13.1 Introduction
Chapter 13 — Oscillations
In the year 1583, a young medical student named Galileo Galilei sat in the cathedral of Pisa, watching a large lamp swing back and forth from the high ceiling. A draught had set it moving, and Galileo — bored, perhaps — began to time the swings. He had no clock, so he used the only reliable timer available to him: his own pulse.
What he noticed changed physics forever. Whether the lamp swung through a wide arc or a small one, each complete swing seemed to take the same amount of time. The size of the swing was shrinking as the lamp slowly came to rest, yet the rhythm held steady. Galileo repeated the observation with pendulums of his own, tying stones to threads of different lengths, and confirmed it: the time of a swing depended on the length of the thread, but almost nothing on how far the bob was pulled aside.
This single insight — that certain motions keep a fixed rhythm — became the beating heart of every clock built for the next three centuries. It also opened a door onto a huge family of motions that repeat themselves: a vibrating guitar string, a bobbing boat, the atoms trembling inside a solid, even the voltage in an AC power line. All of these are governed by the same simple mathematics you are about to learn.
Figure to come
Fig. 13.0 – A pendulum lamp swinging in a cathedral, with a young Galileo seated below timing it against his wrist pulse; dashed arcs showing wide and narrow swings taking equal time.
By the end of this chapter, you will be able to answer each of these questions using clear physical reasoning and a few compact formulas.
Look around you and you will find motion everywhere. Some of it moves in one direction and never comes back the same way — a ball thrown across a field, a car speeding down a highway. You have already studied such motions in earlier chapters: straight-line (rectilinear) motion and the curved flight of a projectile. These are called non-repetitive motions, because the object does not return again and again to where it started.
But there is another large family of motions that do repeat. A planet circling the Sun comes back to the same point in its orbit after a fixed time. A ball whirled on a string returns to the same position again and again. Any motion that repeats itself after equal intervals of time is called periodic motion.
Now think about a slightly different kind of repeating motion — the kind you first met as a child. Rocking in a cradle, or swinging on a swing in a park, you moved forward and backward about a central resting point, over and over. A pendulum in an old wall clock does exactly the same thing: it swings out to one side, comes back, swings to the other side, and returns — endlessly, in a steady rhythm.
This special “to and fro” movement about a central position is called oscillatory motion. The central position is the point where the object would sit quietly if left undisturbed — the resting point of the motion. We call it the mean position.
Examples of this back-and-forth motion are all around us. A boat tied in a river bobs up and down as waves pass under it. The piston inside a steam engine slides back and forth as the engine runs. In each case the object keeps returning through a central point, moving first one way and then the opposite way.
Why should we bother studying such motion so carefully? Because the study of oscillations is one of the true foundations of physics. Once you understand how one object swings back and forth, you hold the key to understanding a huge range of physical phenomena — many of which look nothing like a swing at first glance.
Consider music. In instruments like the sitar, the guitar, or the violin, it is the vibrating strings that create the pleasing sounds we hear. The word “vibrate” here means to move to and fro very rapidly about a mean position — it is oscillation, just fast.
The same idea explains how sound reaches your ear at all. In a drum, the stretched membrane vibrates back and forth. In a telephone or a loudspeaker, a thin sheet called a diaphragm vibrates. These vibrations push against the surrounding air, and the tiny air molecules themselves begin to vibrate to and fro. It is this trembling of air molecules, passed from one to the next, that carries sound across a room.
Figure to come
Fig. 13.1a – A loudspeaker diaphragm vibrating back and forth, with nearby air molecules shown oscillating about their positions, passing the disturbance outward as sound.
Oscillation reaches down even into the material of solids. Inside a solid, the atoms are not perfectly still. Each atom vibrates back and forth about its own fixed resting spot, called its equilibrium position. The hotter the solid, the more energetically its atoms vibrate — in fact, the average energy of these vibrations is directly proportional to the temperature. In this sense, heat itself is tied to oscillation.
Even electricity shows this pattern. The AC (alternating current) power supply that runs the appliances in your home does not push charge steadily in one direction. Instead, the voltage rises to a positive value, falls back, swings to a negative value, and returns — oscillating on either side of a mean value of zero, again and again.
So oscillatory motion is not a small, isolated topic. It is a single idea that appears in music, in sound, in heat, in electricity, and in the structure of matter itself. Learning it once lets you recognise it everywhere.
To describe any periodic or oscillatory motion precisely, physicists use a small set of basic quantities. These include the period (how long one full repetition takes), the frequency (how many repetitions happen per second), the displacement (how far the object is from its mean position), the amplitude (the largest displacement it reaches), and the phase (which tells us exactly where in its cycle the motion is at a given instant). Each of these terms is defined and developed carefully in the next section, and together they form the language we will use for the rest of the chapter.
13.2 Periodic and Oscillatory Motions
Repeating motion is easy to find once you start looking for it. Imagine an insect that climbs steadily up a ramp, slips back to the bottom, and then climbs again, repeating the same journey over and over. If you plotted its height above the ground against time, the graph would rise and fall in an identical pattern again and again, as shown in Fig. 13.1(a).
The same happens in other everyday cases. A child who climbs a step, jumps down, and climbs again traces the repeating height-time pattern of Fig. 13.1(b). And when you bounce a ball between your palm and the floor, its height rises and falls in the repeating pattern of Fig. 13.1(c).
Figure to come
Fig. 13.1 – Three height-versus-time graphs: (a) an insect on a ramp (sawtooth shape), (b) a child on a step (flat-topped shape), (c) a bouncing ball (smooth parabolic arcs), each marked with its period T.
Look closely at the bouncing-ball graph in Fig. 13.1(c). Each curved arc is a piece of a parabola. These arcs come directly from the equations of motion under gravity that you studied earlier in kinematics (Section 2.6). For the downward part of each bounce,
\[h = ut + \frac{1}{2}gt^2\]
and for the upward part,
\[h = ut - \frac{1}{2}gt^2\]
Here \(h\) is the height (in metres, m), \(u\) is the initial speed at the start of that stretch of motion (in m s\(^{-1}\)), \(g\) is the acceleration due to gravity (in m s\(^{-2}\)), and \(t\) is the time (in seconds, s). The only difference between the two equations is the sign of the \(\frac{1}{2}gt^2\) term: gravity speeds the ball up on the way down but slows it down on the way up. Each bounce uses a different value of \(u\), yet the overall pattern still repeats.
As we saw in Section 13.1, any motion that repeats itself after equal intervals of time is periodic motion. The three examples above are all periodic, because their graphs repeat exactly after a fixed time.
Very often, a body in periodic motion has a special resting point somewhere along its path, called its equilibrium position. At this point, the net external force on the body is zero. So if the body is placed there at rest, it will simply stay there forever, undisturbed.
Now suppose you nudge the body slightly away from this equilibrium point. A force at once comes into play that tries to push or pull the body back towards equilibrium. This “come back” force keeps driving the body back and forth through the equilibrium point, and the result is oscillation (or vibration).
A ball resting at the bottom of a bowl shows this clearly. At the bottom it sits in equilibrium. Push it a little way up the curved side and let go, and it rolls back down, overshoots, climbs the other side, and returns — oscillating back and forth about the lowest point until it finally settles.
