Deriving centripetal acceleration

When I was an A-level physics student (many, many years ago, when the world was young LOL) I found the derivation of the centripetal acceleration formula really hard to understand. What follows is a method that I have developed over the years that seems to work well. The PowerPoint is included at the end.

Step 1: consider an object moving on a circular path

Let’s consider an object moving in circular path of radius r at a constant angular speed of ω (omega) radians per second.

The object is moving anticlockwise on the diagram and we show it at two instants which are time t seconds apart. This means that the object has moved an angular distance of ωt radians.

Step 2: consider the linear velocities of the object at these times

The linear velocity is the speed in metres per second and acts at a tangent to the circle, making a right angle with the radius of the circle. We have called the first velocity v1 and the second velocity at the later time v2.

Since the object is moving at a constant angular speed ω and is a fixed radius r from the centre of the circle, the magnitudes of both velocities will be constant and will be given by v = ωr.

Although the magnitude of the linear velocity has not changed, its direction most certainly has. Since acceleration is defined as the change in velocity divided by time, this means that the object has undergone acceleration since velocity is a vector quantity and a change in direction counts as a change, even without a change in magnitude.

Step 3a: Draw a vector diagram of the velocities

We have simply extracted v1 and v2 from the original diagram and placed them nose-to-tail. We have kept their magnitude and direction unchanged during this process.

Step 3b: close the vector diagram to find the resultant

The dark blue arrow is the result of adding v1 and v2. It is not a useful operation in this case because we are interested in the change in velocity not the sum of the velocities, so we will stop there and go back to the drawing board.

Step 3c switch the direction of velocity v1

Since we are interested in the change in velocity, let’s flip the direction of v1 so that it going in the opposite direction. Since it is opposite to v1, we can now call this -v1.

It is preferable to flip v1 rather than v2 since for a change in velocity we typically subtract the initial velocity from the final velocity; that is to say, change in velocity = v2 – v1.

Step 3d: Put the vectors v2 and (-v1) nose-to-tail

Step 3e: close the vector diagram to find the result of adding v2 and (-v1)

The purple arrow shows the result of adding v2 + (-v1); in other words, the purple arrow shows the change in velocity between v1 and v2 due to the change in direction (notwithstanding the fact that the magnitude of both velocities is unchanged).

It is also worth mentioning that that the direction of the purple (v2v1) arrow is in the opposite direction to the radius of the circle: in other words, the change in velocity is directed towards the centre of the circle.

Step 4: Find the angle between v2 and (-v1)

The angle between v2 and (-v1) will be ωt radians.

Step 5: Use the small angle approximation to represent v2-v1 as the arc of a circle

If we assume that ωt is a small angle, then the line representing v2-v1 can be replaced by the arc c of a circle of radius v (where v is the magnitude of the vectors v1 and v2 and v=ωr).

We can then use the familiar relationship that the angle θ (in radians) subtended at the centre of a circle θ = arc length / radius. This lets express the arc length c in terms of ω, t and r.

And finally, we can use the acceleration = change in velocity / time relationship to derive the formula for centripetal acceleration we a = ω2r.


Well, that’s how I would do it. If you would like to use this method or adapt it for your students, then the PowerPoint is attached.

Please Like or leave a comment if you find this useful 🙂

Teaching refraction using a ripple tank

It is a truth universally acknowledged that student misconceptions about waves are legion. Why do so many students find understanding waves so difficult?

David Hammer (2000: S55) suggests that it may, in fact, be not so much a depressingly long list of ‘wrong’ ideas about waves that need to be laboriously expunged; but rather the root of students misconceptions about waves might be a simple case of miscategorisation.

Hammer (building on the work of di Sessa, Wittmann and others) suggests that students are predisposed to place waves in the category of object rather than the more productive category of event.

Thinking of a wave as an object imbues them with a notional permanence in terms of shape and location, as well as an intuitive sense of ‘weightiness’ or ‘mass’ that is permanently associated with the wave.

Looking at a wave through this p-prim or cognitive filter, students may assume that it can be understood in ways that are broadly similar to how an object is understood: one can simply look at or manipulate the ‘object’ whilst ignoring its current environment and without due consideration of its past or its future

For example, students who think that (say) flicking a slinky spring harder will produce a wave with a faster wave speed rather than the wave speed being dependent on the tension in the spring. They are using the misleading analogy of how an object such as a ball behaves when thrown harder rather than thinking correctly about the actual physics of waves.

A series of undulating events…

Hammer suggests that perhaps a more productive cognitive resource that we should seek to activate in our students when learning about waves is that of an event.

An event can be expected to have a location, a duration, a time of occurrence and a cause. Events do not necessarily possess the aspects of permanence that we typically associate with objects; that is to say, an event is expected to be a transient phenomenon that we can learn about by looking, yes, but we have to be looking at exactly the right place at the right time. We also cannot consider them independently of their environment: events have an effect on their immediate environment and are also affected by the environment.

