This episode allows you to familiarise your students with the main features of simple harmonic motion (SHM), before going on to a more mathematical description.

Lesson Summary

• Demonstration: Observing SHM in the lab (20 minutes)
• Discussion: The importance of SHM (10 minutes)
• Demonstration: SHM is isochronous (10 minutes)
• Student activity: Identifying SHM (20 minutes)
• Demonstration: Sinusoidal graphs on a CRO (10 minutes)

Demonstration: Observing SHM in the lab

Start by talking about oscillations. Oscillatory motion is a type of motion that repeats itself in a cyclic fashion.

Set up some visual aids: a pendulum and a mass on a spring. Set them oscillating.

It is easier to see what is going on with a slower oscillation. Tether a dynamics trolley by springs between two retort stands. Add weights to the trolley, so that its period of oscillation is a couple of seconds. Set it oscillating; you will be able to hear the trolley as it speeds up and slows down. If you use a motion sensor connected to a computer it is easy to see the displacement time graph and velocity time graph. You could do the same for a long pendulum.

Set up an oscilloscope with its timebase switched off, so that the spot is central on the screen. Connect to a very slow signal generator, set at a frequency of less than 1 Hz. Watch the up-and-down motion of the spot on the screen. Ask about the shared characteristics of these different oscillations. Points to bring out:

The oscillatory motion repeats itself and thus has similarities with circular motion which also repeatedly returns to its starting point. (This fact will be fruitfully exploited later.)

• Displacement is the greatest at the extremes of the oscillation.
• Velocity is the greatest at the midpoint.
• The oscillating mass is in equilibrium at the midpoint of its oscillation; show this by stopping the mass at this point – it remains stationary.
• When the mass is oscillating, its inertia carries it through the midpoint.

So we have the following pattern of motion:

The mass speeds up as it heads towards the midpoint, has its greatest speed as it passes through the midpoint, and slows down as it continues towards the other extreme of the oscillation. Here, it reverses and starts to accelerate again towards the midpoint.

Of course, not all oscillations are as simple as this, but this is a particularly simple kind, known as simple harmonic motion (SHM). It is relatively easy to analyze mathematically, and many other types of oscillatory motion can be broken down into a combination of SHMs.

Discussion: the importance of SHM

At this point, it is worth saying a few words about the importance of SHM.

All clocks (starting with the pendulum in a grandfather clock) are essentially simple harmonic oscillators; all transmitters and receivers of waves are essentially simple harmonic oscillators; many aspects of engineering design from the massive to the microscopic require a detailed knowledge of SHM (e.g. bridges, earthquake protection of buildings, atomic force microscopes for imaging single atoms). Detailed theories of the behaviour of atoms and molecules (in solids and gases) are applications of SHM. Other aspects of SHM are at the heart of unsolved questions in physics today. Thus although the model oscillators that students meet appear rather basic, they mirror pretty well applications and problems far beyond the school laboratory.

The key aim is to get across how to recognize SHM experimentally as well as the features that are built into the theoretical model of SHM.

Demonstration: SHM is isochronous

Using the apparatus of the earlier demonstration, set one oscillator in motion. Now change the amplitude. What do you notice about the time period of the oscillation? (It doesn’t change.) This is obvious if a motion sensor is used.

Try with the others.

This is a characteristic of SHM; the period is independent of the amplitude, and we say that the motion is isochronic.

Student activity: Identifying SHM

Your students can try out some other oscillating systems; are they isochronic? Do they appear to be SHM?

Set up a circus of experiments (see below). Students in pairs try out a selection and report back to a plenary session.