This leads to an important relationship between the two kinds of motion. Every oscillatory motion is periodic, because a body going to and fro must return to each position after a fixed time. But the reverse is not true: not every periodic motion is oscillatory.
A planet moving around the Sun, or a ball whirled steadily on a string, repeats its motion after a fixed time and so is periodic — yet at no stage does it move to and fro about a mean position. Such circular motion is periodic but not oscillatory.
You may have noticed that we use two words — “oscillation” and “vibration” — for what looks like the same kind of motion. In truth there is no significant physical difference between them. The choice of word is mostly a matter of speed. When the to-and-fro motion is slow, we tend to call it oscillation, as with a branch of a tree swaying in the wind. When it is very rapid, we tend to call it vibration, as with the rapidly moving string of a musical instrument. The underlying physics is exactly the same.
Among all oscillatory motions, one particular kind is the simplest and the most important: simple harmonic motion, usually written as SHM. It arises whenever the restoring force on the oscillating body is directly proportional to how far the body has been displaced from the mean position (which is also the equilibrium position), and always points back towards that mean position.
This is the first, force-based way of describing SHM. A second, equivalent description — in terms of how the displacement changes with time — is developed fully in Section 13.3.
In the real world, oscillations do not continue forever. A swinging pendulum or a plucked string gradually loses energy to friction and other dissipative effects, so the size of its swing shrinks little by little until the body comes to rest at its equilibrium position. This gradual dying-out of oscillation is called damping. A damped oscillation can be kept going only if some outside agency keeps feeding it energy through a repeated periodic push; such a driven motion is called a forced oscillation.
Finally, oscillation is not limited to single objects. Any material medium — a solid, a liquid, a gas — can be pictured as a vast collection of tiny oscillators that are linked, or coupled, to one another. When these coupled oscillators move together, their combined oscillation travels through the medium as a wave. Water waves on a pond, seismic waves inside the Earth, and electromagnetic waves such as light are all examples. We shall study the wave phenomenon in detail in the next chapter.
13.2.1 Period and Frequency
We have already seen that any motion which repeats itself after equal intervals of time is periodic motion. The most natural thing to measure about such a motion is how long one complete repetition takes.
The smallest interval of time after which the motion repeats itself exactly is called the period of the motion. It is denoted by the symbol \(T\), and its SI unit is the second (s).
The word “smallest” in this definition is important. Of course the motion has also repeated after two periods (\(2T\)), three periods (\(3T\)), and so on. But the period is defined as the shortest such repeat time — the time for exactly one complete cycle.
The second is a convenient unit for motions that repeat on a human timescale, but many periodic motions are either far too fast or far too slow for it to be handy. For such cases, other units of time are used.
At the fast end, a quartz crystal vibrates so rapidly that its period is more conveniently expressed in microseconds (\(10^{-6}\) s), abbreviated \(\mu\)s. At the slow end, the planet Mercury takes 88 earth days to complete one orbit of the Sun, and Halley’s comet returns to our skies only once every 76 years.
Instead of asking how long one cycle takes, we can ask the reverse question: how many cycles occur in each unit of time? This quantity is called the frequency of the periodic motion, denoted by the Greek letter \(\nu\) (pronounced “nu”). Since one cycle takes a time \(T\), the number of cycles per second is simply the reciprocal of \(T\):
\[\nu = \frac{1}{T} \tag{13.1}\]
Here \(\nu\) is the frequency and \(T\) is the period in seconds.
From Eq. (13.1), the SI unit of frequency is s\(^{-1}\) (one per second). This unit has been given a special name, the hertz, abbreviated Hz:
\[1 \text{ hertz} = 1 \text{ Hz} = 1 \text{ oscillation per second} = 1 \text{ s}^{-1} \tag{13.2}\]
One more point deserves attention: the frequency \(\nu\) need not be a whole number. A body may repeat its motion, say, 1.25 times every second, giving a frequency of 1.25 Hz. Frequency simply measures how many cycles fit into one second, and that count can perfectly well be a fraction.
13.2.2 Displacement
Back in Class 11 (Section 3.2), you learnt that the displacement of a particle is the change in its position vector — a straightforward idea about where an object is. In the study of oscillations, we stretch the meaning of “displacement” to something broader. Here it stands for the change, with time, of any physical property we are watching, not just position.
The simplest case is still an ordinary change of position. If a steel ball rolls in a straight line along a surface, its distance from the starting point, taken as a function of time, is its position displacement. Where you place the origin from which distance is measured is purely a matter of convenience.
Now consider a block attached to a spring, with the spring’s other end fixed to a rigid wall, resting on a frictionless surface, as shown in Fig. 13.2(a). Here it is most convenient to measure the block’s displacement from its equilibrium position — the point where the spring is neither stretched nor compressed. We call this displacement \(x\).
Figure to come
Fig. 13.2(a) – A block attached to a spring whose other end is fixed to a rigid wall; the block rests on a frictionless surface, with displacement x measured from the equilibrium position.
Displacement need not even be a length. For an oscillating simple pendulum, shown in Fig. 13.2(b), the angle the string makes with the vertical, taken as a function of time, can itself serve as the displacement variable. In this case the “displacement” is an angle.
Figure to come
Fig. 13.2(b) – An oscillating simple pendulum, showing the angular displacement θ measured from the vertical.
This generalisation matters because the displacement variable is not always about position at all. There are many other kinds. The voltage across a capacitor, changing with time in an AC circuit, is a displacement variable. So are the pressure variations in time as a sound wave travels, and the changing electric and magnetic fields in a beam of light. Each of these is a “displacement” in its own context.
Notice from that example that a displacement variable may take both positive and negative values — the block sits on either side of its equilibrium point, and the voltage swings above and below zero. In experiments on oscillations, we simply measure this displacement at many different instants of time.
The displacement can be captured neatly by a mathematical function of time. When the motion is periodic, this function too must repeat in time. One of the simplest periodic functions is
\[f(t) = A\cos\omega t \tag{13.3a}\]
Here \(f(t)\) is the displacement at time \(t\), \(A\) is a constant that fixes the largest value the function can reach, and \(\omega\) is a positive constant that controls how rapidly the function oscillates.
Why is this function periodic, and what is its period? The cosine function returns to the same value every time its angle (its argument) increases by \(2\pi\) radians. So if \(\omega t\) is increased by any whole-number multiple of \(2\pi\), the value of \(f(t)\) is unchanged. The smallest time \(T\) after which the motion repeats is found by demanding that the argument grow by exactly \(2\pi\): that is, \(\omega(t+T) - \omega t = 2\pi\), which gives \(\omega T = 2\pi\), so
\[T = \frac{2\pi}{\omega} \tag{13.3b}\]
With this period, the function satisfies the repeating condition \(f(t) = f(t+T)\).
The same result holds equally well for a sine function, \(f(t) = A\sin\omega t\). Going further, a linear combination of a sine and a cosine of the same \(\omega\), such as
\[f(t) = A\sin\omega t + B\cos\omega t \tag{13.3c}\]
is also a periodic function with the very same period \(T\).
