Resonance is an important physical phenomenon that can appear in a great many different situations. A dramatic example is the Millennium Bridge in London, opened in June 2000 (Figure 18.29). With up to 2000 pedestrians walking on the bridge, it started to sway dangerously. The people also swayed in time with the bridge, and this caused the amplitude of the bridge’s oscillations to increase–this is resonance.
After three days, the bridge was closed. It took engineers two years to analyse the problem and then add ‘dampers’ to the bridge to absorb the energy of its oscillations. The bridge was then reopened and there have been no problems since.
You will have observed a much more familiar example of resonance when pushing a small child on a swing. The swing plus child has a natural frequency of oscillation. A small push in each cycle results in the amplitude increasing until the child is swinging high in the air.

PRACTICAL ACTIVITY 18.3

 

Observing resonance

Resonance can be observed with almost any oscillating system. The system is forced to oscillate at a particular frequency. If the forcing frequency happens to match the natural frequency of oscillation of the system, the amplitude of the resulting oscillations can build up to become very large.

Barton’s pendulums

Barton’s pendulums is a demonstration of this (Figure 18.30). Several pendulums of different lengths hang from a horizontal string. Each has its own natural frequency of oscillation. The ‘driver’ pendulum at the end is different; it has a large mass at the end, and its length is equal to that of one of the others.
When the driver is set swinging, the others gradually start to move. However, only the pendulum whose length matches that of the driver pendulum builds up a large amplitude so that it is resonating.

What is going on here? All the pendulums are coupled together by the suspension. As the driver swings, it moves the suspension, which in turn moves the other pendulums. The frequency of the matching pendulum is the same as that of the driver, and so it gains energy and its amplitude gradually builds up.
The other pendulums have different natural frequencies, so the driver has little effect.
In a similar way, if you were to push the child on the swing once every three-quarters of an oscillation, you would soon find that the swing was moving backwards as you tried to push it forwards, so that your push would slow it down.

A mass–spring system

You can observe resonance for yourself with a simple mass–spring system. You need a mass on the end of a spring (Figure 18.31), chosen so that the mass oscillates up and down with a natural frequency of about $1 Hz$. Now hold the top end of the spring and move your hand up and down rapidly, with an amplitude of a centimetre or two. Very little happens. Now move your hand up and down more slowly, close to $1 Hz$. 
You should see the mass oscillating with gradually increasing amplitude. Adjust your movements to the exact frequency of the natural vibrations of the mass and you will see the greatest effect.

Defining resonance

For resonance to occur, we must have a system that is capable of oscillating freely. We must also have some way in which the system is forced to oscillate. When the forcing frequency matches the natural frequency of the system, the amplitude of the oscillations grows dramatically.
If the driving frequency does not quite match the natural frequency, the amplitude of the oscillations will increase, but not to the same extent as when resonance is achieved. Figure 18.32 shows how the amplitude of oscillations depends on the driving frequency in the region close to resonance.
In resonance, energy is transferred from the driver to the resonating system more efficiently than when resonance does not occur. For example, in the case of the Millennium Bridge, energy was transferred from the pedestrians to the bridge, causing large-amplitude oscillations.

The following statements apply to any system in resonance:
- Its natural frequency is equal to the frequency of the driver.
- Its amplitude is maximum.
- It absorbs the greatest possible energy from the driver.

Resonance and damping

During earthquakes, buildings are forced to oscillate by the vibrations of the Earth. Resonance can occur, resulting in serious damage (Figure 18.33). In regions of the world where earthquakes happen regularly, buildings may be built on foundations that absorb the energy of the shock waves. In this way, the vibrations are ‘damped’ so that the amplitude of the oscillations cannot reach dangerous levels. This is an expensive business, and so far is restricted to the wealthier parts of the world.

Damping is useful if we want to reduce the damaging effects of resonance. Figure 18.34 shows how damping alters the resonance response curve of Figure 18.32. Notice that, as the degree of damping is increased, the amplitude of the resonant vibrations decreases. The resonance peak becomes broader.
There is also an effect on the frequency at which resonance occurs, which becomes lower as the damping increases.

An everyday example of damping can be seen on some doors. For example, a restaurant may have a door leading to the kitchen; this door can swing open in either direction. Such a door is designed to close by itself after someone has passed through it. Ideally, the door should swing back quickly without overshooting its closed position. To achieve this, the door hinges (or the closing mechanism) must be correctly damped. If the hinges are damped too lightly, the door will swing back and forth several times as it closes. If the damping is too heavy, it will take too long to close. With critical damping, the door will swing closed quickly without oscillating.
Critical damping is the minimum amount of damping required to return an oscillator to its equilibrium position without oscillating. Under-damping results in unwanted oscillations; over-damping results in a slower return to equilibrium (see Figure 18.35). A car’s suspension system uses springs to smooth out bumps in the road. It is usually critically damped so that passengers do not experience nasty vibrations every time the car goes over a bump.

