A potential divider circuit is, essentially, a circuit where two or more components are arranged in series.

For non-physicists, these types of circuit can sometimes present problems, so in this post I am going to look in detail at the basic physics involved; and I am going to explain them using the CTM or Coulomb Train Model. (You can find the CTM model explained here.)

In the AQA GCSE Physics (and Combined Science) specifications, students are required to know that:

First, let’s look at the basics of describing electric circuits: current, potential difference and resistance.

**1.0 Using the CTM to explain current, potential difference and resistance**

Pupils tend to start with one concept for electricity in a direct current circuit: a concept labelled ‘current’, or ‘energy’ or ‘electricity’, all interchangeable and having the properties of movement, storability and consumption. Understanding an electrical circuit involvesfirst differentiating the concepts of current, voltage and energybeforerelating them as a system, in which the energy transfer depends upon current, time and the potential difference of the battery.

The notion of current flowing in the circuit is one which pupils often meet in their introduction to a circuit and, because this relates well with their intuitive notions,this concept becomes the primary concept.(Driver 1994: 124 [italics added])

To my mind, the CTM is an excellent “bridging analogy” that helps students visualise the invisible. It is a stepping stone that provides some concrete representations of abstract quantities. In my opinion, it can help students

- move away from analysing circuits in terms of
*just*current. (In my experience, even when students use terms like “potential difference”, in their eyes what they call “potential difference” behaves in a remarkably similar way to current e.g. it “flows through” components.) - understand the difference between current, potential difference and resistance and how important each one is
- begin thinking of a circuit as a whole, interconnected system.

**1.1 The CTM and electric current**

Let’s begin by looking at a very simple circuit: a one ohm resistor connected across a 1 V cell.

Note that it is a good teaching technique to include two ammeters on either side of the component, although the readings on both will be identical. This is to challenge the perennial misconception that electric current is “used up”. Electric charge, according to our current understanding of the universe, is a *conserved quantity* like energy in that it cannot be created or destroyed.

The Coulomb Train Model invites us to picture an electric circuit as a flow of positively charged *coulombs *carrying energy around the circuit in a clockwise fashion as shown below. The coulombs are linked together to form a continuous chain.

The name coulomb is not chosen at random: it is the SI unit of electric charge.

The current in this circuit will be given by *I = V / R* (equation 18 in the list on p.96 of the AQA spec, if you’re keeping track).

Using the AQA mark scheme-friendly FIFA protocol:

The otherwise inexplicable use of the letter “*I*” to represent electric current springs from the work André-Marie Ampère (1775–1836) and the French phrase *intensité de courant* (intensity of current).

From *Q = I t* (equation 17, p.96), current is a flow of electric charge, since *I = Q / t*. That is to say, if a charge of 2 coulombs passes (AQA call this a “charge flow”) in 2 seconds, the current will be …

A current of 1 amp is therefore represented on the CTM as 1 coulomb (or truck) passing by each second.

**1.2 The CTM and Potential Difference**

Potential difference or voltage is essentially the “energy difference” across any two parts of a circuit.

The equation used to define potential difference is not the familiar *V = IR* but rather the less familiar *E = QV* (equation 22 in the AQA list) where *E* is the energy transferred, *Q* is the charge flow (or the number of coulombs passing by in a certain time) and *t* is the time in seconds.

Let’s see what this would look like using the CTM:

For the circuit shown, the voltmeter reading is 1 volt.

Note that on the CTM representation, one joule of energy is *added* to each coulomb as it passes through the cell.

If we had a 1.5 V cell then 1.5 joules would be transferred to each coulomb as it passed through, and so on.

If the voltmeter is moved to a different position as shown above, then the reading is 0 volts. This is because the coulombs at the points “sampled” by the voltmeter have the same amount of energy, so there is zero energy difference between them.

In the position shown above, the voltmeter is measuring the potential difference across the resistor. For the circuit shown (assuming negligible resistance in all other parts of the circuit) the potential difference will be 1 V. In other words, each coulomb is losing one joule of energy as it passes through the resistance.

**1.3 The CTM and Resistance**

In the circuit above, the potential difference across the resistor is 1 V and the current is 1 amp.

Resistance can therefore be thought of as the *potential difference required to drive a current of 1 amp* through that part of the circuit. It can also be thought of as the *energy lost by each coulomb when a current of 1 amp flows* through that part of the circuit; or, e*nergy lost per coulomb per amp*.

**1.4 Summary**

On the diagrams below, the coulombs are moving clockwise.

**2.0 The CTM applied to a potential divider circuit**

A potential divider circuit simply means that at least two resistors are in series so that the potential difference of the cell is shared across the resistors.

**2.1 Two identical resistors**

Now let’s use *V = IR* to check that the potential difference across each separate resistor is indeed half the total supply of 3 V. The resistance of one resistor is one ohm and the current through each one is 1.5 A. So V = 1.5 x 1 = 1.5 V.

But what would happen if we doubled the value of each resistor to 2 ohms?

Well, the current would be smaller: *I = V/R* = 3/4 = 0.75 amps.

The potential difference across each separate resistor would be *V = I R* = 0.75 x 2 = 1.5 V

So, the potential difference is always split equally when two identical resistors are placed in series (although, of course, the total resistance and the current will be different depending on the values of the resistors).

**2.2a Two non-identical resistors**

Let’s consider a circuit with a 2 ohm resistor in series with a 1 ohm resistor.

In this circuit, the total resistance is 1 ohm + 2 ohms = 3 ohms. The current flowing through the circuit is *I = V / R* = 3 / 3 = 1 amp.