Such a combination can always be rewritten as a single sinusoid. To see how, expand a sine of a shifted angle: \(D\sin(\omega t + \phi) = D\sin\omega t\,\cos\phi + D\cos\omega t\,\sin\phi\). Comparing this term by term with \(A\sin\omega t + B\cos\omega t\), we can match the coefficients by setting
\[A = D\cos\phi \quad \text{and} \quad B = D\sin\phi\]
so that Eq. (13.3c) becomes
\[f(t) = D\sin(\omega t + \phi) \tag{13.3d}\]
Here \(D\) and \(\phi\) are constants, given by
\[D = \sqrt{A^2 + B^2} \quad \text{and} \quad \phi = \tan^{-1}\!\left(\frac{B}{A}\right)\]
(You can check the expressions for \(D\) and \(\phi\) by squaring and adding \(A\) and \(B\) to get \(D\), and dividing \(B\) by \(A\) to get \(\tan\phi\).)
Why do sine and cosine functions deserve so much attention? Because of a remarkable mathematical result. Any periodic function whatsoever, no matter how complicated its shape, can be built up as a sum of simple sine and cosine functions of different periods, each multiplied by a suitable coefficient.
13.3 Simple Harmonic Motion
We now turn to the most important of all oscillations. Picture a single particle moving back and forth along the x-axis, about the origin O. It never strays beyond a point \(+A\) on one side or a point \(-A\) on the other, as shown in Fig. 13.3.
Figure to come
Fig. 13.3 – A particle vibrating back and forth about the origin of the x-axis, between the limits +A and -A.
This to-and-fro motion is called simple harmonic if the displacement \(x\) of the particle from the origin changes with time in one particular way — as a cosine function of time:
\[x(t) = A\cos(\omega t + \phi) \tag{13.4}\]
Here \(x(t)\) is the displacement at time \(t\), and \(A\), \(\omega\), and \(\phi\) are constants. These are the same three constants we met in Section 13.2.2; here they finally receive their standard names.
Recall that in Section 13.2 we described SHM through its force: a restoring force directly proportional to displacement and always pointing towards the mean position. Equation (13.4) is the same motion described a different way — through how the displacement changes with time. The two descriptions, force-based and displacement-based, are two faces of one motion, as we will confirm in Section 13.6.
So SHM is not just any periodic motion. It is the special case in which the displacement is a pure sinusoidal (sine or cosine) function of time. This single-frequency, pure-sine character is exactly what makes it the simplest oscillation. And recall Fourier’s theorem from Section 13.2.2: every complicated periodic motion can be built as a sum of such simple harmonic motions. That is why SHM is treated as the fundamental oscillation from which all others are assembled.
Figure 13.4 shows the particle’s position at equal time steps of \(T/4\), at the instants \(t = 0,\ T/4,\ T/2,\ 3T/4,\ T,\ 5T/4\), where \(T\) is the period. Notice in Fig. 13.4 that the motion repeats after a time \(T\) no matter which instant we choose to call \(t = 0\). Notice too that the particle moves fastest as it passes through the centre (\(x = 0\)) and momentarily stops at the extremes (\(x = \pm A\)).
Figure to come
Fig. 13.4 – The particle’s location in SHM at t = 0, T/4, T/2, 3T/4, T, 5T/4; speed maximum at x = 0 and zero at the extremes ±A.
While Fig. 13.4 shows separate snapshots, Fig. 13.5 plots the displacement \(x\) against time \(t\) as one smooth, continuous curve — the familiar wavy shape of a sinusoidal function.
Figure to come
Fig. 13.5 – Displacement as a continuous function of time for simple harmonic motion (a smooth sinusoidal curve).
The three constants \(A\), \(\omega\), and \(\phi\) that characterise a given SHM each have a standard name, summarised in Fig. 13.6. Let us understand them one at a time.
Figure to come
Fig. 13.6 – A summary chart of the standard symbols in Eq. (13.4): x(t) = displacement, A = amplitude, ω = angular frequency, (ωt + φ) = phase, φ = phase constant.
Amplitude. The amplitude \(A\) of an SHM is the magnitude of the maximum displacement of the particle from the mean position. As the cosine function runs through its full range, from \(+1\) to \(-1\), the displacement swings between the extremes \(+A\) and \(-A\).
Two simple harmonic motions may share the same \(\omega\) and \(\phi\) yet have different amplitudes \(A\) and \(B\), as shown by curves 1 and 2 in Fig. 13.7(a).
Figure to come
Fig. 13.7(a) – Displacement versus time from Eq. (13.4) with φ = 0; curves 1 and 2 have two different amplitudes A and B.
Phase. While the amplitude \(A\) stays fixed for a given SHM, the actual state of the particle — its position and velocity — at any instant \(t\) is decided by the quantity \((\omega t + \phi)\) sitting inside the cosine. This time-dependent quantity is called the phase of the motion.
Phase constant. The value of the phase at the starting instant \(t = 0\) is simply \(\phi\). This constant is called the phase constant (or phase angle). If the amplitude is already known, \(\phi\) can be found from the particle’s displacement at \(t = 0\).
Two simple harmonic motions may have identical amplitude \(A\) and identical \(\omega\) but different phase constants \(\phi\), as shown by curves 3 and 4 in Fig. 13.7(b), drawn for \(\phi = 0\) and \(\phi = -\pi/4\).
Figure to come
Fig. 13.7(b) – Two plots from Eq. (13.4) with the same amplitude A; curves 3 and 4 are for φ = 0 and φ = -π/4 respectively.
Angular frequency. Finally, the constant \(\omega\) is closely tied to the period \(T\). To see how, take the simplest case \(\phi = 0\) in Eq. (13.4):
\[x(t) = A\cos\omega t \tag{13.5}\]
Since the motion has period \(T\), the displacement must be the same at time \(t\) and at time \(t + T\); that is, \(x(t) = x(t + T)\):
\[A\cos\omega t = A\cos\omega(t + T) \tag{13.6}\]
The cosine function first repeats itself when its argument increases by exactly \(2\pi\). Therefore we require
\[\omega(t + T) = \omega t + 2\pi\]
which gives \(\omega T = 2\pi\), so
\[\omega = \frac{2\pi}{T} \tag{13.7}\]
The constant \(\omega\) is called the angular frequency of the SHM, and its SI unit is radians per second (rad s\(^{-1}\)).
Because the frequency of oscillation is simply \(\nu = 1/T\) (from Section 13.2.1), we can also write \(\omega = 2\pi/T = 2\pi\nu\). In other words, the angular frequency is \(2\pi\) times the ordinary frequency — it measures how many radians of the cycle are swept out each second.
Two simple harmonic motions may have the same \(A\) and \(\phi\) but different \(\omega\), as seen in Fig. 13.8. There, curve (b) has half the period and therefore twice the frequency of curve (a).
Figure to come
Fig. 13.8 – Plots of Eq. (13.4) for φ = 0 at two different periods; curve (b) has half the period and twice the frequency of curve (a).
13.4 Simple Harmonic Motion and Uniform Circular Motion
There is a surprising and beautiful link between the two kinds of repeating motion we have met: simple harmonic motion, which goes back and forth along a straight line, and uniform circular motion, which goes round and round. In this section we show that SHM is nothing but the “shadow” of uniform circular motion. More precisely, the projection of a uniformly circulating particle onto any diameter of its circle executes SHM.