Using resonance

As we have seen, resonance can be a problem in mechanical systems. However, it can also be useful. For example, many musical instruments rely on resonance.
Resonance is not confined to mechanical systems. It is made use of in, for example, microwave cooking.
The microwaves used have a frequency that matches the natural frequency of vibration of water molecules (the microwave is the ‘driver’ and the molecule is the ‘resonating system’). The water molecules in the food are forced to vibrate and they absorb the energy of the microwave radiation. The water gets hotter and the absorbed energy spreads through the food and cooks or heats it.
Magnetic resonance imaging (MRI) is used in medicine to produce images such as Figure 18.36, showing aspects of a patient’s internal organs. Radio waves having a range of frequencies are used, and particular frequencies are absorbed by particular atomic nuclei. The frequency absorbed depends on the type of nucleus and on its surroundings. By analysing the absorption of the radio waves, a computer-generated image can be produced.
A radio or television also depends on resonance for its tuning circuitry. The aerial picks up signals of many different frequencies from many transmitters. The tuner can be adjusted to resonate at the frequency of the transmitting station you are interested in, and the circuit produces a large-amplitude signal for this frequency only.

Big ideas in physics

This study of simple harmonic motion illustrates some important aspects of physics:
- Physicists often take a complex problem (such as how the atoms in a solid vibrate) and reduce it to a simpler, more manageable problem (such as how a mass–spring system vibrates). This is simpler because we know that the spring obeys Hooke’s law, so that force is proportional to displacement.
- Physicists generally feel happier if they can write mathematical equations that will give numerical answers to problems. The equation $a =  - {\omega ^2}x$, which describes s.h.m., can be solved to give the sine and cosine equations we have considered earlier.
Once physicists have solved one problem like this, they look around for other situations where they can use the same ideas all over again. So the mass–spring theory also works well for vibrating atoms and molecules, for objects bobbing up and down in water, and in many other situations.
- Physicists also seek to modify the theory to fit a greater range of situations. For example, what happens if the vibrating mass experiences a frictional force as it oscillates? (This is damping, as discussed earlier.) What happens if the spring doesn’t obey Hooke’s law? (This is a harder question to answer.)
Your A Level physics course will help you to build up your appreciation of some of these big ideas–fields (magnetic, electric, gravitational), energy and so on.

SUMMARY

Many systems, mechanical and otherwise, will oscillate freely when disturbed from their equilibrium position.
Some oscillators have motion described as simple harmonic motion (s.h.m.). For these systems, graphs of displacement, velocity and acceleration against time are sinusoidal curves–see Figure 18.37.

During a single cycle of s.h.m., the phase changes by $2\pi $ radians. The angular frequency $\omega $ of the motion is related to its period T and frequency f:
In s.h.m., displacement x and velocity ν and acceleration can be represented as functions of time t by equations of the form:

$x - {x_0}\,\sin \,\omega t$ and $v - {v_0}\,\cos \,\omega t$ and $a =  - a\,\sin \,\omega t$

A body executes simple harmonic motion if its acceleration is directly proportional to its displacement from its equilibrium position, and is always directed towards the equilibrium position.
Acceleration a in s.h.m. is related to displacement x by the equation:

$a =  - {\omega ^2}x$

The maximum speed ν0 in s.h.m. is given by the equation:

${v_0} = \omega {x_0}$

The frequency or period of a simple harmonic oscillator is independent of its amplitude.
In s.h.m., there is a regular interchange between kinetic energy and potential energy.
Resistive forces remove energy from an oscillating system. This is known as damping. Damping causes the amplitude to decay with time.
Critical damping is the minimum amount of damping required to return an oscillator to its equilibrium position without oscillating.
When an oscillating system is forced to vibrate close to its natural frequency, the amplitude of vibration increases rapidly. The amplitude is maximum when the forcing frequency matches the natural frequency of the system; this is resonance.
Resonance can be a problem, but it can also be very useful.

EXAM-STYLE QUESTIONS

 

1) A mass, hung from a spring, oscillates with simple harmonic motion.
Which statement is correct? [1]
A: The force on the mass is directly proportional to the angular frequency of the oscillation.
B: The force on the mass is greatest when the displacement of the bob is greatest.
C: The force on the mass is greatest when the speed of the bob is greatest.
D: The force on the mass is inversely proportional to the time period of the oscillation.