Episode 301-1: Oscillation circus (Word, 31 KB)

• Mass on a spring (vertical)
• Mass (large cube of polystyrene) on the end of a slinky spring suspended from the ceiling
• Mass between two springs (vertical, both springs in tension when the mass is at rest)
• Mass between two springs (horizontal, both springs in tension when the mass is at rest – use an air track slider for the mass to have a low friction system)
• Air track slider moving between two buffer springs, or rebounding due to magnetic repulsion
• Vibrating cantilever
• Simple pendulum
• Simple half pendulum (one that bounces off a hard surface when its string is vertical)

Episode 301-2: Mass oscillating between elastic barriers (Word, 36 KB)

Episode 301-3: Oscillating ball (Word, 30 KB)

• Simple pendulum whose string intercepts a peg when vertical, so the length of the pendulum gets shorter for one half of its cycle
• A torsion pendulum

Episode 301-4: Swinging bar or torsion pendulum (Word, 47 KB)

• A ball bouncing off a hard surface
• Ball moving in a semi-circular shaped track (curtain rail)
• Ball moving in a vertical V shaped track (rounded enough at the point of the V to let the ball pass easily)
• Ball moving in a vertical parabolic shaped track (draw out the parabola on a large piece of paper to aid the bending of the track into the parabolic shape)
• A right circular cone on an inclined plane
• A rectangular or square section bar balanced on top of a cylinder – the length of the bar at right angles to the axis of the cylinder
• Water in a U tube
• Hydrometer in water

Demonstration: Sinusoidal graphs on a CRO

Set one or more of the masses oscillating. Ask: what would a displacement-time graph for this look like? You can show the graph using the oscilloscope. Switch on the time base to see the sinusoidal graph as it is traced out. Emphasize that the graph produced is a sine wave or a sinusoidal variation.  Alternatively show the motion with a computer and a motion sensor. A pendulum with about a 2 m string and a suitably large bob, (large coffee can with some sand) works well. The Trolley and spring system works well for slow oscillations. A card may be needed on the trolley so it is picked up correctly by the sensor. Be safe: when fixing a string to a high point, two persons are needed – one to hold the ladder or steps and one to do the job.

(If your signal generator will also generate triangular and square waves, show the oscillations these produce with the timebase off. Ask for predictions of the displacement-time graphs.)

A more challenging question, which you may wish to leave until later: What would a velocity-time graph look like? (Also sinusoidal, but maximum value when displacement is zero.) And acceleration? (Sinusoidal, but 180 ° out of phase with displacement.)

The previous episode laid the qualitative foundations for what follows here: a mathematical approach to simple harmonic motion (SHM), starting from its physical basis and the forces involved.

Lesson Summary

• Demonstration and Discussion: The restoring force in SHM (20 minutes)
• Demonstration and Discussion: Graphical representations of SHM (30 minutes)
• Discussion: Equations of SHM (30 minutes)
• Discussion: Linking SHM to circular motion (10 minutes)
• Student activity: Making a computer model (30 minutes)

Discussion and demonstrations: The restoring force in SHM

So far, we have only considered the characteristics of SHM. Now we can go on to look at the underlying causes of this motion, in terms of forces.

Using a model system, look at the forces involved in SHM. Start with the trolley tethered by springs. Show that it remains stationary at the midpoint of its oscillation; it is in equilibrium. What resultant force acts on it? (Zero.) If it is displaced to the right, in which direction does it start to move? (To the left.) What causes this acceleration? (A resultant force to the left.) You can readily feel the force when you pull the trolley to one side.

So displacement to the right gives a resultant force (and hence acceleration) to the left, and vice versa. We call this force the restoring force (because it ties to restore the mass to its equilibrium position). The greater the displacement, the greater the restoring force.

We can write this mathematically:

Restoring force F  ∝  displacement x

Since the force is always directed towards the equilibrium position, we can say:

F  ∝  -x

or

F = − k × x

Where the minus sign indicates that force and displacement are in opposite directions, and k is a constant (a characteristic of the system).

This is the necessary condition for SHM.

Now it should be clear that we are dealing with vector quantities here; what are they? (Displacement, velocity, acceleration, force.) It makes sense to use a sign convention. We call the midpoint zero; any quantity directed to the right is positive, to the left is negative. (For a vertical oscillation, upwards is positive.)

Think about mass-spring systems. Why might we expect a restoring force which is proportional to displacement? (This is a consequence of Hooke’s law.)

It is harder to see this for a simple pendulum. As the pendulum is displaced to the side, the component of gravity restoring it towards the vertical increases. Force and displacement are (roughly) proportional, and proportionality is closest at small angles. For this reason, it will make sense to start a mathematical analysis with mass-spring systems.