If students think of waves as a series of events propagating through space they are less likely to imbue them with ‘permanent’ properties such as a fixed shape that can be examined at leisure rather than having to be ‘captured’ at one instant. Hammer suggests using a row of falling dominoes to introduce this idea, but you might also care to use this suggested procedure.

You can access an editable copy of the slides that follow in Google Jamboard format by clicking on this link.

Teaching Refraction Step 1: Breaking = bad waves

I like to start by anchoring the idea of changing wave speed in a context that students may be familiar with: waves on a beach. However, we should try and separate the general idea of an undulating water wave from that of a breaking wave. Begin by asking this question:

Give thirty seconds thinking time and then ask students to hold up either one or two fingers on 3-2-1-now! to show their preferred answer. (‘Finger voting’ is a great method for ensuring that every student answers without having to dig out those mini whiteboards).

The correct answer is, of course, the top diagram. This is because the bottom diagram shows a breaking wave.

Teaching Refraction Step 2: Why do waves ‘break’?

In short, because waves slow down as they hit the beach. The top part of the wave is moving faster than the bottom so the wave breaks up as it slides off the bottom part. In effect, the wave topples over because the bottom is moving more slowly than the top part.

The correct answer is ‘two fingers’

It is important that students appreciate that although the wavelength of the wave does change, the frequency of the wave does not. The frequency of the wave depends on the weather patterns that produced the wave in the deep ocean many hundreds or thousand of miles away. The slope of the beach cannot produce more or fewer waves per second. In other words, the frequency of a wave depends on its history, not its current environment.

All the beach can do is change the wave speed, not the wave frequency.

Teaching Refraction Step 3: the view from above

We can check our students’ understanding by asking them to comment or annotate a diagram similar to the one below.

Some good questions to ask — before the wavelength annotations are added — are:

  • Are we viewing the waves from above or from the side? (From above.)
  • Can we tell where the crests of the waves are? (Yes, where the line of foam are.)
  • Can we tell where the troughs of the waves are? (Yes, midway between the crests.)
  • Can we measure the wavelength of the waves? (Yes, the crest to crest distance.)
  • Can we tell if the waves are speeding up or slowing down as they reach the shore? (Yes, the waves are bunching together which suggests that slow down as they reach the shore.)

Teaching Refraction Step 4: Understanding the ripple tank

Physics teachers often assume that the operation and principles of a ripple tank are self-evident to students. In my experience, they are not and it is worth spending a little time exploring and explaining how a ripple tank works.

Teaching Refraction Step 5: the view from the side

Teaching Refraction Step 6: Seeing refraction in the ripple tank (1)

It’s a good idea to first show what happens when the waves hit the boundary at right angles; in other words, when the direction of travel of the waves is parallel to the normal line.

I like to add the annotations live with the class using Google Jamboard. (The questions can be covered with a blank box until you are ready to show them to the students.)

You can access an animated, annotable version of this and the other slides in this post in Google Jamboard format by clicking on this link.

Teaching Refraction Step 7: Seeing refraction in the ripple tank (2)

The next step is to show what happens when the water waves arrive at the boundary at an angle i; in other words, the direction of travel of the waves makes an angle of i degrees with the normal line.

Again, I like to add the annotations live using Google Jamboard.


References

Hammer, D. (2000). Student resources for learning introductory physicsAmerican Journal of Physics68(S1), S52-S59.

Wittmann, M. C., Steinberg, R. N., & Redish, E. F. (1999). Making sense of how students make sense of mechanical wavesThe physics teacher37(1), 15-21.

We all adore Caloric

We all adore a Kia-Ora

Advertising slogan for ‘Kia-Ora’ orange drink (c. 1985)

Energy is harder to define than you would think. Nobel laureate Richard Feynman defined ‘energy’ as

a numerical quantity which does not change when something happens. It is not a description of a mechanism, or anything concrete; it is just a strange fact that we can calculate some number and when we finish watching nature go through her tricks and calculate the number again, it is the same. […] It is important to realize that in physics today, we have no knowledge of what energy is. […] It is an abstract thing in that it does not tell us the mechanism or the reasons for the various formulas.

Feynman Lectures on Physics, Vol 1, Lecture 4 Conservation of Energy (1963)

Current secondary school science teaching approaches to energy often picture energy as a ‘quasi-material substance’.

By ‘quasi-material substance’ we mean that ‘energy is like a material substance in how it behaves’ (Fairhurst 2021) and that some of its behaviours can be modelled as, say, an orange liquid (see IoP 2016).

The eight energy stores as suggested by the IoP

And yet, sometimes these well-meaning (and, in my opinion, effective) approaches can draw some dismissive comments from some physicists.

