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12.1Simple harmonic oscillations
Oscillations refer to back and forth motion that repeats. A typical example of an oscillation is provided by the simple pendulum, i.e. a mass attached to a vertical string. When the mass is displaced slightly sideways and then released, the mass begins to oscillate. In an oscillation the motion is periodic, i.e. repetitive, and the body moves back and forth around an equilibrium position.
Two important characteristic of oscillatory motion, shown in Figures 12.1a and 12.1b, are:
âthe period, T, which is the time taken to complete one full oscillation
âthe amplitude of the oscillation, which is the maximum displacement away from the equilibrium position.

Figure 12.1: a A full oscillation lasts for one period. At the end of a time interval equal to one period T, the system is in the same state as the state it started from. b The red arrows represent velocities.

The frequency f of the oscillations is the number of oscillations per second. The unit of frequency is the inverse second, sâ1, and is called the hertz (Hz).
Since we have one oscillation in a time equal to the period T, the number of oscillations in 1 s is 1T and so:
f=1T.
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CHECK YOURSELF 1
A particle performs oscillations completing five full oscillations in 4 s. What is the period and frequency of the oscillations?
It can be shown that the period of a pendulum only depends on the length of the string and not the mass of the bob. It is given by:
T=2ÏLgâââ
CHECK YOURSELF 2
The length of a pendulum is increased by a factor of 4. By what factor does the period change?
In real life examples of oscillations can be found everywhere including the motion of a diving board as a diver prepares to dive, a tight guitar string that is set in motion by plucking the string, or a skyscraper under the action of the wind or an earthquake.
For a system to oscillate it is necessary to have a restoring force, i.e. a force that tends to bring the system back towards its equilibrium position when the system is displaced away from equilibrium.
A very special kind of oscillation is called simple harmonic motion (SHM). In SHM, the acceleration is proportional to and opposite to the displacement. Mathematically, these two conditions can be stated as a â âx or a = âÏ2x
(Since F = ma, this is equivalent to saying that the restoring force is also proportional to and opposite to the displacement.)
The constant Ï is known as the angular frequency and is related to the period T by the relation Ï=2ÏT. Since f=1T we also have that Ï = 2Ïf. Angular frequency is measured in radians per second.
A simple system that undergoes simple harmonic oscillations is that of a block attached to a horizontal spring on a frictionless table. The top diagram in Figure 12.2 shows the equilibrium position of the system: the spring has its natural length.

Figure 12.2: The massâspring system. The net force on the body is proportional to the displacement and opposite in direction.

In the lower diagram the block has been displaced by an amount x away from the equilibrium position. A tension force of magnitude kx develops in the spring directed opposite to the displacement; this is the net force on the block. By Newtonâs second law we then have
ma=âkx
The minus sign is there because the force is opposite to the displacement x.
This means that the acceleration is a=âkmx. Comparing with a = âÏ2x we see that for this case Ï2=km and so the period is
T=2ÏÏ=2Ïkmââââ=2Ïmkââââ
CHECK YOURSELF 3
The mass at the end of a spring oscillates with period T. What will be the period if the mass is halved and the spring constant is doubled?
Displacementâtime graphs
Figure 12.3 shows two of the many possible graphs to describe the displacement of an object performing simple harmonic oscillations. In mathematics, cosine and sine functions are called harmonic functions, which is why these oscillations are called harmonic.
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Figure 12.3: The variation of displacement with time in SHM. Two possibilities include a a cosine or b a sine depending on when the clock is started, i.e. what the initial position and velocity are.

What can we tell from these graphs? The maximum extension past equilibrium is 4.0 cm so this is the amplitude of the motion. One full oscillation takes 2.0 s so this is the period of the motion. The frequency is f=1T=0.50Â Hz. Notice that the amplitude stays constant as time goes on: this is a characteristic of SHM.
In Figure 12.3 the amplitude was 4.0 cm. What would happen if the amplitude was 2.0 cm? The answer is shown by the blue curve in Figure 12.4 where we have also shown the curve in 12.3a for comparison.

Figure 12.4: The blue curve has a smaller amplitude but the same period.