A simple experiment, shown in Fig. 13.9, makes this connection easy to picture. Tie a ball to the end of a string and whirl it in a horizontal circle about a fixed point, keeping the angular speed constant. The ball is now in uniform circular motion in the horizontal plane.
Figure to come
Fig. 13.9 – A ball whirled in a horizontal circle; viewed edge-on (in the plane of the circle) its motion appears as straight-line to-and-fro motion, i.e. SHM.
Now change how you look at it. View the ball edge-on — from the side, with your eye in the plane of the circle. You no longer see a circle at all. Instead the ball appears to move back and forth along a straight horizontal line, with the centre of rotation as the midpoint of that line. You would see exactly the same to-and-fro motion by watching the ball’s shadow cast on a wall placed perpendicular to the plane of the circle. In both cases, what you are really watching is the ball’s motion projected onto a single diameter of the circle.
Let us now put this into mathematics, using Fig. 13.10. Suppose a particle P moves uniformly on a circle of radius \(A\) with angular speed \(\omega\), going round anticlockwise. At the starting instant \(t = 0\), the position vector \(\overrightarrow{OP}\) makes an angle \(\phi\) with the positive direction of the x-axis. In a time \(t\), the particle sweeps through a further angle \(\omega t\), so that \(\overrightarrow{OP}\) now makes an angle \((\omega t + \phi)\) with the positive x-axis.
Figure to come
Fig. 13.10 – The reference circle of radius A; particle P at angle (ωt + φ), with its projection P′ on the x-axis and initial angle φ marked.
Next, drop a perpendicular from P onto the x-axis. The foot of this perpendicular is the point P′, the projection of P on the x-axis. As P travels around the circle, P′ slides back and forth along the x-axis between \(+A\) and \(-A\). From the right-angled triangle in Fig. 13.10, the x-coordinate of P′ (the side adjacent to the angle) is the radius times the cosine of the angle:
\[x(t) = A\cos(\omega t + \phi)\]
But this is exactly the defining equation of SHM, Eq. (13.4). So we have proved it: as the particle P moves uniformly on the circle, its projection P′ on a diameter of that circle executes simple harmonic motion. The rounded, endless circular motion, seen edge-on, is precisely the back-and-forth of SHM.
Because this circle is used as a tool to understand the straight-line motion, the moving particle P and the circle it travels on are given special names.
We are free to project the motion of P onto any diameter we like. If instead we project onto the y-axis, the y-coordinate of the projection is the radius times the sine of the angle:
\[y = A\sin(\omega t + \phi)\]
This is also an SHM, with the very same amplitude \(A\) as the projection on the x-axis, but differing from it in phase by \(\pi/2\).
One caution is essential here. This link between circular motion and SHM is purely geometric — it is about the shape of the motion, not about the forces. The force acting on a particle in linear SHM is the restoring force, directed back and forth along the line of motion towards the mean position. It is quite different from the centripetal force that keeps the reference particle P moving in its circle, which always points inward towards the centre. The two motions look connected on paper, but the forces behind them are not the same.
13.5 Velocity and Acceleration in Simple Harmonic Motion
In the previous section we saw that the displacement in SHM is the x-projection of a reference particle moving on a circle. The same reference-circle trick lets us find the velocity and acceleration of the SHM as well — we simply project the velocity and acceleration of the reference particle onto the x-axis.
Start with velocity. A particle P in uniform circular motion has a speed equal to its angular speed \(\omega\) multiplied by the radius \(A\) of the circle:
\[v = \omega A \tag{13.8}\]
Here \(v\) is the speed (in m s\(^{-1}\)), \(\omega\) is the angular speed (in rad s\(^{-1}\)), and \(A\) is the radius (in m).
The direction of this velocity \(\vec{v}\) at any instant is along the tangent to the circle at the point where P happens to be. To get the velocity of the shadow particle P′ on the x-axis, we project this tangential velocity onto the x-axis. From the geometry of Fig. 13.11, this x-projection works out to
\[v(t) = -\omega A\sin(\omega t + \phi) \tag{13.9}\]
Figure to come
Fig. 13.11 – The velocity v(t) of the projection particle P′ is the x-projection of the tangential velocity v of the reference particle P.
The negative sign tells us that at this instant the velocity of P′ points opposite to the positive direction of the x-axis. Equation (13.9) gives the instantaneous velocity of any particle executing SHM whose displacement is given by Eq. (13.4).
We did not really need the geometry at all. Since velocity is the rate of change of displacement, we can get the same result simply by differentiating the displacement, Eq. (13.4), with respect to time:
\[v(t) = \frac{d}{dt}\,x(t) \tag{13.10}\]
Carrying out this differentiation on \(x(t) = A\cos(\omega t + \phi)\) gives exactly \(v(t) = -\omega A\sin(\omega t + \phi)\), confirming Eq. (13.9).
There is a very useful companion relation hidden in these two equations. If we square the displacement and the velocity and combine them using \(\sin^2\theta + \cos^2\theta = 1\), the time drops out, and we are left with the speed written directly in terms of the displacement:
\[v = \pm\,\omega\sqrt{A^2 - x^2}\]
This shows at a glance that the speed is largest at the centre (\(x = 0\)), where it equals \(\omega A\), and drops to zero at the extremes (\(x = \pm A\)) — a fact worth remembering for numericals.
Now for acceleration. We again borrow from the reference circle. The reference particle P, moving uniformly in its circle, has a centripetal acceleration of magnitude \(v^2/A\), which equals \(\omega^2 A\), always directed inward towards the centre — that is, along the line PO. Projecting this acceleration onto the x-axis gives the acceleration of P′, shown in Fig. 13.12:
\[a(t) = -\omega^2 A\cos(\omega t + \phi) = -\omega^2 x(t) \tag{13.11}\]
Figure to come
Fig. 13.12 – The acceleration a(t) of the projection particle P′ is the x-projection of the centripetal acceleration a of the reference particle P.
Just as with velocity, we can reach Eq. (13.11) without any geometry, by differentiating the velocity, Eq. (13.9), with respect to time:
\[a(t) = \frac{d}{dt}\,v(t) \tag{13.12}\]
Equation (13.11) contains the single most important property of SHM. It says that the acceleration is directly proportional to the displacement, with a minus sign. The minus sign means the acceleration always points opposite to the displacement — that is, back towards the centre.
To see this clearly, note the signs. When \(x(t) > 0\) (particle to the right of centre), Eq. (13.11) gives \(a(t) < 0\) (acceleration to the left). When \(x(t) < 0\) (particle to the left of centre), \(a(t) > 0\) (acceleration to the right). So whatever the value of \(x\) between \(-A\) and \(+A\), the acceleration is always aimed back at the centre.
To compare the three quantities side by side, let us take the simple case \(\phi = 0\) and write them together:
\[x(t) = A\cos\omega t, \qquad v(t) = -\omega A\sin\omega t, \qquad a(t) = -\omega^2 A\cos\omega t\]
Their plots against time are shown in Fig. 13.13. All three vary sinusoidally with the same period \(T\); only their peak values and their phases differ. The displacement \(x\) swings between \(-A\) and \(+A\); the velocity swings between \(-\omega A\) and \(+\omega A\); and the acceleration swings between \(-\omega^2 A\) and \(+\omega^2 A\). The quantities \(\omega A\) and \(\omega^2 A\) are called the velocity amplitude and the acceleration amplitude, respectively.