The Simpsons Comic Book Guy character saying "Picturing energy as a quasi-material substance? That teaching approach smacks of the oh-so-discredited 'Caloric' theory of energy to me . . ."
The Simpsons’ Comic Book Guy weighs in the ‘Teaching Energy’ debate

What was the ‘Caloric Theory of Energy’?

To begin with, there was never a ‘Caloric Theory of Energy’ since the concept of energy had not been developed yet; but the Caloric Theory of Heat was an important step along the way.

Caloric was an invisible, weightless and self-repelling fluid that moved from hot objects to cold objects. Antoine Lavoisier (1743-1794) supposed that the total amount of caloric in the universe was constant: in other words, caloric was thought to be a conserved quantity.

Caloric was thought to be a form of ‘subtle matter’ that obeyed physical laws and yet was so attenuated that it was difficult to detect. This seems bizarre to our modern sensibilities and yet Caloric Theory did score some notable successes.

  • Caloric explained how the volume of air changed with temperature. Cold air would absorb caloric and thus expand.
  • The Carnot cycle which describes the maximum efficiency of a heat engine (i.e. a mechanical engine powered by heat) was developed by Sadi Carnot (1796-1832) on the basis of the Caloric Theory

Why Caloric Theory was replaced

It began with Count Rumford in 1798. He published some observations on the manufacturing process of cannons. Cannon barrels had to be drilled or bored out of solid cylinders of metal and this process generated huge quantities of heat. Rumford noted that cannons that had been previously bored produced as much heat as cannons that were being freshly bored for the first time. Caloric Theory suggested that this should not be the case as the older cannons would have lost a great deal of caloric from being previously drilled.

The fact that friction could seemingly generate limitless quantities of caloric strongly suggested that it was not a conserved quantity.

We now understand from the work James Prescott Joule (1818-1889) and Rudolf Clausius (1822-1888) that Caloric Theory had only a part of the big picture: it is energy that is the conserved quantity, not caloric or heat.

As Feynman puts it:

At the time when Carnot lived, the first law of thermodynamics, the conservation of energy, was not known. Carnot’s arguments [using the Caloric Theory] were so carefully drawn, however, that they are valid even though the first law was not known in his time!

Feynman Lectures on Physics, Vol 1, Lecture 44 The Laws of Thermodynamics

In other words, the Caloric Theory is not automatically wrong in all respects — provided, that is, it is combined with the principle of conservation of energy, so that energy in general is conserved, and not just the energy associated with heat.

We now know, of course, that heat is not a form of attenuated ‘subtle matter’ but rather the detectable, cumulative result of the motion of quadrillions of microscopic particles. However, this is a complex picture for novice learners to absorb.

Caloric Theory as a bridging analogy

David Hammer (2000) argues persuasively that certain common student cognitive resources can serve as anchoring conceptions because they align well with physicists’ understanding of a particular topic. An anchoring conception helps to activate useful cognitive resources and a bridging analogy serves as a conduit to help students apply these resources in what is, initially, an unfamiliar situation.

The anchoring conception in this case is students’ understanding of the behaviour of liquids. The useful cognitive resources that are activated when this is brought into play include:

  • the idea of spontaneous flow e.g. water flows downhill;
  • the idea of measurement e.g. we can measure the volume of liquid in a container; and
  • the idea of conservation of volume e.g. if we pour water from a jug into an empty cup then the total volume remains constant.

The bridging analogy which serves as a channel for students to apply these cognitive resources in the context of understanding energy transfers is the idea of ‘energy as a quasi-material substance’ (which can be considered as an iteration of the ‘adapted’ Caloric Theory which includes the conservation of energy).

The bridging analogy helps students understand that:

  • energy can flow spontaneously e.g. from hot to cold;
  • energy can be measured and quantified e.g. we can measure how much energy has been transferred into a thermal energy store; and
  • energy does not appear or disappear: the total amount of energy in a closed system is constant.

Of course, a bridging analogy is not the last word but only the first step along the journey to a more complete understanding of the physics involved in energy transfers. However, I believe the ‘energy as a quasi-material substance’ analogy is very helpful in giving students a ‘sense of mechanism’ in their first encounters with this topic.

Teachers are, of course, free not to use this or other bridging analogies, but I hope that this post has persuaded even my more reluctant colleagues that they need a more substantive argument than a knee jerk ‘energy-as-substance = Caloric Theory = BAD’.


References

Fairhurst P. (2021), Best Evidence in Science Teaching: Teaching Energy. https://www.stem.org.uk/sites/default/files/pages/downloads/BEST_Article_Teaching%20energy.pdfhttps://www.stem.org.uk/sites/default/files/pages/downloads/BEST_Article_Teaching%20energy.pdf [Accessed April 2022]

Hammer, D. (2000). Student resources for learning introductory physicsAmerican Journal of Physics68(S1), S52-S59.