2) The bob of a simple pendulum has a mass of $0.40 kg$. The pendulum oscillates with a period of $2.0 s$ and an amplitude of $0.15 m$.
At one point in its cycle it has a potential energy of $0.020 J$.
What is the kinetic energy of the pendulum bob at this point? [1]
A: $0.024 J$
B: $0.044 J$
C: $0.14 J$
D: $0.18 J$

3) State and justify whether the following oscillators show simple harmonic motion:
a: a basketball being bounced repeatedly on the ground. [2]
b: a guitar string vibrating [2]
c: a conducting sphere vibrating between two parallel, oppositely charged metal plates [1]
d: the pendulum of a grandfather clock. [2]
[Total: 7]

4) The pendulum of a clock is displaced by a distance of $4.0 cm$ and it oscillates in s.h.m. with a period of $1.0 s$.
a: Write down an equation to describe the displacement x of the pendulum bob with time t. [2]
b: Calculate:
i- the maximum velocity of the pendulum bob [2]
ii- its velocity when its displacement is $2.0 cm$. [1]
[Total: 5]

5) A $50 g$ mass is attached to a securely clamped spring. The mass is pulled downwards by $16 mm$ and released, which causes it to oscillate with s.h.m. of time period of $0.84 s$.
a: Calculate the frequency of the oscillation. [1]
b: Calculate the maximum velocity of the mass. [1]
c: Calculate the maximum kinetic energy of the mass and state at which point in the oscillation it will have this velocity. [2]
d: Write down the maximum gravitational potential energy of the mass (relative to its equilibrium position). You may assume that the damping is negligible. [1]
[Total: 5]

6) In each of the three graphs, a, b and c in Figure 18.38, give the phase difference between the two curves:
i- as a fraction of an oscillation [1]
ii- in degrees [1]
iii- in radians. [1]
[Total: 3]

Figure 18.38

7) a: Determine the frequency and the period of the oscillation described by this graph. [2]

b: Use a copy of the graph and on the same axes sketch:
i- the velocity of the particle [1]
ii- the acceleration of the particle. [2]
[Total: 5]

8) These graphs show the displacement of a body as it vibrates between two points.

a: State and explain whether the body is moving with simple harmonic motion. [1]
b: Make a copy of the three graphs.
i- On the second set of axes on your copy show the velocity of the body as it vibrates. [1]
ii- On the third set of axes on your copy, show the acceleration of the
body. [2]
[Total: 4]

9) This diagram shows the piston of a small car engine that oscillates in the cylinder with a motion that approximates simple harmonic motion at 4200 revs per minute (1 rev = 1 cycle). The mass of the piston is $0.24 kg$.

a: Explain what is meant by simple harmonic motion. [2]
b: Calculate the frequency of the oscillation. [1]
c: The amplitude of the oscillation is $12.5 cm$. Calculate:
i- the maximum speed at which the piston moves [2]
ii- the maximum acceleration of the piston [2]
iii- the force required on the piston to produce the maximum acceleration. [1]
[Total: 8]

10) This diagram shows a turntable with a rod attached to it a distance $15 cm$ from the centre. The turntable is illuminated from the side so that a shadow is cast on a screen.

A simple pendulum is placed behind the turntable and is set oscillating so that it has an amplitude equal to the distance of the rod from the centre of the turntable.
The speed of rotation of the turntable is adjusted. When it is rotating at 1.5 revolutions per second the shadow of the pendulum and the rod are found to move back and forth across the screen exactly in phase.
a: Explain what is meant by the term in phase. [1]
b: Write down an equation to describe the displacement x of the pendulum from its equilibrium position and the angular frequency of the oscillation of the pendulum. [1]
c: The turntable rotates through ${60^ \circ }$ from the position of maximum displacement shown in the diagram.
i- Calculate the displacement (from its equilibrium position) of the pendulum at this point. [3]
ii- Calculate its speed at this point. [2]
iii- Through what further angle must the turntable rotate before it has this speed again? [1]
[Total: 8]

11) When a cricket ball hits a cricket bat at high speed it can cause a standing wave to form on the bat. In one such example, the handle of the bat moved with a frequency of $60 Hz$ with an amplitude of $2.8 mm$.
The vibrational movement of the bat handle can be modelled on simple harmonic motion.
a: State the conditions for simple harmonic motion. [2]
b: Calculate the maximum acceleration of the bat handle. [2]
Given that the part of the bat handle held by the cricketer has a mass of $0.48 kg$, calculate the maximum force produced on his hands. [1]
d: The oscillations are damped and die away after about five complete cycles.
Sketch a displacement–time graph to show the oscillations. [2]
[Total: 7]

12) Seismometers are used to detect and measure the shock waves that travel through the Earth due to earthquakes.
This diagram shows the structure of a simple seismometer. The shock wave will cause the mass to vibrate, causing a trace to be drawn on the paper scroll.

a: The frequency of a typical shock wave is between 30 and $40 Hz$. Explain why the natural frequency of the spring–mass system in the seismometer should be very much less than this range of frequencies. [3]
This graph shows the acceleration of the mass against its displacement when the seismometer is recording an earthquake.

b: What evidence does the graph give that the motion is simple harmonic? [2]
c: Use information from the graph to calculate the frequency of the oscillation. [4]
[Total: 9]