Figure to come
Fig. 13.13 – Displacement, velocity, and acceleration of a particle in SHM plotted against time; same period T, but differing in phase.
Looking at Fig. 13.13, compare the phases. The velocity curve is shifted from the displacement curve by a phase of \(\pi/2\) (a quarter cycle), so velocity peaks where displacement is zero. The acceleration curve is shifted from the displacement curve by a phase of \(\pi\) (half a cycle), so acceleration is exactly opposite in sign to displacement — which is just what the minus sign in \(a = -\omega^2 x\) tells us.
13.6 Force Law for Simple Harmonic Motion
So far we have described simple harmonic motion in two ways: through its displacement, \(x(t) = A\cos(\omega t + \phi)\) (Eq. 13.4), and through its acceleration, \(a = -\omega^2 x\) (Eq. 13.11). We now bring force into the picture, using Newton’s second law of motion.
For a particle of mass \(m\), Newton’s second law says the force is mass times acceleration, \(F = ma\). Substituting the SHM acceleration from Eq. (13.11), \(a = -\omega^2 x(t)\), gives
\[F(t) = ma = -m\omega^2 x(t)\]
The combination \(m\omega^2\) is just a constant for a given oscillator, so we give it a single name \(k\) and write
\[F(t) = -k\,x(t) \tag{13.13}\]
where
\[k = m\omega^2 \tag{13.14a}\]
and, rearranging this to make \(\omega\) the subject,
\[\omega = \sqrt{\frac{k}{m}} \tag{13.14b}\]
Here \(F\) is the force (in newtons, N), \(m\) is the mass (in kg), \(x\) is the displacement from the mean position (in m), \(\omega\) is the angular frequency (in rad s\(^{-1}\)), and \(k\) is the constant defined above (in N m\(^{-1}\)).
Look closely at the minus sign in Eq. (13.13). Exactly as with the acceleration, it means the force always points back towards the mean position, opposing whatever displacement the particle has. Because it always acts to bring the particle back home to its mean position, this force is called the restoring force. You can picture it acting on the familiar block-and-spring system of Fig. 13.2(a): pull the block out by \(x\) and the spring pulls it back with a force proportional to \(x\).
The constant \(k\) is called the force constant (or spring constant). From Eq. (13.14a), \(k = m\omega^2\), and its SI unit is newton per metre (N m\(^{-1}\)). It measures how strong the restoring force is for a given displacement.
We can now keep a promise made in Section 13.3. Simple harmonic motion can be defined in two completely equivalent ways: either by the displacement equation, Eq. (13.4), or by the force law, Eq. (13.13). The two are linked by calculus. Going from the displacement to the force required us to differentiate twice — once to get velocity, again to get acceleration, which Newton’s law turns into force. Going the other way, integrating the force law twice brings us right back to the displacement equation. Either statement fully captures SHM.
Notice also that the force in Eq. (13.13) is proportional to the first power of \(x\) — it is a linear relationship. A particle oscillating under such a force is called a linear harmonic oscillator. In the real world, the restoring force may contain small extra terms proportional to \(x^2\), \(x^3\), and so on; a particle governed by such a force is called a non-linear oscillator.
13.7 Energy in Simple Harmonic Motion
A particle in SHM is never at rest for long, and its stored energy is never fixed in one form. As it swings, its energy keeps shifting back and forth between two kinds — kinetic and potential — while the two together stay constant. Both the kinetic and the potential energy vary between zero and a maximum value during each oscillation.
Let us start with kinetic energy. In Section 13.5 we found that the velocity of a particle in SHM is a periodic function of time, and that it falls to zero at the extreme positions. The kinetic energy \(K\), defined as \(\tfrac{1}{2}mv^2\), therefore also changes with time. Substituting the SHM velocity \(v = -\omega A\sin(\omega t + \phi)\) from Eq. (13.9),
\[K = \frac{1}{2}mv^2 = \frac{1}{2}m\,\omega^2 A^2 \sin^2(\omega t + \phi)\]
Recall from Eq. (13.14a) that \(k = m\omega^2\). Using this to replace \(m\omega^2\) by \(k\) gives the neat form
\[K = \frac{1}{2}k\,A^2 \sin^2(\omega t + \phi) \tag{13.15}\]
Here \(K\) is the kinetic energy (in joules, J), \(m\) the mass (kg), \(v\) the speed (m s\(^{-1}\)), \(A\) the amplitude (m), \(\omega\) the angular frequency (rad s\(^{-1}\)), and \(k\) the force constant (N m\(^{-1}\)).
The kinetic energy is thus a periodic function of time. It is zero when the displacement is maximum (at the extremes, where the particle momentarily stops) and largest when the particle is at the mean position (where it moves fastest). One subtle point: because \(K\) depends on \(v^2\), the sign of the velocity does not matter — so the kinetic-energy pattern repeats after only half a period, \(T/2\).
Now for the potential energy. In Chapter 6 you learnt that potential energy can be defined only for a conservative force — a force for which the work done depends only on the endpoints, not on the path. The spring force \(F = -kx\) is exactly such a conservative force. The work you do in stretching the spring by a distance \(x\) gets stored as potential energy, and that stored energy is
\[U = \frac{1}{2}k\,x^2 \tag{13.16}\]
Since the displacement in SHM is \(x = A\cos(\omega t + \phi)\), squaring it and inserting into Eq. (13.16) gives the potential energy as a function of time:
\[U(x) = \frac{1}{2}k\,x^2 = \frac{1}{2}k\,A^2 \cos^2(\omega t + \phi) \tag{13.17}\]
So the potential energy, too, is periodic with period \(T/2\). It is zero at the mean position (where the spring is relaxed) and maximum at the extreme displacements (where the spring is stretched or compressed the most) — the exact opposite of the kinetic energy.
Now add the two together. Using Eqs. (13.15) and (13.17), the total mechanical energy \(E\) of the system is
\[E = U + K = \frac{1}{2}k\,A^2\cos^2(\omega t + \phi) + \frac{1}{2}k\,A^2\sin^2(\omega t + \phi)\]
\[= \frac{1}{2}k\,A^2\left[\cos^2(\omega t + \phi) + \sin^2(\omega t + \phi)\right]\]
By the familiar trigonometric identity \(\sin^2\theta + \cos^2\theta = 1\), the bracket equals unity. Therefore
\[E = \frac{1}{2}k\,A^2 \tag{13.18}\]
This is a remarkable result. The total mechanical energy of a harmonic oscillator does not depend on time at all — it stays fixed throughout the motion, exactly as we expect for motion under any conservative force. Notice too that it depends on the square of the amplitude: a larger swing stores more energy.
The way the kinetic and potential energies change with time and with position is shown in Fig. 13.16, plotted against time in part (a) and against displacement in part (b).
Figure to come
Fig. 13.16 – (a) Kinetic, potential, and total energy as functions of time; (b) the same three energies as functions of displacement; K and U repeat with period T/2, while the total energy stays constant.