Institute of Physics (2016), Physics Narrative: Shifting Energy Between Stores. Available from https://spark.iop.org/collections/shifting-energy-between-stores-physics-narrative [Accessed April 2022]

Dual Coding and Equations of Motion for GCSE

Necessity may well be the mother of invention, but teacher desperation is often the mother of new pedagogy.

For an unconscionably long time, I think that I failed to adequately help students understand the so called ‘equations of motion’ (the mathematical descriptions of uniformly accelerated motion using the standard v, u, a, s and t notation) because I suffered from the ‘curse of knowledge’: I did not find the topic hard, so I naturally assumed that students wouldn’t either. This, sadly, proved not to be the case, even when the equation was printed on the equation sheet!

What follows is a summary of a dual coding technique that I have found really helpful in helping students become confident with problems involving the ‘equations of motion’. This is especially true at GCSE, where students encounter formulas such as

Equation of motion (v^2 - u^2 = 2as)

for the first time.

A dual coding convention for representing motion

Table to show dual coding convention

Applying dual coding to an equation of motion problem

EXAMPLE: A car is travelling at 6.0 m/s. As the car passes a lamp-post it accelerates up to a velocity of 14.2 m/s over a distance of 250 m. Calculate a) the acceleration; and b) the time taken for this change.

The problem can be represented using the dual coding convention as shown below.

Diagram showing car accelerating past a lamp-post using the dual coding convention outlined in the blog

Note that the arrow for v is longer than the arrow for u since the car has a positive acceleration; that is to say, the car in this example is speeding up. Also, the convention has a different style of arrow for acceleration, emphasising that it is an entirely different type of quantity from velocity.

We can now answer part (a) using the FIFA calculation system.

Working showing solution for part (a)

Part (b) can be answered as shown below.

Working for part b of the problem

Visualising Stopping Distance Questions

These can be challenging for many students, as we often seem to grabbing numbers and manipulating them without rhyme or reason. Dual coding helps make our thinking explicit,

EXAMPLE: A driver has a reaction time of 0.7 seconds. The maximum deceleration of the car is 2.6 metres per second squared. Calculate the overall stopping distance when the car is travelling at 13 m/s.

We need to calculate both the thinking distance and the braking distance to solve this problem

The acceleration of the car is zero during the driver’s reaction time, since the brakes have not been applied yet.

We visualise the first part of the problem like this.

Using s=vt we find that the thinking distance is 9.1 m.

Now let’s look at the second part of the question.

There are three things to note:

  • Since the car comes to a complete stop, the final velocity v is zero.
  • The acceleration is negative (shown as a backward pointing arrow) since we are talking about a deceleration: in other words, the velocity gradually decreases in size from the initial velocity value of u as the car traverses the distance s.
  • Since the second part of the question does not involve any consideration of time we have omitted the ‘clock’ symbols for the second part of the journey.

We can now apply FIFA:

Working for braking distance part of problem

Since the acceleration arrow points in the opposite direction to the positive arrow, we enter it as a negative value. When we get to the third line of the Fine Tune stage, we see that a negative 169 divided by negative 4.4 gives positive 38.408 — in other words, the dual coding convention does the hard work of assigning positive and negative values!

And finally, we can see that the overall stopping distance is 9.1 + 38.4 = 47.5 metres,

Conclusion

I have found this form of dual coding extremely useful in helping students understand the easily-missed subtleties of motion problems, and hope other science teachers will give it a try.


If you have enjoyed this post, you might enjoy these also:

Signposting Fleming’s Left Hand and Right Hand Rules

Students undoubtedly find electromagnetism tricky, especially at GCSE.

I have found it helpful to start with the F = BIl formula.

Introducing the “magnetic Bill” formula

This means that we can use the streamlined F-B-I mnemonic — developed by no less a personage than Robert J. Van De Graaff (1901-1967) of Van De Graaff generator fame — instead of the cumbersome “First finger = Field, seCond finger = Current, thuMb = Motion” convention.

Students find the beginning steps of applying Fleming’s Left Hand Rule (FLHR) quite hard to apply, so I print out little 3D “signposts” to help them. You can download file by clicking the link below.

You can see how they are used in the photo below.

Photograph to show how to use the FLHR signpost

Important tip: it can be really helpful if students label the arrows on both sides, such as in the example shown below. (The required precision in double sided printing defeated me!)

Another example of the FLHR signpost in use

Signposts for Fleming’s Right Hand Rule are included on the template.


Other posts I have written on magnetism include:

Feel free to dip into these 🙂

Misconceptions and p-prims at ResearchED 2021

Many thanks to all those who attended my talk on “Dealing with Misconceptions: the p-prim and refining raw intuitions approach” and for the stimulating discussions afterwards!

And especially huge thanks to Bill Wilkinson for his help in sorting out some tech issues!

The PowerPoint can be downloaded below.

You can watch a short summary of the talk here.