So the potential difference across the 2 ohm resistor is *V = IR* = 1 x 2 = 2 V and the potential difference across the one ohm resistor is *V = IR* = 1 x 1 = 1 V.

Note that the resistor with the largest value gets the largest “share” of the potential difference.

**2.2b Two non-identical resistors (different order)**

Now let’s reverse the order of the resistors.

The current remains unchanged because the total resistance of the circuit is still the same.

Note that the largest resistor still gets the largest share of the potential difference, whichever way round the resistors are placed.**2.3 In Defence of the CTM and Donation Models**

Many Physics teachers prefer “rope models” to so-called “donation models” like the CTM.

And it is perfectly true that rope models have some good points such as the ability to easily explain AC and a more accurate approximation of what happens when current starts to flow or stops flowing. The difficulty in their use, in my opinion, is that you are using concepts that many students barely understand (e.g. friction to model resistance) to explain how very unfamiliar concepts such as potential difference work. Also, the vagueness of some of the analogs is unhelpful: for example, when we compare potential difference to “push”, are we talking about the net resultant force on the rope or simply the force needed to balance the frictional force and keep it moving at a steady speed?

To my way of thinking, the CTM has the advantage of encouraging quantitative thinking about current, potential difference and resistance almost from the moment of first teaching. Admittedly, it cannot cope with AC — but then again, we model AC as a *direct *current when we use RMS values. Now admittedly, rope models are far better at picturing what happens in the initial fractions of a second when a current starts to flow after closing a switch. Be that as it may, the CTM comes into its own when we consider the “steady state” of current flow after the initial surge currents.

One of the frequent criticisms (which is usually considered quite damning) of this type of model is “How do the coulombs know how much energy to drop off at each resistor?”

For example, in the diagram above, how do the coulombs “know” to drop off 1 J at the first resistor and 2 J at the second resistor?

The answer is: they don’t. Rather, the energy loss is due to the nature of the resistor: think of a resistor as a tunnel lined with strip curtains. A coulomb loses only a small amount of its excess energy passing through a low value resistor, but a much larger amount passing through a higher value resistor, as modelled below.

FWIW I therefore commend the use of the CTM to all interested parties.

__References__

Driver, R., Squires, A., Rushworth, P., & Wood-Robinson, V. (1994). *Making sense of secondary science: Research into children’s ideas*. Routledge.

The AQA GCSE Required Practical on Acceleration (see pp. 21-22 and pp. 55-57) has proved to be problematic for many teachers, especially those who do not have access to a working set of light gates and data logging equipment.

In version 3.8 of the Practical Handbook (pre-March 2018), AQA advised using the following equipment featuring a linear air track (LAT). The “vacuum cleaner set to blow”, (or more likely, a specialised LAT blower), creates a cushion of air that minimises friction between the glider and track.

However, in version 5.0 (dated March 2018) of the handbook, AQA put forward a very different method where schools were advised to video the motion of the car using a smartphone in an effort to obtain precise timings at the 20 cm, 40 cm and other marks.

It is possible that AQA published the revised version in response to a number of schools contacting them to say. “We don’t have a linear air track. Or light gates. Or a ‘vacuum cleaner set to blow’.”

The weakness of the “new” version (at least in my opinion) is that it is *not quantitative*: the method suggested merely records the times at which the toy car passed the lines. Many students may well be able to *indirectly deduce* the relationship between resultant force and acceleration from this raw timing data; but, to my mind, it would be cognitively less demanding if they were able to compare *measurements* of resultant force and acceleration instead.

We simplify the AQA method as above: we simply time how long the toy car takes to complete the whole journey from start to finish.

If a runway of one metre or longer is set up, then the total time for the journey of the toy car will be 20 seconds or so for the smallest accelerating weight: this makes manual timing perfectly feasible.

Important note: the length of the runway will be limited by the height of the bench. As soon as the weight stack hits the floor, the toy car will no longer experience an accelerating force and, while it may continue at a constant speed (more or less!) it will no longer be accelerating. In practice, the best way to sort this out is to pull the toy car back so that the weight stack is just below the pulley and mark this position as the start line; then slowly move the toy car forward until the weight stack is just touching the floor, and mark this position as the finish line. Measure the distance between the two lines and this is the length of your runway.

In addition, the weight stack should feature very small masses; that is to say, if you use 100 g masses then the toy car will accelerate very quickly and manual timing will prove to be impossible. In practice, we found that adding small metal washers to an improvised hook made from a paper clip worked well. We found the average mass of the washers by placing ten of them on a scale.

Then input the data into this spreadsheet (click the link to download from Google Drive) and the software should do the rest (including plotting the graph!).

To confirm the straight line and directly proportional relationship between accelerating force and acceleration, bear in mind that the *total mass of the accelerating system must remain constant *in order for it to be a “fair test”.

The parts of our system that are accelerating are the toy car, the string and the weight stack. The total mass of the accelerating system shown below is 461 g (assuming the mass of the hook and the string are negligible).

The accelerating (or resultant) force is the weight of 0.2 g mass on the hook, which0 can be calculated using W = mg and will be equal to 0.00196 N or 1.96 mN.

In the second diagram, we have increased the mass on the weight stack to 0.4 g (and the accelerating force to 0.00392 N or 3.92 mN) but note that the total mass of the accelerating system is still the same at 461 g.

In practice, we found that using blu-tac to stick a matchbox tray to the roof of the car made managing and transferring the weight stack easier.