Observe from Fig. 13.16 that both the kinetic and the potential energy are always positive. Kinetic energy can never be negative, since it depends on the square of the speed. The potential energy is kept positive by the choice of the undetermined constant in its definition. Both energies peak twice during each period of the SHM. At the mean position (\(x = 0\)) the energy is entirely kinetic; at the extremes (\(x = \pm A\)) it is entirely potential. Everywhere in between, kinetic energy grows at the expense of potential energy, or the reverse — but their sum never changes.
13.8 The Simple Pendulum
Recall from the chapter opener that Galileo, watching a lamp swing from a cathedral ceiling and timing it against his own pulse, noticed that its swings kept a steady rhythm even as they grew smaller. The swinging system he was watching was a kind of pendulum, and its periodic motion is what we now study carefully.
You can build a pendulum of your own very easily and see this rhythm for yourself.
We shall now show that this periodic motion is, in fact, simple harmonic — provided the swings are small. Consider an idealised simple pendulum: a small, heavy bob of mass \(m\) tied to an inextensible, massless string of length \(L\), whose upper end is fixed to a rigid support. The bob swings back and forth in a vertical plane about the vertical line through the support, as shown in Fig. 13.17(a).
Figure to come
Fig. 13.17(a) – A pendulum bob oscillating to and fro about its mean (lowest) position, hanging from a rigid support by a string of length L.
To find the forces, we draw a kind of free-body diagram, shown in Fig. 13.17(b). Only two forces act on the bob: the tension \(T\) along the string, and the pull of gravity \(mg\) acting vertically downward. Let \(\theta\) be the angle the string makes with the vertical. When the bob hangs at its mean position, \(\theta = 0\).
Figure to come
Fig. 13.17(b) – Free-body diagram of the bob: tension T along the string, weight mg downward resolved into mg cosθ along the string and mg sinθ perpendicular to it, with angle θ marked.
The weight \(mg\) can be split into two useful components: a part \(mg\cos\theta\) acting along the string, and a part \(mg\sin\theta\) acting perpendicular to the string (along the direction of motion, tangent to the arc).
Because the bob moves along a circular arc of radius \(L\) centred at the support, it has two kinds of acceleration. There is a radial (centre-seeking) acceleration \(\omega^2 L\) directed towards the support, and there is also a tangential acceleration, which appears because the bob speeds up and slows down as it moves along the arc — its motion around the arc is not uniform. The radial acceleration is supplied by the net radial force \(T - mg\cos\theta\), while the tangential acceleration is supplied by the tangential force \(mg\sin\theta\).
It turns out to be much more convenient to work with torque about the support rather than force. The reason is that the radial forces — the tension \(T\) and the component \(mg\cos\theta\) — both act along the string, whose line passes right through the support, so they exert zero torque about it. (Torque is force times the perpendicular distance from the axis; a force whose line of action passes through the axis has no moment arm, and hence no torque.) The entire torque about the support therefore comes from the tangential component \(mg\sin\theta\), which acts at a distance \(L\) from the support:
\[\tau = -L\,(mg\sin\theta) \tag{13.19}\]
The negative sign appears because this torque always tries to reduce the angular displacement — it is a restoring torque, pulling the bob back towards the mean position.
By Newton’s law of rotational motion — the rotational analogue of \(F = ma\) — this torque equals the moment of inertia times the angular acceleration:
\[\tau = I\alpha \tag{13.20}\]
where \(I\) is the moment of inertia of the system about the support and \(\alpha\) is the angular acceleration. Combining Eqs. (13.19) and (13.20),
\[I\alpha = -m\,g\sin\theta\,L \tag{13.21}\]
so that
\[\alpha = -\frac{mgL}{I}\sin\theta \tag{13.22}\]
Equation (13.22) is not yet the equation of SHM, because the restoring quantity is proportional to \(\sin\theta\), not to \(\theta\) itself. We can simplify it by assuming the displacement \(\theta\) is small. The sine of an angle can be written as an infinite series,
\[\sin\theta = \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} \pm \cdots \tag{13.23}\]
where \(\theta\) is measured in radians. When \(\theta\) is small, the terms \(\theta^3/3!\), \(\theta^5/5!\), and beyond become extremely tiny compared with \(\theta\), so we may keep just the first term and write \(\sin\theta \approx \theta\). Equation (13.22) then becomes
\[\alpha = -\frac{mgL}{I}\,\theta \tag{13.24}\]
Just how good is this approximation? Table 13.1 lists the angle \(\theta\) in degrees, the same angle in radians, and the value of \(\sin\theta\). Notice that even for \(\theta\) as large as 20 degrees, \(\sin\theta\) is very nearly equal to \(\theta\) expressed in radians.
Table 13.1 — \(\sin\theta\) as a function of angle \(\theta\)
| \(\theta\) (degrees) | \(\theta\) (radians) | \(\sin\theta\) |
|---|---|---|
| 0 | 0 | 0 |
| 5 | 0.087 | 0.087 |
| 10 | 0.174 | 0.174 |
| 15 | 0.262 | 0.259 |
| 20 | 0.349 | 0.342 |
Equation (13.24) is mathematically identical in form to Eq. (13.11), \(a = -\omega^2 x\), except that the variable is now the angular displacement \(\theta\) instead of the linear displacement \(x\). This proves that, for small \(\theta\), the motion of the bob is simple harmonic. Comparing the two equations, the quantity playing the role of \(\omega^2\) is \(mgL/I\), so
\[\omega = \sqrt{\frac{mgL}{I}}\]
and the period is
\[T = 2\pi\sqrt{\frac{I}{mgL}} \tag{13.25}\]
Now we use the fact that the string is massless, so the whole moment of inertia comes from the point bob at distance \(L\) from the support. For a point mass, \(I = mL^2\). Substituting this into Eq. (13.25),
\[T = 2\pi\sqrt{\frac{mL^2}{mgL}} = 2\pi\sqrt{\frac{L}{g}} \tag{13.26}\]
This is the well-known formula for the time period of a simple pendulum.
Look carefully at what Eq. (13.26) contains — and what it does not. The period depends only on the length \(L\) and on \(g\). The mass \(m\) has cancelled out entirely, and the amplitude of the swing does not appear at all. So the period is decided purely by how long the pendulum is and by the strength of gravity, and this is exactly why Galileo found that the width of the swing hardly mattered.
This also settles what happens as a real pendulum slowly dies down. Friction and air resistance gradually drain its energy, so the amplitude of the swing shrinks little by little — that is what changes. But because the period does not depend on amplitude, each swing still takes the same time as the swing gets smaller — the rhythm stays the same. That steady rhythm, holding fast while the swing fades, is precisely what makes a pendulum such a good timekeeper.
13.9 Summary
The motion that repeats itself is called periodic motion.
The period \(T\) is the time required for one complete oscillation, or cycle. It is related to the frequency \(\nu\) by
\[T = \frac{1}{\nu}\]
The frequency \(\nu\) of periodic or oscillatory motion is the number of oscillations per unit time. In the SI, it is measured in hertz:
\[1 \text{ hertz} = 1 \text{ Hz} = 1 \text{ oscillation per second} = 1\ \text{s}^{-1}\]
- 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
\[\omega = \frac{2\pi}{T} = 2\pi\nu \quad \text{(angular frequency)}\]
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.