The references are below with links to freely available copies (where I’ve been able to find them).

I think Redish and Kuo (2015) is an excellent introduction to the Resources Framework.


DiSessa, A. A. (1988). Knowledge in pieces. In G. Forman & P. B. Pufall (Eds.), Constructivism in the computer age (pp. 49–70). Lawrence Erlbaum Associates, Inc.

DiSessa, A. A. (1993). Toward an epistemology of physics. Cognition and instruction10(2-3), 105-225.

Hammer, D. (2000). Student resources for learning introductory physics. American Journal of Physics68(S1), S52-S59.

Nelmes, A. (2004). Putting conceptions in their place: using analogy to cue and strengthen scientifically correct conceptions.

Redish, E. F., & Kuo, E. (2015). Language of physics, language of math: Disciplinary culture and dynamic epistemology. Science & Education24(5), 561-590.

Measuring the radius of the Earth in 240 BC

The brain is wider than the sky,
For, put them side by side,
The one the other will include
With ease, and you beside.

Emily Dickinson, ‘The Brain’

Most science teachers find that ‘Space’ is one of the most enduringly fascinating topics for many students: the sense of wonder engendered as our home planet becomes lost in the empty vastness of the Solar System, which then becomes lost in the trackless star-studded immensity of the Milky Way galaxy, is a joy to behold.

But a common question asked by students is: How do we know all this? How do we know the distance to the nearest star to the Sun is 4 light-years? Or how do we know the distance to the Sun? Or the Moon?

I admit, with embarrassment, that I used to answer with a casual and unintentionally-dismissive ‘Oh well, scientists have measured them!’ which (though true) must have sounded more like a confession of faith rather than a sober recounting of empirical fact. Which, to be fair, it probably was; simply because I had not yet made the effort to find out how these measurements were first taken.

The technological resources available to our ancestors would seem primitive and rudimentary to our eyes but, coupled with the deep well of human ingenuity that I like to think is a hallmark of our species, it proved not just ‘world-beating’ but ‘universe-beating’.

I hope you enjoy this whistle stop tour of this little-visited corner of the scientific hinterland, and choose to share some these stories with your students. It is good to know that the brain is indeed ‘wider than the sky’.

I have presented this in a style and format suitable for sharing and discussing with KS3/KS4 students (11-16 year olds).

Mad dogs and Eratosthenes go out in the midday Sun…

To begin at the beginning: the first reliable measurement of the size of the Earth was made in 240 BC and it all began (at least in this re-telling) with the fact that Eratosthenes liked talking to tourists. (‘Err-at-oss-THen-ees’ with the ‘TH’ said as in ‘thermometer’ — never forget that students of all ages often welcome help in learning how to pronounce unfamiliar words)

Alexandria (in present day Egypt) was a thriving city and a tourist magnet. Eratosthenes made a point of speaking to as many visitors as he could. Their stories, taken with a pinch of salt, were an invaluable source of information about the wider world. Eratosthenes was chief librarian of the Library of Alexandria, regarded as one of the Seven Wonders of the World at the time, and considered it his duty to collect, catalogue and classify as much information as he could.

One visitor, present in Alexandria on the longest day of the year (June 21st by our calendar), mentioned something in passing to Eratosthenes that the Librarian found hard to forget: ‘You know,’ said the visitor, ‘at noon on this day, in my home town there are no shadows.’

How could that be? pondered Eratosthenes. There was only one explanation: the Sun was directly overhead at noon on that day in Syene (the tourist’s home town, now known as Aswan).

The same was not true of Alexandria. At noon, there was a small but noticeable shadow. Eratosthenes measured the angle of the shadow at midday on the longest day. It was seven degrees.

No shadows at Syene, but a 7 degree shadow at Alexandria at the exact same time. Again, there was only one explanation: Alexandria was ’tilted’ by 7 degrees with respect to Syene.

Seven degrees of separation

The sphericity of the Earth had been recognised by astronomers from c. 500 BC so this difference was no surprise to Eratosthenes, but what he realised that since he was comparing the length of shadows at two places on the Earth’s surface at the same time then the 7o wasn’t just the angle of the shadow: 7o was the angle subtended at the centre of the Earth by radial lines drawn from both locations.

Eratosthenes paid a person to pace out the distance between Alexandria and Syene. (This was not such an odd request as it sounds to our ears: in the ancient world there were professionals called bematists who were trained to measure distances by counting their steps.)

It took the bematist nearly a month to walk that distance and it turned out to be 5000 stadia or 780 km by our measurements.

Eratosthenes then used a simple ratio method to calculate the circumference of the Earth, C:

Then:

The modern value for the radius of the Earth is 6371 km.

Ifs and buts…

There is still some debate as to the actual length of one Greek stadium but Eratosthenes’ measurement is generally agreed to within 1-2% of the modern value.