Personal note: as a beginning teacher, I demonstrated the linear air track version of this experiment to an A-level Physics class and ended up *disconfirming* Newton’s Second Law instead of confirming it; I was both embarrassed and immensely puzzled until an older, wiser colleague pointed out that the variables had been well and truly confounded by *not* keeping the total mass of the accelerating system constant.

It was embarrassing and that’s why I always harp on about this aspect of the experiment.

This can be considered as “deep background” rather than necessary information, but I, for one, consider it really interesting.

Acceleration is the rate of change of a rate of change. Velocity is the rate of change of displacement with time and acceleration is the rate of change of velocity.

Interested individuals may care to delve into higher derivatives like *jerk*, *snap*, *crackle* and *pop* (I kid you not — these are the technical terms). *Jerk* is the rate of change of acceleration and hence can be defined as (takes a deep breath) the rate of change of a rate of change of rate of change. More can be found in the fascinating article by Eager, Pendrill and Reistad (2016) linked to above.

But on a much more prosaic level, acceleration can be defined as *a = (v – u) / t* where *v* is the final instantaneous velocity, *u* is the inital instantaneous velocity and *t* is the time taken for the change.

The instantaneous velocity is the velocity at a momentary instant of time. It is, if you like, the velocity indicated by the needle on a speedometer at a single instant of time and is different from the average velocity which is calculated from the total distance travelled divided the time taken.

This can be shown in diagram form like this:

However, our experiment is simplified because we made sure that the toy car was *stationary* when the timer was zero; in other words, we ensured *u* = 0 m/s.

This simplifies *a = (v – u) / t* to *a = v / t.*

But how can we find *v*, the instantaneous velocity at the end of the journey when we have no direct means of measuring it, such as a speedometer or a light gate?

Let’s assume that, for the toy car, the jerk is zero (again, let me emphasize that *jerk* is a technical term defined as the rate of change of acceleration).

This means that the acceleration is constant.

This fact allows us to calculate the average velocity using a very simple formula: average velocity = (*u* + *v*) / *t .*

But remember that *u* = 0 so average velocity = *v* / 2 .

More pertinently for us, provided that *u* = 0 and jerk = 0, it allows us to calculate a value for *v* using *v* = 2 x (average velocity) .

The spreadsheet linked to above uses this formula to calculate *v* and then uses *a = v / t*.

This could be done as a demonstration or, since only basic equipment is needed, a class experiment. Students may need access to computers running the spreadsheet during the experiment or soon afterwards. We found that one laptop shared between two groups was sufficient.

** First experiment** (relationship between force and acceleration): set up as shown in the diagram. Place washers totalling a mass of 0.8 g (or similar) and washers totalling a mass of 0.2 g on the hook or weight stack. Hold the toy car stationary at the start line. Release and start the timer. Stop the timer. Input data into the spreadsheet and repeat with different mass on the hook.

It can be useful to get students to manually “check” the value of a calculated by the spreadsheet to provide low stakes practice of using the acceleration formula.

** Second experiment** (relationship between mass and acceleration). Keep the accelerating force constant with (say) 0.6 g on the hook or weight stack. Hold the toy car stationary at the start line. Release and start the timer. Stop the timer. Input data into the second tab on the spreadsheet and repeat with 100 g added to the toy car (possibly blu-tac’ed into place).

This blog post grew in the telling. Please let me know if you try the methods outlined here and how successful you found them

Eager, D., Pendrill, A. M., & Reistad, N. (2016). Beyond velocity and acceleration: jerk, snap and higher derivatives. *European Journal of Physics*, *37*(6), 065008.

My own understanding of this topic has undergone some changes thanks to some fascinating (and ongoing) discussions via EduTwitter.

What I suggest is that we look at the phenomena in question through two lenses:

- a macroscopic lens, where we focus on things we can sense and measure directly in the laboratory
- a microscopic lens, where we focus on using the particle model to explain phase changes such as melting and freezing.

The ** enojis** for thermal energy stores (as suggested by the Institute of Physics) look like this (Note: ‘enoji’ = ‘energy’ + ’emoji’; and that the IoP do not use the term):

In many ways, they are an excellent representation. Firstly, energy is represented as a “quasi-material entity” in the form of an orange liquid which can be shifted between stores, so the enoji on the left could represent an aluminium block *before* it is heated, and the one on the right *after* it is heated. Secondly, it also attempts to make clear that the so-called *forms of energy* are labels added for human convenience and that energy is the same basic “stuff” whether it is in the thermal energy store or the kinetic energy store. Thirdly, it makes the link between kinetic theory and thermal energy stores explicit: the particles in a hot object are moving faster than the particles in the colder object.

However, I think the third point is not necessarily an advantage as I believe it will muddy the conceptual waters when it comes to talking about internal energy later on.

If I was a graphic designer working for the IoP these are the enojis I would present:

In other words, a change in the thermal energy store is always associated with a temperature change. To increase the temperature of an object, we need to shift energy into the thermal energy store. To cool an object, energy needs to be shifted out of the thermal energy store.

This has the advantage of focusing on the directly observable macroscopic properties of the system and is, I think, broadly in line with the approach suggested by the AQA specification.

Internal energy is the “hidden” energy of an object.

The “visible” energies associated with an object would include its kinetic energy store if it is moving, and its gravitational potential energy store if it is lifted above ground level. But there is also a deeper, macroscopically-invisible store of energy associated with the particles of which the object is composed.

To understand internal energy, we have to look through our microscopic lens.