The particle velocity and acceleration during SHM as functions of time are given by
\[v(t) = -\omega A\sin(\omega t + \phi) \quad \text{(velocity)}\] \[a(t) = -\omega^2 A\cos(\omega t + \phi)\] \[= -\omega^2 x(t) \quad \text{(acceleration)}\]
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.
The force acting in a simple harmonic motion is proportional to the displacement and is always directed towards the centre of motion.
A particle executing simple harmonic motion has, at any time, kinetic energy \(K = \tfrac{1}{2}mv^2\) and potential energy \(U = \tfrac{1}{2}kx^2\). 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.
A particle of mass \(m\) oscillating under the influence of Hooke’s law restoring force given by \(F = -kx\) exhibits simple harmonic motion with
\[\omega = \sqrt{\frac{k}{m}} \quad \text{(angular frequency)}\] \[T = 2\pi\sqrt{\frac{m}{k}} \quad \text{(period)}\]
Such a system is also called a linear oscillator.
- The motion of a simple pendulum swinging through small angles is approximately simple harmonic. The period of oscillation is given by
\[T = 2\pi\sqrt{\frac{L}{g}}\]
Table of Physical Quantities
| Physical quantity | Symbol | Dimensions | Unit | Remarks |
|---|---|---|---|---|
| Period | \(T\) | \([\text{T}]\) | s | The least time for motion to repeat itself |
| Frequency | \(\nu\ (\text{or } f)\) | \([\text{T}^{-1}]\) | s\(^{-1}\) | \(\nu = \dfrac{1}{T}\) |
| Angular frequency | \(\omega\) | \([\text{T}^{-1}]\) | s\(^{-1}\) | \(\omega = 2\pi\nu\) |
| Phase constant | \(\phi\) | Dimensionless | rad | Initial value of phase of displacement in SHM |
| Force constant | \(k\) | \([\text{MT}^{-2}]\) | N m\(^{-1}\) | Simple harmonic motion \(F = -kx\) |
13.10 Points to Ponder
The period \(T\) is the least time after which motion repeats itself. Thus, motion repeats itself after \(nT\) where \(n\) is an integer.
Every periodic motion is not simple harmonic motion. Only that periodic motion governed by the force law \(F = -kx\) is simple harmonic.
Circular motion can arise due to an inverse-square law force (as in planetary motion) as well as due to a 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 \(A\).
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.
From point 4 above, given the amplitude or energy, the phase of motion is determined by the initial position or initial velocity.
A combination of two simple harmonic motions with arbitrary amplitudes and phases is not necessarily periodic. It is periodic only if the 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 an infinite number of harmonic motions with appropriate amplitudes.
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).
The motion of a simple pendulum is simple harmonic for small angular displacement.
For the motion of a particle to be simple harmonic, its displacement \(x\) must be expressible in either of the following forms:
\[x = A\cos\omega t + B\sin\omega t\] \[x = A\cos(\omega t + \alpha), \qquad x = B\sin(\omega t + \beta)\]
The three forms are completely equivalent (any one can be expressed in terms of any of the other two forms).
Thus, damped simple harmonic motion is not strictly simple harmonic. It is approximately so only for time intervals much less than \(2m/b\), where \(b\) is the damping constant.
13.11 NCERT Questions
- Which of the following examples represent periodic motion?
- A swimmer completing one (return) trip from one bank of a river to the other and back.
- A freely suspended bar magnet displaced from its N–S direction and released.
- A hydrogen molecule rotating about its centre of mass.
- An arrow released from a bow.
- Which of the following examples represent (nearly) simple harmonic motion and which represent periodic but not simple harmonic motion?
- the rotation of earth about its axis.
- motion of an oscillating mercury column in a U-tube.
- motion of a ball bearing inside a smooth curved bowl, when released from a point slightly above the lower most point.
- general vibrations of a polyatomic molecule about its equilibrium position.
- Fig. 13.18 depicts four \(x\)–\(t\) plots for linear motion of a particle. Which of the plots represent periodic motion? What is the period of motion (in case of periodic motion)?
Figure to come
Fig. 13.18 – Four x–t plots for a particle: (a) a smooth curve that keeps rising and never repeats; (b) a sawtooth-type curve repeating with marks at t = −3, −1, 0, 1, 3 s; (c) an irregular repeating pattern of bumps with marks at t = 1, 4, 7, 10, 13 s; (d) a smooth wavy (sinusoidal) curve with marks at t = −3, −2, −1, 0, 1, 2, 3 s.
Which of the following functions of time represent (a) simple harmonic, (b) periodic but not simple harmonic, and (c) non-periodic motion? Give the period for each case of periodic motion (\(\omega\) is any positive constant):
- \(\sin\omega t - \cos\omega t\)
- \(\sin^3\omega t\)
- \(3\cos\left(\dfrac{\pi}{4} - 2\omega t\right)\)
- \(\cos\omega t + \cos 3\omega t + \cos 5\omega t\)
- \(\exp(-\omega^2 t^2)\)
- \(1 + \omega t + \omega^2 t^2\)
A particle is in linear simple harmonic motion between two points, A and B, 10 cm apart. Take the direction from A to B as the positive direction and give the signs of velocity, acceleration and force on the particle when it is
- at the end A,
- at the end B,
- at the mid-point of AB going towards A,
- at 2 cm away from B going towards A,
- at 3 cm away from A going towards B, and
- at 4 cm away from B going towards A.
Which of the following relationships between the acceleration \(a\) and the displacement \(x\) of a particle involve simple harmonic motion?
- \(a = 0.7x\)
- \(a = -200x^2\)
- \(a = -10x\)
- \(a = 100x^3\)
The motion of a particle executing simple harmonic motion is described by the displacement function, \[x(t) = A\cos(\omega t + \phi).\] If the initial (\(t = 0\)) position of the particle is 1 cm and its initial velocity is \(\omega\) cm/s, what are its amplitude and initial phase angle? The angular frequency of the particle is \(\pi\) s\(^{-1}\). If instead of the cosine function, we choose the sine function to describe the SHM: \(x = B\sin(\omega t + \alpha)\), what are the amplitude and initial phase of the particle with the above initial conditions.
A spring balance has a scale that reads from 0 to 50 kg. The length of the scale is 20 cm. A body suspended from this balance, when displaced and released, oscillates with a period of 0.6 s. What is the weight of the body?
A spring having with a spring constant 1200 N m\(^{-1}\) is mounted on a horizontal table as shown in Fig. 13.19. A mass of 3 kg is attached to the free end of the spring. The mass is then pulled sideways to a distance of 2.0 cm and released.
Figure to come
Fig. 13.19 – A spring lying on a horizontal table, fixed to a rigid wall at its left end, with a block of mass 3 kg attached to its free right end.
Determine (i) the frequency of oscillations, (ii) maximum acceleration of the mass, and (iii) the maximum speed of the mass.
- In Exercise 13.9, let us take the position of mass when the spring is unstreched as \(x = 0\), and the direction from left to right as the positive direction of \(x\)-axis. Give \(x\) as a function of time \(t\) for the oscillating mass if at the moment we start the stopwatch (\(t = 0\)), the mass is
- at the mean position,
- at the maximum stretched position, and
- at the maximum compressed position. In what way do these functions for SHM differ from each other, in frequency, in amplitude or the initial phase?