We see that even though the amplitude changed, the period stayed the same. This is another characteristic of SHM.
In SHM
âthe period does not depend on amplitude
âthe amplitude stays constant as time goes on.
Let us have another look at Figure 12.3a. What can we say about the velocity in this oscillation? We know that velocity is the gradient of a displacementâtime graph. So we can say something about velocity by looking at the gradient of the graph Figure 12.3a at each point. At t = 0 the gradient is zero, so the velocity is also zero. As t increases the gradient becomes negative and at t = 1.0 s it becomes zero again. The gradient has its largest magnitude at t = 0.5 s. From t = 1.0 s to t = 2.0 s the gradient is positive. Its magnitude is largest at t = 1.5 s. Figure 12.5a shows the velocity plotted against time. Figure 12.5b does the same for the displacement in Figure 12.3b.

Figure 12.5: a The variation of velocity v with time t in SHM for the displacement graph of Figure 12.3a. b The variation of velocity v with time t in SHM for the displacement graph of Figure 12.3b.

In the same way, we can get information about acceleration by looking at the gradient of the velocityâtime graph. But it is much simpler to recall that a â âx. This means that the accelerationâtime graph has the opposite shape to the displacementâtime graphâthe peaks on the displacementâtime graph become troughs on the accelerationâtime graph (of course the scale on the vertical axis will be different). This is shown in Figure 12.6.
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Figure 12.6: a The variation of acceleration a with time t in SHM for the displacement graph of Figure 12.3a. b The variation of acceleration a with time t in SHM for the displacement graph of Figure 12.3b.

Since F = ma, a graph of force versus time would have the same shape as the graph in Figure 12.6.
Figure 12.7 shows all three (displacement, velocity and acceleration) as functions of time.

Figure 12.7: The variation with time of displacement, velocity and acceleration in SHM on the same axes.

Accelerationâdisplacement graph
Since the acceleration is proportional to and opposite to the displacement a graph of acceleration versus displacement gives a straight line through the origin with a negative slope, as shown in Figure 12.8. The graph corresponds to the motion in Figure 12.3.

Figure 12.8: The graph of acceleration a versus displacement x is a straight line through the origin with a negative slope.

(Since F = ma, a graph of force versus displacement would have the same shape as the graph in Figure 12.8.)
Since a = âÏ2x, the gradient (slope) of the acceleration versus displacement graph is âÏ2. The gradient of the graph in Figure 12.8 is âÏ2=â40â404.0â(â4.0)=â10Â sâ2, Ï=10âââÂ rad sâ1 and so the period is T=2Ï10ââââ2.0Â s in agreement with Figure 12.3.
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WORKED EXAMPLE 12.1
State and explain whether graphs I, II and III in Figure 12.9 represent simple harmonic oscillations.

Figure 12.9: For worked example 12.1.

Answer
Graph I does not show SHM, since the period does not stay constant as time goes on. Graph II does, since the acceleration is proportional and opposite to the displacement (straight-line graph through origin with negative slope). Graph III does not, since the amplitude does not stay constant.
CHECK YOURSELF 4
Figure 12.10 shows the standard representation of SHM. The red arrows represent velocity.
On a copy of the diagram, put blue arrows at each step to represent acceleration.

Figure 12.10: For check yourself 4.

WORKED EXAMPLE 12.2
The graph in Figure 12.11 shows the variation with time of the displacement of a particle from a fixed equilibrium position.

Figure 12.11: Graph showing the variation with time of the displacement of a particle performing SHM.

aUse the graph to determine the period of the motion.
bOn a copy of the graph, mark:
ia point where the velocity is zero (label this with the letter Z)
iia point where the velocity is positive and has the largest magnitude (label this with the letter V)
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iiia point where the acceleration is positive and has the largest magnitude (label this with the letter A).
Answer
aThe period is read off the graph as T = 0.20 s.
biThe velocity is zero at any point where the displacement is at a maximum or a minimum.
iiFor example at t = 0 s, 0.20 s or t = 0.40 s.
iiiFor example at t = 0.15 s, 0.35 s.
Phase difference
Figure 12.12 shows two harmonic oscillations. They have the same amplitude and period so are in fact identical. So why do the graphs look different? The reason is that the observer who is observing the oscillations started the stopwatch at different times; they started it when the displacement was zero for the blue curve and when the displacement was â1.0 cm for the red curve. We say there is a phase difference between the two oscillations. Phase difference, ÎÏ, is an angle and is given by: ÎÏ=ÎtTÃ2Ï (in radians)
where the shift Ît is the separation in time between two neighbouring peaks and T is the period.

Figure 12.12: Two graphs of simple harmonic oscillations with a phase difference between them.

In Figure 12.12, two neighbouring peaks are separated by 0.25 s and the period is 1.0 s. The phase difference is then ÎÏ=ÎtTÃ2Ï=0.251.0Ã2Ï=Ï2.
CHECK YOURSELF 5
What is the phase difference between the two oscillations shown in Figure 12.13?