The *Oxford Dictionary of Physics* (2015) defines internal energy as:

The total of the kinetic energies of the atoms and molecules of which a system consists and the potential energies associated with their mutual interactions. It does not include the kinetic and potential energies of the system as a whole nor their nuclear energies or other intra-atomic energies.

In other words, we can equate the internal energy to the sum of the kinetic energy of each individual particle added to the sum of the potential energy due to the forces between each particle. In the simple model below, the intermolecular forces between each particle are modelled as springs, so the potential energy can be thought as stretching and squashing the “springs”. (Note: try not to talk about “bonds” in this context as it annoys the hell out of chemists, some of whom have been known to kick like a mule when provoked!)

We can never measure or calculate the value of the absolute internal energy of a system in a particular state since energy will be shifting from kinetic energy stores to potential energy stores and vice versa moment-by-moment. What is a useful and significant quantity is the *change in the internal energy, *particularly when we are considering phase changes such as solid to liquid and so on.

This means that internal energy is * not* synonymous with thermal energy; rather, the thermal energy of a system can be taken as being a

As Rod Nave (2000) points out in his excellent web resource Hyperphysics, what we think of as the thermal energy store of a system (i.e. the sum of the translational kinetic energies of small point-like particles), is often an extremely small part of the total internal energy of the system.

My excellent Edu-tweeting colleague @PhysicsUK has pointed out that there is indeed a discrepancy between the equations presented by AQA in their specification and on the student equation sheet.

If a change in thermal energy is always associated with a change in temperature (macroscopic lens) then we should not use the term to describe the energy change associated with a change of state when there is no temperature change (microscopic lens).

@PhysicsUK reports that AQA have ‘fessed up to the mistake and intend to correct it in the near future. Sooner would be better than later, please, AQA!

**References**

Nave, R. (2000). *HyperPhysics*. Georgia State University, Department of Physics and Astronomy.

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“It * is* cheese. (Caerphilly.)”

Main research finding of the first manned Welsh mission to the Moon, as reported by Max Boyce c. 1974.

When writing *A Brief History Of Time*, it is said that a literary agent warned Stephen Hawking that each mathematical equation that he included in the final draft would halve its eventual sales.

And one of the complex equations that Hawking wished to include? A summary of Newton’s 2nd Law: *F=ma.*

In other words, the force *F* acting on an object is equal to the mass *m* of the object multiplied by the acceleration experienced by the object.

(In the end, Hawking opted to include only Einstein’s *E=mc** ^{2}* in what turned out to be the ultra bestselling

@SciByDegrees wrote an interesting post arguing that the old Physics teacher’s standby of summarising Newton’s Second Law of Motion (N2) as *F=ma* is wrong.

The gist of his argument (and it’s a hard argument to counter) is that *F=d(mv)/dt* is a far better expression of the law than the *F=ma* version because it covers a wider range of circumstances.

This states N2 in terms of momentum, where momentum is the product of mass m multiplied by the velocity v. More exactly, it says that force acting on an object is equal to the object’s rate of change of momentum: or, if you prefer, the change in momentum divided by the time taken for the change is equal to the force.

This is the version of N2 stated in most dictionaries of Physics. For example, the *Oxford Dictionary of Physics* (2015) p. 383.

I know, because on reading @SciByDegrees’ post I immediately looked up N2 in the *Dictionary* with the express intention of countering the argument. Imagine my consternation and horror when I found that I was wrong. (Actually, not that much consternation and horror: I am fairly inured to being wrong as it happens fairly often…)

The argument suggests that, just like *V=IR* is not a statement of Ohm’s Law unless R has a fixed value (like a fixed length of wire at a constant temperature), F=ma is not a sufficient statement of N2 unless the mass *m* is constant.

For example, if we consider a rocket capable of producing a steady 1000 N of thrust; at *t*=0 its mass is (say) 10 kg so its acceleration is 100 m/s^{2}. However at *t*=1 s its mass has decreased by 1 kg so the acceleration is now 111 m/s^{2} even though the thrust is still 1000 N so obviously *F* is not proportional to *m* so *F* does not equal *ma* in this situation.

Richard Feynman (1965) wrote along similar lines in his justly famous *Lectures on Physics*:

Thus at the beginning we take several things for granted. First, that the mass of an object is

constant; it isn’t really, but we shall start out with the Newtonian approximation that mass is constant, the same all the time, and that, further, when we put two objects together, their massesadd. These ideas were of course implied by Newton when he wrote his equation, for otherwise it is meaningless. For example, suppose the mass varied inversely as the velocity; then the momentum wouldnever changein any circumstance, so the law means nothing unless you know how the mass changes with velocity. At first we say,it does not change.

However, I think Feynman is considerably oversimplifying what Newton said here. Dare one suppose that Feynman, who had an enviable natural facility for talking intelligently and arrestingly about nearly any subject under the Sun, had perhaps skimped a little on his background reading?

Incidentally, does anyone think that physicists (especially physics educators — myself included) are beginning to treat Feynman as the medieval scholastics are reputed to have treated Aristotle? That is to say, he is regarded as the final word on everything; or, at least, everything physics-related in the case of Feynman.

George Smith (2008) points out that:

The modern

F=maform of Newton’s second law nowhere occurs in any edition of thePrincipia[ . . . ] Instead, it has the following formulation in all three editions:A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed. In the body of thePrincipiathis law is applied both to discrete cases, in which an instantaneous impulse such as from impact is effecting the change in motion, and to continuously acting cases, such as the change in motion in the continuous deceleration of a body moving in a resisting medium. Newton thus appears to have intended his second law to be neutral between discrete forces (that is, what we now call impulses) and continuous forces.