- Figures 13.20 correspond to two circular motions. The radius of the circle, the period of revolution, the initial position, and the sense of revolution (i.e. clockwise or anti-clockwise) are indicated on each figure.
Figure to come
Fig. 13.20 – Two reference circles: (a) radius 3 cm, period T = 2 s, particle P starting at the lowest point (on the −y axis); (b) radius 2 m, period T = 4 s, particle P starting on the +x axis, with the sense of revolution indicated on each.
Obtain the corresponding simple harmonic motions of the \(x\)-projection of the radius vector of the revolving particle P, in each case.
- Plot the corresponding reference circle for each of the following simple harmonic motions. Indicate the initial (\(t = 0\)) position of the particle, the radius of the circle, and the angular speed of the rotating particle. For simplicity, the sense of rotation may be fixed to be anticlockwise in every case: (\(x\) is in cm and \(t\) is in s).
- \(x = -2\sin\left(3t + \dfrac{\pi}{3}\right)\)
- \(x = \cos\left(\dfrac{\pi}{6} - t\right)\)
- \(x = 3\sin\left(2\pi t + \dfrac{\pi}{4}\right)\)
- \(x = 2\cos\pi t\)
- Figure 13.21(a) shows a spring of force constant \(k\) clamped rigidly at one end and a mass \(m\) attached to its free end. A force \(F\) applied at the free end stretches the spring. Figure 13.21(b) shows the same spring with both ends free and attached to a mass \(m\) at either end. Each end of the spring in Fig. 13.21(b) is stretched by the same force \(F\).
Figure to come
Fig. 13.21 – (a) A spring fixed to a wall at one end with a mass m at its free end, pulled by a force F; (b) the same spring with a mass m at each free end, each end pulled outward by an equal force F.
a. What is the maximum extension of the spring in the two cases?
b. If the mass in Fig. (a) and the two masses in Fig. (b) are released, what is the period of oscillation in each case?The piston in the cylinder head of a locomotive has a stroke (twice the amplitude) of 1.0 m. If the piston moves with simple harmonic motion with an angular frequency of 200 rad/min, what is its maximum speed?
The acceleration due to gravity on the surface of moon is 1.7 m s\(^{-2}\). What is the time period of a simple pendulum on the surface of moon if its time period on the surface of earth is 3.5 s? (\(g\) on the surface of earth is 9.8 m s\(^{-2}\))
A simple pendulum of length \(l\) and having a bob of mass \(M\) is suspended in a car. The car is moving on a circular track of radius \(R\) with a uniform speed \(v\). If the pendulum makes small oscillations in a radial direction about its equilibrium position, what will be its time period?
A cylindrical piece of cork of density of base area \(A\) and height \(h\) floats in a liquid of density \(\rho_l\). The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with a period \[T = 2\pi\sqrt{\frac{h\rho}{\rho_l g}}\] where \(\rho\) is the density of cork. (Ignore damping due to viscosity of the liquid).
One end of a U-tube containing mercury is connected to a suction pump and the other end to atmosphere. A small pressure difference is maintained between the two columns. Show that, when the suction pump is removed, the column of mercury in the U-tube executes simple harmonic motion.
13.12 Check Your Concepts
Every oscillatory motion is periodic, but every periodic motion is not oscillatory. Explain this statement, and give one clear example of a motion that is periodic but not oscillatory.
Is there any fundamental physical difference between an “oscillation” and a “vibration”? On what basis do we usually choose one word over the other? Support your answer with one example of each.
The frequency of a periodic motion need not be a whole number. Explain what a frequency of, say, 1.25 Hz means physically, and describe a situation where such a value could arise.
State Fourier’s theorem. Explain why sine and cosine functions are regarded as the fundamental building blocks of every periodic motion.
A student claims that the amplitude of a simple harmonic motion can be a negative number. Do you agree? Justify your answer using the relation between amplitude and phase constant.
Two simple harmonic motions have the same amplitude and the same angular frequency but different phase constants. Describe how their displacement–time graphs are related. Does changing the phase constant change the shape or size of the graph?
In simple harmonic motion, the speed is maximum at the mean position while the acceleration is maximum at the extreme positions. Explain, using physical reasoning, why the two maxima occur at opposite points of the swing.
The acceleration of a particle in SHM is given by \(a = -\omega^2 x\). Explain the physical significance of the negative sign, and state why this makes the force a “restoring” force.
Explain how the projection of uniform circular motion onto a diameter gives simple harmonic motion. Then explain why the force acting in linear SHM is nevertheless different in nature from the centripetal force of the circular motion.
In simple harmonic motion, the displacement repeats after a time \(T\), but the kinetic energy and the potential energy each repeat after \(T/2\). Give the reason for this difference.
Explain why the time period of a simple pendulum (swinging through small angles) does not depend on either the mass of the bob or the amplitude of the swing.
Assertion–Reason: “A pendulum clock that keeps correct time at sea level will run slow if carried to the top of a high mountain.” State whether this is correct and give the physical reason, referring to the pendulum period formula.
13.13 Practice with Numericals
A particle completes 240 oscillations in one minute. Calculate its frequency and its time period.
Express the periodic function \(x = \sin\omega t + \sqrt{3}\,\cos\omega t\) as a single sinusoidal function of the form \(A\cos(\omega t + \phi)\) or \(D\sin(\omega t + \phi)\). State its amplitude and its period.
The displacement of a particle in SHM is \(x = 6\cos\!\left(3\pi t + \dfrac{\pi}{6}\right)\) cm, with \(t\) in seconds. Find (a) the amplitude, (b) the angular frequency, (c) the period, (d) the frequency, and (e) the initial phase.
A particle executes SHM with an amplitude of 5 cm and a period of 0.4 s. Calculate its maximum speed and its maximum acceleration.
A particle in SHM has an amplitude of 10 cm. At what distance from the mean position is its speed equal to half of its maximum speed?
A body of mass 0.1 kg is attached to a spring of force constant 40 N m\(^{-1}\) and set into SHM. Find its angular frequency, time period, and frequency.
A block of mass 2 kg is attached to a spring of force constant 128 N m\(^{-1}\), pulled 4 cm from the mean position on a frictionless surface, and released. Find (a) the total mechanical energy, and (b) the kinetic and potential energies when the block is 2 cm from the mean position.
A particle executes SHM with amplitude \(A\). At what displacement from the mean position is the kinetic energy equal to twice the potential energy?
A seconds pendulum (time period 2 s on Earth) is taken to a planet where the acceleration due to gravity is one-fourth of its value on Earth. Find the new time period of the pendulum.
A simple pendulum has a length of 0.8 m. Calculate its time period of oscillation. (Take \(g = 9.8\) m s\(^{-2}\).)
A reference particle moves anticlockwise on a circle of radius 4 cm with a period of 3 s, starting on the positive \(x\)-axis at \(t = 0\). Write the equation \(x(t)\) for the simple harmonic motion of its projection on the \(x\)-axis, stating the amplitude and angular frequency.
A spring stretches by 5 cm when a mass of 2 kg is hung from it. If the mass is then set into vertical oscillation, find the time period of the motion. (Take \(g = 9.8\) m s\(^{-2}\).)