Sadly, none of the copies of the book where Eratosthenes explained his method called On the measure of the earth have survived from antiquity so the version presented here is a simplified one outlined by Cleomedes in a later book. For further details, readers are directed to the excellent Wikipedia article on Eratosthenes.

Astronomer Carl Sagan also memorably explained this method in his 1980 TV documentary series Cosmos.

You might want to read…

This is part of a series exploring how humans ‘measured the size of the sky’:

Part 2: How Aristarchus measured the distance between the Earth and the Moon

Part 3: How Aristarchus measured the distance between the Earth and the Sun

Binding energy: the pool table analogy

Nuclear binding energy and binding energy per nucleon are difficult concepts for A-level physics students to grasp. I have found the ‘pool table analogy’ that follows helpful for students to wrap their heads around these concepts.

Background

Since mass and energy are not independent entities, their separate conservation principles are properly a single one — the principle of conservation of mass-energy. Mass can be created or destroyed , but when this happens, an equivalent amount of energy simultaneously vanishes or comes into being, and vice versa. Mass and energy are different aspects of the same thing.

Beiser 1987: 29

E = mc2

There, I’ve said it. This is the first time I have directly referred to this equation since starting this blog in 2013. I suppose I have been more concerned with the ‘andallthat‘-ness side of things rather than E=mc2. Well, no more! E=mc2 will form the very centre of this post. (And about time too!)

The E is for ‘rest energy’: that is to say, the energy an object of mass m has simply by virtue of being. It is half the energy that would be liberated if it met its antimatter doppelganger and particles and antiparticles annihilated each other. A scientist in a popular novel sternly advised a person witnessing an annihilation event to ‘Shield your eyes!’ because of the flash of electromagnetic radiation that would be produced.

Well, you could if you wanted to, but it wouldn’t do much good since the radiation would be in the form of gamma rays which are to human eyes what the sound waves from a silent dog whistle are to human ears: beyond the frequency range that we can detect.

The main problem is likely to be the amount of energy released since the conversion factor is c2: that is to say, the velocity of light squared. For perspective, it is estimated that the atomic bomb detonated over Hiroshima achieved its devastation by directly converting only 0.0007 kg of matter into energy. (That would be 0.002% of the 38.5 kg of enriched uranium in the bomb.)

Matter contains a lot of energy locked away as ‘rest energy’. But these processes which liberate rest energy are mercifully rare, aren’t they?

No, they’re not. As Arthur Beiser put it in his classic Concepts of Modern Physics:

In fact, processes in which rest energy is liberated are very familiar. It is simply that we do not usually think of them in such terms. In every chemical reaction that evolves energy, a certain amount of matter disappears, but the lost mass is so small a fraction of the total mass of the reacting substances that it is imperceptible. Hence the ‘law’ of conservation of mass in chemistry.

Beiser 1987: 29

Building a helium atom

The constituents of a helium nucleus have a greater mass when separated than they do when they’re joined together.

Here, I’ll prove it to you:

The change in mass due to the loss of energy as the constituents come together is appreciable as a significant fraction of its original mass. Although 0.0293/4.0319*100% = 0.7% may not seem like a lot, it’s enough of a difference to keep the Sun shining.

The loss of energy is called the binding energy and for a helium atom it corresponds to a release of 27 MeV (mega electron volts) or 4.4 x 10-12 joules. Since there are four nucleons (particles that make up a nucleus) then the binding energy per nucleon (which is a guide to the stability of the new nucleus) is some 7 MeV.

But why must systems lose energy in order to become more stable?

The Pool Table Analogy for binding energy

Imagine four balls on a pool table as shown.

The balls have the freedom to move anywhere on the table in their ‘unbound’ configuration.

However, what if they were knocked into the corner pocket?

To enter the ‘bound’ configuration they must lose energy: in the case of the pool balls we are talking about gravitational potential energy, a matter of some 0.30 J per ball or a total energy loss of 4 x 0.30 = 1.2 joules.

The binding energy of a pool table ‘helium nucleus’ is thus some 1.2 joules while the ‘binding energy per nucleon’ is 0.30 J. In other words, we would have to supply 1.2 J of energy to the ‘helium nucleus’ to break the forces binding the particles together so they can move freely apart from each other.

Just as a real helium nucleus, the pool table system becomes more stable when some of its constituents lose energy and less stable when they gain energy.


Reference

Beiser, A. (1987). Concepts of modern physics. McGraw-Hill Companies.

A Gnome-inal Value for ‘g’

The Gnome Experiment Kit from precision scale manufacturers Kern and Sohn.

. . . setting storms and billows at defiance, and visiting the remotest parts of the terraqueous globe.