This, I think, supports my contention that *F=ma* is as good a modern reformulation of Newton’s 2nd Law as any other.

If we go back to the rocket example, the instantaneous acceleration at *t*=0 and *t*=1 s can be calculated using *F=ma* (provided we take account of the change in *m*, of course). In effect, we are considering the change in motion due an instantaneous impulse here.

Please note that I would cheerfully concede that *F=d(mv)/dt* would yield a better and more productive analysis of rocket motion if we are considering the *continuous* action of the force over time rather than at isolated instants.

The analogy with V=IR is useful here. *V* is always equal to *I* times *R* but *V* is only directly proportional to *I* over a continuous range of values of *I* for a limited set of conductors we call Ohmic conductors whose resistance *R* is fixed over a range of physical conditions. Likewise, *F* is always equal to *m* times *a* but *F* is only directly proportional to *a* for a continuous range of values of *a* when we are considering a system whose mass is fixed.

As *V=IR* is neutral with respect to whether R is fixed is not, I believe that *F=ma* is neutral with respect to whether *m* is fixed or not.

What is Newton’s Second Law? Is it a definition of force? Is it a definition of mass? Or is it an empirical proposition linking force, mass and acceleration?

Brian Ellis (1965) argues that it partakes of all three:

Consider how Newton’s second law is actually used. In some fields it is unquestionably true that Newton’s second law is used to define a scale of force. How else, for example, can we measure interplanetary gravitational forces? But it is also unquestionably true that Newton’s second law is sometimes used to define a scale of mass. Consider, for example, the use of mass spectrography. And in yet other fields, where force, mass and acceleration are all easily and independently measurable, Newton’s second law of motion functions as an empirical correlation between these three quantities. Consider, for example, the application of Newton’s second law in ballistics and rocketry [ . . .] To suppose that Newton’s second law of motion, or any law for that matter, must have a unique role that we can describe generally and call the logical status is an unfounded and unjustifiable supposition.

In some senses, I suppose we might like those unfortunate nations in *Gulliver’s Travels* who fought a long and bitter war over the question of whether one should eat a boiled egg from the pointed or rounded end:

During the course of these troubles, the emperors of Blefusca … accusing us of making a schism in religion, by offending against a fundamental doctrine of our great prophet Lustrog, in the fifty-fourth chapter of the Blundecral (which is their Alcoran). This, however, is thought to be a mere strain upon the text; for the words are these: ‘that all true believers break their eggs at the convenient end.’ And which is the convenient end, seems, in my humble opinion to be left to every man’s conscience.

*REFERENCES*

Ellis, Brian. “The origin and nature of Newton’s laws of motion.” *Beyond the edge of certainty* (1965): 29-68.

Feynman, R. P., Leighton, R. B., & Sands, M. (1965). The Feynman Lectures on Physics; vol. 1 (Accessed from http://www.feynmanlectures.caltech.edu/I_09.html on 11/4/19)

Smith, George, “Newton’s *Philosophiae Naturalis Principia Mathematica*“, *The Stanford Encyclopedia of Philosophy *(Winter 2008 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/win2008/entries/newton-principia/>.ewto

And if two men ride of a horse, one must ride behind.

*— *Shakespeare*, Much Ado About Nothing*

Sir Isaac Newton stands in popular estimation as the foremost intellect of his age; or perhaps, of *any* age. If a person is never truly dead while their name is spoken, then Sir Isaac stands with us still: partially overshadowed by Einstein at the dawn of the twentieth century, maybe, but never totally eclipsed.

But in the roiling intellectual cauldron of the Age of Enlightenment, even such a venerable polymath as Newton had some serious competition. As Newton himself modestly observed in a letter to a contemporary in 1676: “If I have seen a little further it is by standing on the shoulders of Giants.”

Except that one interpretation has it that the letter was not intended to be modest, but was rather a combative dig at the man to whom it was addressed: Robert Hooke, a man of but “middling” stature and, as a result of a childhood illness, also a hunchback. Not one of the “Giants” with broad philosophic shoulders to whom Newton felt indebted to, then.

In popular estimation, therefore, it would appear that Hooke is fated always to sit behind Newton. At GCSE and A-level, students learn of Newton’s Laws of Motion, the eponymous unit of force, and his Law of Universal Gravitation.

And what do they learn of Hooke? They learn of his work on springs. They learn of “Hooke’s Law”: that is, the force exerted by a spring is directly proportional to its extension.

Ut tensio, sic vis.[As extension, so is the force.]

— Robert Hooke,

Lectures de Potentia Restituvia[1678]

Newton has all the laws of motion on Earth and in Heaven in the palm of his hand, and Hooke has springs. Perhaps, then, Hooke deserves to be forever second on the horse of eternal fame?

But look closer. To what objects or classes of object can we apply Hooke’s Law? The answer is: *all of them*.

Hooke’s Law applies to everything solid: muscle, bone, sinew, concrete, wood, ice, crystal and stone. Stretch them or squash them, and they will deform in exact proportion to the size of the force applied to them.

That is, if one power stretch or bend it one space, two will bend it two, and three will bend it three, and so forward.

The major point being that Hooke’s Law is as universal as gravity: it is baked into the very fabric of the universe: it is a direct consequence of the interactions between atoms.

Now before I wax too lyrical, it must be pointed out that Hooke’s Law is a first-order linear approximation: it fails when the deforming force increases beyond a certain limit, and that limit is unique to each material. But within the limits of its domain indicated by the red circle above, it reigns supreme.