Discussion and demonstrations: Graphical representations of SHM

We need to get to a point where we can develop the equation F = − k × x to a = −  ω ² × x , where a is the acceleration and ω is the angular velocity associated with the SHM. To do this, we develop the graphical representation of SHM.

Consider first the tethered trolley at its maximum displacement. Its velocity is zero; as you release it, its acceleration is maximum. Show how the trolley accelerates towards the midpoint. Sketch the displacement-time graph for this quarter of the oscillation. What happens next? The trolley decelerates as it moves to maximum negative displacement. Extend the graph. Continue for the second half of the oscillation, so that you have one cycle of a sine graph.

Below this graph, draw the corresponding velocity-time graph. Deduce velocity from the gradient of the displacement graph. Show how maximum velocity occurs when displacement is zero.

Below this, sketch the acceleration-time graph.

Episode 302-1: Snapshots of the motion of a simple harmonic oscillator (Word, 413 KB)

Episode 302-2: Step by step through the dynamics (Word, 172 KB)

Episode 302-3: Graphs of simple harmonic motion (Word, 228 KB)

Discussion: Equations of SHM

These graphs can be represented by equations. For displacement:

x = A × sin(2 π ft) or x = A × sin( ω t)

Explain that f is the frequency of the oscillation, and is related to the period T by f = 1 T . The amplitude of the oscillation is A .

Velocity: v = 2 π ft  ×  A × sin(2 π ft) or v =  ω  ×  A × sin( ω t)

Acceleration: a = (2 π ft)²  ×  A × sin(2 π ft) or v =  ω ²  ×  A × sin( ω t)

Depending on your students’ mathematical knowledge, you may be able to explain where these equations come from. You may simply have to indicate their plausibility.

Also point out that in some text books the sin and cos functions are written in terms of the real physical frequency f rather than ω.

Comparing the equations for displacement and acceleration gives:

a = −  ω ² × x

and applying Newton’s second law gives:

F = − m  ×  ω ² × x

These are the fundamental conditions which must be met if a mass is to oscillate with SHM. If, for any system, we can show that F ∝ – x then we have shown that it will execute SHM, and its frequency will be given by:

ω  = 2 π f , so ω  =     F mx therefore ω is related to the restoring force per unit mass per unit displacement.

Discussion: Linking SHM to circular motion

If your specification requires you to explore SHM with reference to motion in the auxiliary circle (or if you wish to adopt this approach anyway), then this is a good point to do so. It has the merit of showing the link between SHM and circular motion, but for many students it may simply add confusion.

Episode 302-4: A language to describe oscillations (Word, 121 KB)

See also this tutorial on the University of Guelph website,which links ω for a circle and SHM.

Simple harmonic motion and circular motion, University of Guelph

Student activity: Making a computer model

It will help students to grasp the relationships between displacement, velocity and acceleration in SHM if they make a computer model.

Episode 302-5: Build your own simple harmonic oscillator (Word, 70 KB)

This episode and the next focus on practical simple harmonic motion (SHM) systems.

Lesson Summary

• Discussion: Hooke’s law leads to SHM (15 minutes)
• Student experiment: Testing the relationship T = 2 π     m k} (30 minutes)
• Student activity: Using a computer model (20 minutes)
• Discussion: Modelling in physics (10 minutes)
• Worked example: Applying the relationship T = 2 π     m k} (10 minutes)
• Student questions: Calculations involving T = 2 π     m k} (30 minutes)

Discussion: Hooke’s law leads to SHM

If your specification requires it, here is where you can derive the expression for the period of a mass-spring system.

We have ω  = 2 π f, so ω  =     F mx as the requirement for SHM, and F = k × x from Hooke’s law.

It is easiest to deal with a horizontal mass-spring system first (because you can ignore gravity).

Student experiment: Testing the relationship

Students can test the relationship

T = 2 π     m k for a mass-spring system. (Note that this expression is independent of g.)

Students may find that there is a systematic error, caused by the finite mass of the spring. Try modifying the simple theory to take into account the mass of the spring m_S :

T = 2 π     m  +  m_S k

Episode 303-1: Loaded spring oscillator (Word, 59 KB)

Student activity: Using a computer model

They can also use a computer model.