Samuel Johnson, The Rambler, 17 April 1750

That an object in free fall will accelerate towards the centre of our terraqueous globe at a rate of 9.81 metres per second per second is, at best, only a partial and parochial truth. It is 9.81 metres per second per second in the United Kingdom, yes; but the value of both acceleration due to free fall and the gravitational field strength vary from place to place across the globe (and in the SI System of measurement, the two quantities are numerically equal and dimensionally equivalent).

For example, according to Hirt et al. (2013) the lowest value for g on the Earth’s surface is atop Mount Huascarán in Peru where g = 9.7639 m s-2 and the highest is at the surface of the Arctic Ocean where g = 9.8337 m s-2.

Why does g vary?

There are three factors which can affect the local value of g.

Firstly, the distribution of mass within the volume of the Earth. The Earth is not of uniform density and volumes of rock within the crust of especially high or low density could affect g at the surface. The density of the rocks comprising the Earth’s crust varies between 2.6 – 2.9 g/cm3 (according to Jones 2007). This is a variation of 10% but the crust only comprises about 1.6% of the Earth’s mass since the density of material in the mantle and core is far higher so the variation in g due this factor is probably of the order of 0.2%.

Secondly, the Earth is not a perfect sphere but rather an oblate spheroid that bulges at the equator so that the equatorial radius is 6378 km but the polar radius is 6357 km. This is a variation of 0.33% but since the gravitational force is proportional to 1/r2 let’s assume that this accounts for a possible variation of the order of 0.7% in the value of g.

Thirdly, the acceleration due to the rotation of the Earth. We will look in detail at the theory underlying this in a moment, but from our rough and ready calculations above, it would seem that this is the major factor accounting for any variation in g: that is to say, g is a minimum at the equator and a maximum at the poles because of the Earth’s rotation.


The Gnome Experiment

In 2012, precision scale manufacturers Kern and Sohn used this well-known variation in the value of g to embark on a highly successful advertising campaign they called the ‘Gnome Experiment’ (see link 1 and link 2).

Whatever units their lying LCD displays show, electronic scales don’t measure mass or even weight: they actually measure the reaction force the scales exert on the item in their top pan. The reading will be affected if the scales are accelerating.

In diagram A, the apple is not accelerating so the resultant upward force on the apple is exactly 0.981 N. The scales show a reading of 0.981/9.81 = 0.100 000 kg = 100.000 g (assuming, of course, that they are calibrated for use in the UK).

In diagram B, the apple and scales are in an elevator that is accelerating upward at 1.00 metres per second per second. The resultant upward force must therefore be larger than the downward weight as shown in the free body diagram. The scales show a reading of 1.081/9.81 – 0.110 194 kg = 110.194 g.

In diagram C, the the apple and scales are in an elevator that is accelerating downwards at 1.00 metres per second per second. The resultant upward force must therefore be smaller than the downward weight as shown in the free body diagram. The scales show a reading of 0.881/9.81 – 0.089 806 kg = 89.806 g.


Never mind the weight, feel the acceleration

Now let’s look at the situation the Kern gnome mentioned above. The gnome was measured to have a ‘mass’ (or ‘reaction force’ calibrated in grams, really) of 309.82 g at the South Pole.

Showing this situation on a diagram:

Looking at the free body diagram for Kern the Gnome at the equator, we see that his reaction force must be less than his weight in order to produce the required centripetal acceleration towards the centre of the Earth. Assuming the scales are calibrated for the UK this would predict a reading on the scales of 3.029/9.81= 0.30875 kg = 308.75 g.

The actual value recorded at the equator during the Gnome Experiment was 307.86 g, a discrepancy of 0.3% which would suggest a contribution from one or both of the first two factors affecting g as discussed at the beginning of this post.

Although the work of Hirt et al. (2013) may seem the definitive scientific word on the gravitational environment close to the Earth’s surface, there is great value in taking measurements that are perhaps more directly understandable to check our comprehension: and that I think explains the emotional resonance that many felt in response to the Kern Gnome Experiment. There is a role for the ‘artificer’ as well as the ‘philosopher’ in the scientific enterprise on which humanity has embarked, but perhaps Samuel Johnson put it more eloquently:

The philosopher may very justly be delighted with the extent of his views, the artificer with the readiness of his hands; but let the one remember, that, without mechanical performances, refined speculation is an empty dream, and the other, that, without theoretical reasoning, dexterity is little more than a brute instinct.

Samuel Johnson, The Rambler, 17 April 1750

References

Hirt, C., Claessens, S., Fecher, T., Kuhn, M., Pail, R., & Rexer, M. (2013). New ultrahigh‐resolution picture of Earth’s gravity fieldGeophysical research letters40(16), 4279-4283.

Jones, F. (2007). Geophysics Foundations: Physical Properties: Density. University of British Columbia website, accessed on 2/5/21.

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Nature Abhors A Change In Flux

Aristotle memorably said that Nature abhors a vacuum: in other words. he thought that a region of space entirely devoid of matter, including air, was logically impossible.