How do you calculate how much a steel beam will bow when a kitten walks across it? Hooke’s Law. How could we model the stresses on the bones of a galloping dinosaur? Hooke’s Law. How can we calculate how much Mount Everest bends when it is buffeted by wind? Hooke’s Law.

Time to re-evaluate the seating order on Shakespeare’s horse, mayhap?

]]>Students and non-specialist teachers alike wonder: whence the * half*?

This post is intended to be a diagrammatic answer to this question using a Singapore Bar Model approach: so pedants, please avert your eyes.

I am indebted to Ben Rogers’ recent excellent post on showing momentum using the Bar Model approach for starting me thinking along these lines.

Imagine pushing an object with a mass ** m** with a constant force

(On the diagram, I’ve used the SUVAT dual coding conventions that I suggested in a previous post.)

So let’s consider the work done on the object by the force:

Step 1: work done = force x distance moved in the direction of the force

Step 2:* W_{d }*=

But remember ** s** =

Step 3:* W_{d }*=

And also remember that ** F** =

Step 4: * W_{d }*=

Also remember that a = change in velocity / time, so ** a** = (

Step 5: * W_{d }*=

The * t*s cancel so:

Step 6: * W_{d }*=

Since this is the work done *on* the object by the force, it is equal to the energy transferred to the kinetic energy store of the object. In other words, it is the energy the object has gained *because* it is moving — its kinetic energy, no less: * E_{k }= mv^{2}*.

On a Singapore Bar Model diagram this can be represented as follows:

The kinetic energy is represented by the *volume* of the bar.

But wait: * E_{k}=mv^{2}*!?!?

That’s just wrong: where did the half go?

Houston, we have a problem.

The problem lies with Step 3 above. We *wrongly**assumed* that the object has a constant velocity over the whole of the distance ** s**.

It doesn’t because it is accelerating: it starts off moving slowly and ends up moving at the maximum, final velocity ** v** when it has travelled the total distance

So Step 3 should read:

But remember that ** s** = (average velocity) x

Because the object is accelerating at a constant rate, the average velocity is (* v* +

Step 3:* W_{d}*=

And also remember that ** F** =

Step 4: * W_{d}*=

Also remember that a = change in velocity / time, so ** a** = (

Step 5: * W_{d }*=

The * t*s cancel so:

Step 6: * W_{d}*=

Based on this, of course, *E _{k }= ½mv^{2}*

(Phew! Houston, we no longer have a problem.)

Using the Bar Model representation, the volume of the bar which is above the blue plane represents the kinetic energy of an object of mass ** m** moving at a velocity

The reason it is ** half the volume** of the bar and not the full volume (as in the incorrect Part the First analysis) is because we are considering the work done by a constant force accelerating an object which is initially at rest; the velocity of the object increases gradually from zero as the force acts upon it. It therefore takes a longer time to cover the distance

So there we have it, ** E_{k }= ½mv^{2 }**by a rather circuitous method.

But why go “all around the houses” in this manner? For exactly the same reason as we might choose to go by the path less travelled on some of our other journeys: quite simply, we might find that we enjoy the view.

]]>Many years ago, I was taught the dual coding technique outlined below to help with SUVAT problems. Of course, it wasn’t referred to as “dual coding” back then, but dual coding it most definitely is.

I found it a very useful technique at the time and I still find it useful to this day. And what is more, it is in my opinion a pedagogically powerful procedure. I genuinely believe that this technique helps students understand the complexities and nuances of SUVAT because it brings many things which are usually *implicit* out into the open and makes them *explicit*.

BOSWELL. ‘He says plain things in a formal and abstract way, to be sure: but his method is good: for to have clear notions upon any subject, we must have recourse to analytick arrangement.’

JOHNSON. ‘Sir, it is what every body does, whether they will or no.

But sometimes things may be made darker by definition. I see a cow, I define her,Animal quadrupes ruminans cornutum.But a goat ruminates, and a cow may have no horns.Cowis plainer.— Boswell’s

Life of Johnson(1791)

As I see it, the enduring difficulty with SUVAT problems is that such things can indeed be made *darker by definition.* Students are usually more than willing to accept the formal definitions of *s*, *u*, *v*, *a* and *t* and can apply them to straightforward and predictable problems. However, the robotic death-by-algorithm approach fails all too frequently when faced with even minor variations on a theme.

Worse still, students often treat acceleration, displacement and velocity as nearly-synonymous interchangeable quantities: they are all lumped together in that naive “intuitive physics” category called MOVEMENT.

The approach that follows attempts to make students plainly see differences between the SUVAT quantities and, hopefully, as make them as *plain as a cow* (to borrow Dr Johnson’s colourful phrasing).

Problem: a motorcycle accelerates from rest at 0.8 m/s^{2} for a time of 6.0 seconds. Calculate (a) the distance travelled; and (b) the final velocity.

Please note:

- We are using the AQA-friendly convention of substituting values before rearrangement. (Some AQA mark schemes award a mark for the correct substitution of values into an expression; however, the mark will not be awarded if the expression is incorrectly rearranged. Weaker students are strongly encouraged to substitute before rearrangement, and this is what I model.)
- A later time is indicated by the movement of the hands on the clock.

So far, so blindingly obvious, some might say.

But I hope the following examples will indicate the versatility of the approach.

Problem: A coin is dropped from rest takes 0.84 s to fall a distance of 3.5 m so that it strikes the water at the bottom of a well. With what speed must it be thrown vertically so that it takes exactly 1.5 s to hit the surface of the water?