Figure 12.13: For check yourself question 5.

SCIENCE IN CONTEXT
The measurement of time depends on regular oscillations. In early time-keeping devices the oscillations were mechanical, for example the swinging of a pendulum. Now they are electrical oscillations in electronic circuits. The need for internationally accepted measures of time is essential for communications, travel, electricity supply and practically all other aspects of modern life. Time is what these devices measure, but nobody knows what time really is!
NATURE OF SCIENCE
In real life we observe many oscillations for which the period is not independent of amplitude, for example the waving of a branch in the wind or the bouncing of a ball released above the ground. These do not obey the simple SHM equations in this section. But the general principles of physics we have met in this section govern many oscillations in the world about us, from water waves in the deep ocean to the vibration of a carâs suspension system. The idea of the simple harmonic oscillator and the mathematics of SHM give physicists powerful tools to describe all periodic oscillations.
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Links
In ChaptersÂ 13â16 we will see that simple harmonic oscillations are the key for understanding waves. When a mechanical wave travels through a medium the particles of the medium will oscillate about their equilibrium positions with simple harmonic motion. The frequency of the oscillations is the frequency of the wave.
TEST YOUR UNDERSTANDING
1State what is meant by:
aan oscillation
ba simple harmonic oscillation.
2A ball goes back and forth along a horizontal floor, bouncing off two vertical walls. No resistance forces are present. Suggest whether this motion is simple harmonic.
3A ball keeps bouncing vertically off the floor. No resistance forces are present. Suggest whether this motion is simple harmonic.
4Figure 12.14 shows the variation with displacement of the acceleration of a particle that is performing oscillations.

Figure 12.14: For question 4.

Explain:
ahow it is known that the particle is performing oscillations
bwhy the oscillations are not simple harmonic
cwhy the oscillations are simple harmonic for small amplitudes.
5What is the length of a pendulum of period 1.0 s?
6The length of the metallic thread of a pendulum increases by 4.0% due to a temperature increase. By what percentage does the period of the pendulum change? (Use ideas of propagation of uncertainties.)
7A mass is performing simple harmonic oscillations at the end of a spring of spring constant 240 Nmâ1. The period of oscillation is 1.2 s. What is the mass?
8In a spacecraft in orbit, everything feels weightless. How could an astronaut measure her mass?
9A student investigates the pendulum equation T=2ÏLgâââ.
aHe plots T on the vertical axis. What variable should he plot on the horizontal axis in order to get a straight line?
bWhat would the gradient be?
cAnother student plots T2 on the vertical axis. What variable should she plot on the horizontal axis in order to get a straight line?
dWhat would the gradient be?
10Look at Figure 12.15.
aWhat is the phase difference between the two oscillations?
bWhat is the angular frequency of each oscillation?
cWhat is the acceleration of a point on the blue curve at t = 1 s?

Figure 12.15: For question 10.

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11The graph (Figure 12.16) shows the variation with time of the position of a particle executing SHM.

Figure 12.16: For question 11.

aState a time at which
ithe velocity is zero
iithe velocity is positive and a maximum
iiithe acceleration is zero
ivthe acceleration is positive and a maximum.
bCalculate the frequency and angular frequency of oscillations.
12Figure 12.17 shows part of the graph of the variation with time of the displacement of a particle executing SHM.

Figure 12.17: For question 12.

On the graph mark intervals of time for which the velocity and the acceleration are:
ain the same direction (mark this interval with the letter S)
bin opposite directions (mark this interval with the letter O).
13The graph (Figure 12.18) shows the variation with time of the velocity of a particle in SHM.

Figure 12.18: For question 13.

aEstimate to one significant figure the amplitude of the displacement, explaining your reasoning.
bMark with the letter M a point where the acceleration is positive and a maximum.
cMark with the letter E a point where the particle is passing through the equilibrium position.
dAt t = 1.0 s, is the displacement positive, negative or zero?
14The graph (Figure 12.19) shows the variation with displacement of the acceleration of a particle.

Figure 12.19: For question 14.

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aExplain why the motion is SHM.
bMark with the letter V a point on the graph where the speed is a maximum.
cDetermine the period of the motion.
dThe mass of the oscillating body is 0.15 kg. Calculate the restoring force when the displacement is â 0.20 m.
eHow would the graph change if the amplitude of oscillations is halved?