Aristotle turned out to be wrong in that regard, as he was in numerous others (but not quite as many as we – secure and perhaps a little complacent and arrogant as we look down our noses at him from our modern scientific perspective – often like to pretend).

An amusing version which is perhaps more consistent with our current scientific understanding was penned by D. J. Griffiths (2013) when he wrote: Nature abhors a change in flux.

Magnetic flux (represented by the Greek letter phi, Φ) is a useful quantity that takes account of both the strength of the magnetic field and its extent. It is the total ‘magnetic flow’ passing through a given area. You can also think of it as the number of magnetic field lines multiplied by the area they pass through so a strong magnetic field confined to a small area might have the same flux (or ‘effect’) as weaker field spread out over a large area.


Lenz’s Law

Emil Lenz formulated an earlier statement of the Nature abhors a change of flux principle when he stated what I think is the most consistently underrated laws of electromagnetism, at least in terms of developing students’ understanding:

The current induced in a circuit due to a change in a magnetic field is directed to oppose the change in flux and to exert a mechanical force which opposes the motion.

Lenz’s Law (1834)

This is a qualitative rather than a quantitive law since it is about the direction, not the magnitude, of an induced current. Let’s look at its application in the familiar A-level Physics context of dropping a bar magnet through a coil of wire.


Dropping a magnet through a coil in pictures

Picture 1

In picture 1 above, the magnet is approaching the coil with a small velocity v. The magnet is too far away from the coil to produce any magnetic flux in the centre of the coil. (For more on the handy convention I have used to draw the coils and show the current flow, please click on this link.) Since there is no magnetic flux, or more to the point, no change in magnetic flux, then by Faraday’s Law of Electromagnetic Induction there is no induced current in the coil.

Picture 2

in picture 2, the magnet has accelerated to a higher velocity v due to the effect of gravitational force. The magnet is now close enough so that it produces a magnetic flux inside the coil. More to the point, there is an increase in the magnetic flux as the magnet gets closer to the coil: by Faraday’s Law, this produces an induced current in the coil (shown using the dot and cross convention).

To ascertain the direction of the current flow in the coil we can use Lenz’s Law which states that the current will flow in such a way so as to oppose the change in flux producing it. The red circles show the magnetic field lines produced by the induced current. These are in the opposite direction to the purple field lines produced by the bar magnet (highlighted yellow on diagram 2): in effect, they are attempting to cancel out the magnetic flux which produce them!

The direction of current flow in the coil will produce a temporary north magnetic pole at the top of the coil which, of course, will attempt to repel the falling magnet; this is ‘mechanical force which opposes the motion’ mentioned in Lenz’s Law. The upward magnetic force on the falling magnet will make it accelerate downward at a rate less than g as it approaches the coil.

Picture 3

In picture 3, the purple magnetic field lines within the volume of the coil are approximately parallel so that there will be no change of flux while the magnet is in this approximate position. In other words, the number of field lines passing through the cross-sectional area of the coil will be approximately constant. Using Faraday’s Law, there will be no flow of induced current. Since there is no change in flux to oppose, Lenz’s Law does not apply. The magnet will be accelerating downwards at g.

Picture 4

As the magnet emerges from the bottom of the coil, the magnetic flux through the coil decreases. This results in a flow of induced current as per Faraday’s Law. The direction of induced current flow will be as shown so that the red field lines are in the same direction as the purple field lines; Lenz’s Law is now working to oppose the reduction of magnetic flux through the coil!

A temporary north magnetic pole is generated by the induced current at the lower end of the coil. This will produce an upward magnetic force on the falling magnet so that it accelerates downward at a rate less than g. This, again, is the ‘mechanical force which opposes the motion’ mentioned in Lenz’s Law.


Dropping a magnet through a coil in graphical form

This would be one of my desert island graphs since it is such a powerfully concise summary of some beautiful physics.

The graph shows the reversal in the direction of the current as discussed above. Also, the maximum induced emf in region 2 (blue line) is less than that in region 4 (red line) since the magnet is moving more slowly.

What is more, from Faraday’s Law (where ℇ is the induced emf and N is total number of turns of the coil), the blue area is equal to the red area since:

and N and ∆Φ are fixed values for a given coil and bar magnet.

As I said previously, there is so much fascinating physics in this graph that I think it worth exploring in depth with your A level Physics students 🙂

Other news

If you have enjoyed this post, then you may be interested to know that I’ve written a book! Cracking Key Concepts in Secondary Science (co-authored with Adam Boxer and Heena Dave) is due to be published by Corwin in July 2021.

References

Lenz, E. (1834), “Ueber die Bestimmung der Richtung der durch elektodynamische Vertheilung erregten galvanischen Ströme”, Annalen der Physik und Chemie107 (31), pp. 483–494

Griffiths, David (2013). Introduction to Electrodynamics. p. 315.