Episode 303-2: Oscillating freely (Word, 26 KB)

and the model

Episode 303-3: Modelling springs and masses (Word, 24 KB)

Discussion: Modelling in physics

The study of SHM may well be the first occasion that students meet detailed mathematical modelling. It may be worth spelling out for them what is happening. There is a physical behaviour we want to understand. First we simplify the actual situation to an idealised physical model by making assumptions, e.g. no friction, pendulum strings or springs that have no mass, etc. Then we make a mathematical model to represent the physical model. The mathematical model is then analysed (solved) and has to be interpreted in terms of the physical model. Experiments try to mirror the physical model but they cannot do this exactly (e.g. make a pendulum string as light as possible while still being strong enough to support the bob). So care is needed when comparing the theory with experimental measurements.

Worked examples: Applying the relationship

A vibrating atom in a solid can be modelled as a mass m between two tensioned springs, the springs representing the interatomic forces.

For typical interatomic forces k = 60 N m^-1

Mass of an atom (Na in NaCl) ~ 3.8 × 10^-26 kg

Estimate the natural vibration frequency of atoms.

f = 1T

f = 12 π  ×      2k m

f ~9 × 10¹² Hz, which is in the IR region of the electromagnetic spectrum. We will return to atomic vibrations when discussing resonance.

Student questions: Calculations involving T = 2 π     m k

These questions reinforce basic ideas about SHM.

Episode 303-4: Harmonic oscillators (Word, 35 KB)

This episode reinforces many of the fundamental ideas about simple harmonic motion (SHM).

Lesson Summary

• Student experiment: Measuring the restoring force (20 minutes)
• Student experiment: Testing the relationship
T = 2 π     l g (30 minutes)
• Student activity: Using an applet of a pendulum (30 minutes)
• Discussion: Gravitational and inertial mass (10 minutes)
• Student questions: Calculations involving pendulums (30 minutes)

Note a complication: a simple pendulum shows SHM only for small amplitude oscillations.

Student experiment: Measuring the restoring force

Measure the restoring force for a simple pendulum.

Episode 304-1: The simple pendulum (Word, 57 KB)

Student experiment: Testing the relationship

Test the relationship T = 2 π     l g for a simple pendulum. Students could decide for themselves which measurements to make, which quantities to vary, and how to process and interpret the results. Encourage them to look for deviations from linear behaviour, arising from large-amplitude oscillations.

Student activity: Using an applet of a pendulum

Investigate a virtual pendulum; this allows you to vary g . You can also force the pendulum, which is useful later when studying resonance.

NB: the analysis of the data uses log-log plots, so this may not be suitable for all students.

Episode 304-2: Virtual pendulum (Word, 31 KB)

Discussion: Gravitational and inertial mass

The fact that the period of a simple pendulum is independent of the mass of the bob is an example of the Principle of Equivalence – something still not understood today and being tested by very sophisticated experiments involving astronomical measurements on the one hand and how single atoms fall due to gravity on the other.

The basic puzzle is why the m in Fm = a (where m is the inertial mass which determines how an object responds to any unbalanced force) has exactly the same magnitude as the m in mg (where the m is the gravitational mass, the source of the gravitational force).

In deriving the equation for the period of a simple pendulum, we have used both, and used the fact that numerically they cancel out.

Student questions: Calculations involving pendulums

These questions reinforce basic ideas about SHM.

Episode 304-3: Pendulum (Word, 25 KB)

Qualitatively, students will appreciate that there is a continuous change in the ways that energy is stored during simple harmonic motion (SHM). Here, they can also learn about the mathematical basis for calculating energy.

Lesson Summary

• Demonstration: An experimental displacement-time graph (10 minutes)
• Discussion: Maximum values of quantities in SHM (15 minutes)
• Student questions: Practice with the equations (30 minutes)
• Discussion: Changes in the ways energy is stored in SHM (20 minutes)
• Student questions: Changes in the ways energy is stored a pendulum (20 minutes)

Demonstration: An experimental displacement-time graph

Use a water pendulum to draw a large displacement-time graph for a pendulum. You could ask a group of students to prepare this in advance and demonstrate it to the class.