Another advantage of this method is that it makes assigning positive and negative directions to the SUVAT vectors easy as it becomes a matter of simply comparing the directions of each vector quantity (that is to say, *s*, *u*, *v* and *a*) with the arbitrarily selected positive direction arrow when we substitute values into the expression.

But what would happen if we’d selected a different positive direction arrow?

Problem: A well is 3.5 m deep so that a coin dropped from rest takes 0.84 s to strike the surface of the water. With what speed must it be thrown so that it takes exactly 1.5 s to hit the surface of the water?

The answer is, of course, numerically equal to the previous answer. However, following the arbitrarily selected down is positive convention, we have a negative answer.

Let’s look at this typical problem from AQA.

We could annotate the diagram like this:

Guiding our students through the calculation:

Some trad-inclined teachers have embraced the motto: *Just tell ’em!*

It’s a good motto, to which dual coding can add the welcome corollary: *Just show ’em!*

The famous phrase is, of course, from physicist Eugene Wigner (1960: 2):

My principal aim is to illuminate it from several sides. The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it.

Further exploration of the above problem using dual coding can, I believe, give A-level students a glimpse of the truth of Wigner’s phrase.

In the calculation above, we found that when *s* = -1.8 m, *v* could have a value of plus or minus 6.90 m/s. Since we were interested in the velocity of the kite boarder at the end of the journey, we concluded that it was the negative root that was significant for our purposes.

But does the positive root have any physical significance? Why yes, it does. It indicates the other possible value of *v* when *s* = -1.8 m.

The displacement was -1.8 m at only one point on the *real* journey. However, if the kite boarder had *started their projectile motion from the level of the water surface* instead of from the top of the ramp, their vertical velocity at this point would have been +6.9 m/s.

The fact that the kite boarder did *not* start their journey from this point is immaterial. Applying the mathematics not only tells us about their actual journey, but all other possible journeys that are consistent with the stated parameters and the subset of the laws of physics that we are considering in this problem — and that, to me, borders enough on the mysterious to bring home Wigner’s point.

This information allows us to annotate our final diagram as below (bearing in mind, of course, that the real journey of the kite boarder started from the top of the ramp and not from the water’s surface as shown).

Let me end on a more cheerful note. The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.

Wigner 1960: 9

Wigner, E. (1960). The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Communications in Pure and Applied Mathematics; Vol. 13, No. 1.

]]>Small beer for some in the edu-blogosphere perhaps, but I am well chuffed.

And my most popular blogs were (in reverse order):

- 5) IoP Energy for Busy Teachers. This is yet another of my periodic tilts at the Institute of Physics’ revised schema for teaching energy, including some attempted humour.
- 4) The FBI and Gang Sign for Physicists. I am am at a loss to explain why this random stream-of-edu-conciousness post from 2016 seems to get a slow but steady stream of readers, mainly from the U.S.A. for some reason
- 3) Teaching Electric Circuits? Climb On Board The Coulomb Train! This, I have to admit, is one of my personal favourites. Although a persuasive case can be made for the rope model (I’m looking at you, @PhysicsUK and this), the CTM (Coulomb Train Model) is still the best IMHO. For example, which one would be the best when we’re considering RMS values, huh?
- 2) Two posts on applying the Singapore Bar Model to GCSE Science Topics and ditching those horrible, horrible formula triangles. The first was on Magnification and the second on Kinetic Energy.
- 1) And my all-time most-viewed blog post is …
**MARKOPALYPSE NOW****!!!**(And thanks to Adam Boxer’s A Chemical Orthodoxy for a link that generated many of the views.)

Bubbling under, we have my contribution to the #CurriculumInScience symposium, Using P-Prims For Fun And Profit. I’m hoping this will get a few more views in the New Year.

And on that note: thanks for reading this far and Happy New Year everyone.

]]>“She said she was going to join a church as soon as she decided which one was right. She never did decide. She did develop a terrific hankering for a crucifix, though. And she bought one from a Santa Fe gift shop during a trip the little family made out West during the Great Depression. Like so many Americans,

”she was trying to construct a life that made sense from things she found in gift shops.— Kurt Vonnegut,

Slaughterhouse-Five[emphasis added]

It was never supposed to be like this, of course. Many of the great thinkers of the past conceived of the human mind as a vast pyramid: either an inverted pyramid resting on an apex consisting of a single, unfalsifiable thought such as “I think therefore I am” as Rationalists such as Descartes posited; or, alternatively, as a pyramid resting on a base of simple sense-impressions as Empiricists such as Locke suggested.

The truths emerging from modern cognitive science indicate that things are a good deal more complicated and messier than either the Rationalists or Empiricists supposed.

In fact, all of us are closer to Mrs Pilgrim in Vonnegut’s Slaughterhouse-Five than we would generally like to admit: the uncomfortable truth is that we are all closer to opportunistic concept-grubbing, “gift shop”-magpies than the systematic pyramid-masons of either Rationalist or Empiricist thought. Each and every one of us is, to a greater or lesser degree, “trying to construct a life that makes sense” from random things that we find lying around in real or metaphorical gift shops.

Perhaps (of all people!) Dashiell Hammett put it best:

“Nobody thinks clearly, no matter what they pretend. Thinking’s a dizzy business, a matter of catching as many of those foggy glimpses as you can and fitting them together the best you can. That’s why people hang on so tight to their beliefs and opinions; because, compared to the haphazard way in which they arrived at, even the goofiest opinion seems wonderfully clear, sane, and self-evident. And if you let it get away from you, then you’ve got to dive back into that foggy muddle to wangle yourself out another to take its place.”

