13.6 Force law for simple harmonic motion

Newton's second law turns the acceleration of Eq. (13.11) into a force. For a particle of mass $m$ in SHM,

Writing $k$ for the positive constant $m{\omega }^2$,

so the period of a mass $m$ held by a spring of spring constant $k$ is

Because it always pulls the body back towards the mean position, this force is the restoring force. Notice that SHM can now be recognised in two ways. You may be given the motion, $x=A\cos (\omega t+\varphi ),$ or you may be given the force, $F=-kx$. They carry the same information: two differentiations take you from the motion to the force, and two integrations (a skill for later) take you back, with $A$ and $\varphi$ appearing as the two constants of integration. In problems the force route is usually quicker: find the net force for a small displacement, and if it comes out as $-(\text{constant})\times (\text{displacement}),$ the motion is SHM.

Only this force law gives SHM. A ball bouncing on a floor repeats its motion, but the force on it, steady gravity plus a sharp kick at each bounce, has nothing to do with $-kx$, so the motion is periodic without being simple harmonic. A body moving under a force exactly proportional to $x$ is called a linear harmonic oscillator. Why is this one case so common? Near the bottom of almost any potential-energy valley, the force can be written as $-kx$ plus smaller terms in $x^2$, $x^3$ and so on. For small swings those extra terms hardly matter, so atoms in a crystal, a floating boat and a swinging lamp all behave, to a good approximation, as linear oscillators. When the extra terms do matter, the oscillator is called non-linear.

Notice something remarkable in $T=2\pi \sqrt{m\mathrm{/}k}:$ the amplitude does not appear. A spring oscillator takes the same time for a wide swing as for a narrow one, because a wider swing brings a proportionally larger restoring force. Planetary orbits behave quite differently. By Kepler's third law a bigger orbit takes longer, with $T^2$ proportional to the cube of the orbit's size.

The block on a horizontal spring is the standard example, but the force law also covers springs that hang vertically. Gravity then does not spoil the SHM; it only moves the equilibrium point (Fig. 13.12).

Two springs can share one mass. Side by side (in parallel), they stretch by the same amount and their forces add. End to end (in series), they carry the same force and their stretches add. Applying the force law to each arrangement gives the single spring constant that would do the same job:

For example, a $2.0\mathrm{k}\mathrm{g}$ block held by springs of $300\mathrm{N}\mathrm{/}\mathrm{m}$ and $600\mathrm{N}\mathrm{/}\mathrm{m}$ in parallel oscillates with $T=2\pi \sqrt{2.0\mathrm{/}900}\approx 0.30\mathrm{s}\mathrm{.}$ With the same springs in series, $k=200\mathrm{N}\mathrm{/}\mathrm{m}$ and $T\approx 0.63\mathrm{s}\mathrm{.}$ A block between two springs fixed to opposite walls is a parallel arrangement: a displacement stretches one spring and squashes the other, and both forces point back towards the middle.

13.7 Energy in simple harmonic motion

An oscillator constantly trades one kind of energy for another. At the ends of the swing it stops for an instant, so it has no kinetic energy; at the centre it moves fastest. Meanwhile the spring is most stretched or squashed at the ends and relaxed at the centre. Each kind of energy rises from nothing to a peak and falls back again, while their sum never changes.

No $t$ survives in Eq. (13.18): with no friction, nothing can drain or add energy, so the total is locked in from the moment of release. It grows as the amplitude squared, so doubling the swing stores four times the energy. Combining Eqs. (13.16) and (13.18) also gives the kinetic energy at any position, $K=\frac{1}{2}k(A^2-x^2),$ which matches the speed formula of Section 13.5.

Plotted against $x$, the potential energy is a parabola, a bowl whose rim reaches the total-energy line exactly at $x=\pm A$ (Fig. 13.13). At any position the gap between the bowl and the line is the kinetic energy. The particle can never go beyond $\pm A$, because there $U$ would exceed $E$ and $K$ would have to be negative, which is impossible. Plotted against time, $K$ and $U$ are two humps that take turns, each peaking twice per period.

In both graphs nothing dips below zero. For $K$ that is automatic, since it depends on $v^2$. For $U$ it is a consequence of our choice of zero at the mean position: only changes of $U$ are physical, and putting the zero elsewhere would shift the bowl up or down without changing the motion.