Episode 305-1: The water pendulum (Word, 36 KB)

Discussion: Maximum values of quantities in SHM

Refer back to the sine and cosine equations for SHM. Show that the maximum values of displacement, velocity and acceleration are given by (the term in front of sin or cos):

• Maximum displacement = A
• Maximum velocity =  ω  ×  A
• Maximum acceleration =  ω ²  ×  A

Compare these relationships with the equations for circular motion:

• Displacement = r
• Velocity =  ω  ×  r
• Acceleration =  ω ²  ×  r

If you have adopted the auxiliary circle approach earlier, the parallels should be clear.

Student questions: Practice with the equations

It will help to provide some more practice in using the equations and analyzing motion.

Episode 305-2: Oscillators (Word, 70 KB)

Discussion: Energy changes in SHM

Think about the changes in the ways energy is stored in a mechanical oscillator. Recap that, as it passes through its equilibrium position, its speed and hence its energy stored kinetically are a maximum. At the maximum displacement, the speed and hence the energy stored kinetically are both zero. The potential energy will be a maximum when the speed is zero and vice versa. In a spring, the potential energy is the energy stored elastically; in a pendulum it is energy stored gravitationally. Assuming that there is no friction or air drag the total energy E of the oscillator must remain constant

For the mass and spring system, the work done stretching a spring by an amount x is the area under the force extension graph, 12 kx². The PE-extension graph is a parabola.

The energy stored kinetically will be zero at +  A and a maximum when x is 0, so its graph is an inverted version of the graph showing energy stored elastically. At any position energy stored kinetically  +  energy stored elastically is a constant E, where E  =  KE_max  =  PE_max.

PE_max  ∝  A² , so the total energy E of SHM is proportional to amplitude².

Episode 305-3: Energy stored in a stretched spring (Word, 79 KB)

Draw a graph from x from −  A to +  A , to show energy stored kinetically, potential energy (energy stored elastically or gravitationally), and total energy. You can also draw graphs of KE and PE against time.

Episode 305-4: Energy flow in an oscillator (Word, 71 KB)

Student questions: Energy of a pendulum

Some useful questions on energy of a pendulum.

Episode 305-5: Energy and pendulums (Word, 54 KB)

The ideas here are fairly straightforward. If work is done on an oscillating system (by friction), its amplitude will decrease. This usually follows an exponential decay pattern.

Lesson Summary

• Demonstration: Water in a U-tube (10 minutes)
• Demonstration: A damped spring (10 minutes)
• Student experiment: Investigating damped simple harmonic motion (SHM) (30 minutes)
• Discussion: Exponential decay of amplitude (10 minutes)
• Discussion: Using damping (10 minutes)

Demonstration: Water in a U-tube

Show how water oscillates in a U-tube. The oscillation dies away quite quickly.

Energy will be dissipated (stored thermally) as a result of an oscillation. Friction due to viscosity does work on the oscillator, and the oscillator will return to its position of stable equilibrium. This is called damping .

Demonstration: A damped spring

Set up a suspended mass-spring system with a damper – a piece of card attached horizontally to the mass to increase the air drag. Alternatively, clamp a springy metal blade (e.g. hacksaw blade) firmly to the bench. Attach a mass to the free end, and add a damping card.

Show how the amplitude decreases with time.

Safety

If the hacksaw blade could be used as a weapon, have the teeth ground off by workshop staff.

Student experiment: Investigating damped SHM

Students can design their own arrangement for investigating damped SHM. They can investigate the pattern of decrease of amplitude (exponential decay), and whether the frequency is affected by damping.

Plot amplitude versus number of swings or against time. Use the graph to check for exponential decay of the amplitude using the constant ratio property.

The decay of the amplitude is exponential and careful measurement of the frequency may show that it is slightly less than the undamped value. (This can be accounted for theoretically.)

Discussion: Exponential decay of amplitude

Link this observation to other examples of exponential decay if already taught (RC circuit, radioactive decay, absorption of light by glass, a hot body heating the surroundings etc.) to suggest what type of physics is taking place.

Exponential decay of the amplitude A implies that:

Rate of decay of maximum amplitude A  ∝  present value of A .

Large amplitude implies high maximum velocity, which implies greater drag.

Recall that PE_MAX  ∝  A², so the total energy of an oscillator  ∝  A².

When the amplitude has decayed to 12 its original value, the energy of the oscillator is now 14 the initial value. Energy has been dissipated, and is now stored thermally.

Discussion: Using damping

For some oscillators (e.g. clocks) we want minimum damping; for others (e.g. a vehicle shock absorbing system) we want them to return to equilibrium as fast as possible. The latter requires a unique value of the damping (critical damping) so that the system returns to equilibrium without overshooting; i.e. it gets to equilibrium in the minimum time without oscillating at all. Overdamping prevents over-shooting and thus any oscillations, but by making the damping large enough the system can take as long as you want to regain its starting equilibrium position.

Simple harmonic oscillators show resonance if they are forced to vibrate at their natural frequency. This is a phenomenon of great importance in many aspects of science.

Lesson Summary

• Discussion: Resonance as a phenomenon (10 minutes)
• Demonstration: Barton’s pendulums (10 minutes)
• Student activity: An applet of a forced pendulum (20 minutes)
• Student experiment: A selection of model systems (30 minutes)
• Student questions: Questions on resonance (40 minutes)
• Discussion: The effect of damping on resonance (10 minutes)
• Demonstration and student reading: The Tacoma Narrows bridge disaster (30 minutes)

Discussion: Resonance as a phenomenon

An oscillator can be forced to vibrate with increasing amplitude; to do this; energy must be shifted in the right way.

A child on a park swing is the classic example that all can visualise. The push must come at the same natural frequency as the oscillating pendulum-like swing and at the right point in the swing’s cycle.

So the way that energy is shifted into system must be tuned to the oscillator, or the oscillator must be able to be tuned to the way that energy is being shifted. Matching up the natural frequency and the forcing frequency results in a resonant system. The fundamental resonant frequency is synonymous with the natural frequency of an oscillator.

Resonance can lead to very large oscillation amplitudes that can result in damage. E.g. buildings etc need to have their natural frequency very different from the likely vibration frequencies due to earthquakes.

Demonstration: Barton’s pendulums

Barton’s pendulums are a famous demonstration of a resonance effect.

Episode 307-1: Barton’s pendulums (Word, 38 KB)

Student activity: An applet of a forced pendulum

Investigate a virtual pendulum which can be forced.

Episode 307-2: Forced oscillations (Word, 20 KB)

Student experiment: A selection of model systems

Students can be allocated to one of the following experiments (duplication is easy), followed by a brief plenary session where each system is demonstrated to the whole class.

Episode 307-3: Book on a string (Word, 41 KB)

Episode 307-4: Resonance of a milk bottle (Word, 91 KB)

Episode 307-5: Resonance of a hacksaw blade (Word, 66 KB)

Episode 307-6: Resonance of a mass on a spring (Word, 98 KB)

Student questions: Questions on resonance

Episode 307-7: Oscillator energy and resonance (Word, 59 KB)

Episode 307-8: Resonance in car suspension systems (Word, 204 KB)

Episode 307-9: Car suspension (Word, 21 KB)

Discussion: The effect of damping on resonance

If a resonant system is forced at frequencies above or below the resonant (natural) frequency f₀ , the amplitude of oscillation will be reduced. The resonance curve peaks at f₀ . You may need to discuss how the shape of the curve depends on the degree of damping.

Episode 307-10: Resonance (Word, 47 KB)

Demonstration and student reading: The Tacoma Narrows bridge disaster.

The Tacoma Narrows bridge disaster is generally described as a consequence of resonance. However, the full details of the mechanism are still debated. If possible show a video of the bridge as it collapsed in high winds on 7 November 1940. However, it seems more than likely that it is an example of positive feedback, a sort of inverse damping which created this effect.

Episode 307-11: Tacoma Narrows bridge disaster (Word, 100 KB)

Episode 307-12: Tacoma Narrows: Re-evaluating the evidence (Word, 57 KB)