— Dashiell Hammett,The Dain Curse

Andrea diSessa (1996) recounts a series of interviews with “J”, a freshman undergraduate student of Physics at university. During one interview, J was asked to explain the physics of throwing a ball up into the air. She recounted a near-perfect, professional physicist-level analysis of the phenomenon, noting (correctly) that after the ball leaves the hand the “only force acting on it is gravity”. However, when diSessa asked the seemingly innocuous question about what happens at the peak of the toss:

Rather than produce a straightforward answer, J proceeded to reformulate her description of the toss. The reformulation is not instantaneous . . . Strikingly she winds up with an “impetus theory” account of the toss. “Your hand imparts a force that at first overcomes gravity, but gradually dies away. At the peak, there a balance of forces, which is broken as the internal force fades further and gravity takes over.”

In other words, even a student of Physics, educated to a much higher level of domain-specific knowledge than the typical layperson, can be persuaded to retreat back into the ”foggy muddle” with surprising ease. In other words, even the very best of us can snatch defeat from the jaws of victory all too easily.

diSessa (1988) explains this and similar models as part of the KIP model (Knowledge in Pieces). For example:

intuitive physics is a fragmented collection of ideas, loosely connected and reinforcing, having none of the commitment or systematicity that one attributes to theories.

The basic “atom” or building block of this empirical model is the p-prim or phenomenological primitive.

P-prims are elements of intuitive knowledge that constitute people’s “sense of mechanism”, their sense of which happenings are obvious, which are plausible, which are implausible, and how one can explain or refute real or imagined possibilities. [diSessa 2018: 69]

P-prims are abstractions of familiar events that come to serve as explanations as they are applied to a wider range of contexts. The p indicates that they originate from the phenomenologically-rich and lived experience of human beings; the prim indicates that they are primitive in the sense that they sufficient explanations of phenomena. Once a p-prim is invoked, usually no further explanation is required or possible: “That’s just the way it is.” Examples of p-prims suggested by diSessa [1996: 716] are:

- The “Ohm’s Law” p-prim: the idea that an outcome increases as a “force” increases, but decreases as the “resistance” increases.
- The “Balance and Equilibrium” p-prim: systems which are “in balance” will be stable; systems which are “out of balance” will naturally and spontaneously return to equilibrium.
- The “Blocking and Guiding” p-prim: solid and stable objects can stop objects moving without applying a force; tubes and railway tracks can also “guide” moving objects without applying any force.
- The “Dying Away” p-prim: lack of motion or activity is the natural state of inanimate objects; if disturbed, they will naturally return to this state as the perturbation “dies away’

P-prims are subconceptual: they comprise a fluid and changeable layer below concepts and beliefs. Humans may have hundreds if not thousands of p-prims. There is no strict hierarchy: we may shift from one p-prim to another with simply a shift of attention. Where multiple p-prims conflict, one facet of the situation may cue the application of a particular p-prim rather than another. [see diSessa 1996: 715]

In his hugely-influential The Structure of Scientific Revolutions (1970), Thomas Kuhn suggested that scientific progress had two distinct phases:

**Normal Science**, where essentially scientists engaged in puzzle-solving activity but where the guiding paradigm or disciplinary matrix of the science is more or less accepted without question. An example might be pre-Copernican astronomy where astronomers made observations and predictions without questioning the geocentric model of the Solar System;**Revolutionary Science**, where scientists realise their previously-successful paradigm is no longer able to adequately explain observed phenomena. An example might be the rejection of the Newtonian paradigm and the acceptance of Einsteinian relativistic physics in the early 1900s.

Scientific progress was thus viewed as a gestalt switch between two incommensurable systems of knowledge. One either sees a “Newtonian”-duck, or a “Relativistic”-rabbit. One cannot see both simultaneously.

Kuhn’s work was immensely influential (perhaps overly influential) in a number of spheres; in the context of education, the heady seductiveness of Kuhn’s approach directly influenced what diSessa [2014: 5] dubs the “misconceptions movement”.

Broadly speaking, proponents thought that students had deeply entrenched but false beliefs. The solution seemed obvious: these false beliefs were barriers to learning that had to be rooted out and overcome (c.f. the Ohm’s Law p-prim above!) . Students had to be persuaded to ditch their false beliefs and accept the correct ones.

But what was the nature of these false beliefs? diSessa [2014:7] argues that some like Carey (1985) drew explicit parallels with Kuhn’s work, arguing that children undergo a paradigm-shift at about 10-years-old when they recognise that inanimate objects do not have intentions and begin to think of “alive” as describing a set of mechanistic processes. Others (argues diSessa) like McCloskey (1983) supposed that students begin school physics with a well-formed, coherent and articulate theory (with parallels to early medieval scientists such as Buridan and Galileo) that directly competes with and interferes with their acceptance of Newtonian physics.

However, all of these approaches can be categorised as being part of the “Misconceptions Movement”.

A positive influence of misconceptions studies was bringing the importance of educational research into practical instructional circles. Teachers saw vivid examples of students responding to apparently simple conceptual questions in incorrect ways. Poor performance in response to basic questions, often years into instruction, could not be dismissed.

[diSessa 2014: 6]

Another hugely positive influence of Misconceptions research was that it showed that students were not “blank slates” and that prior knowledge had a strong influence on future learning.

However, according to diSessa the misconceptions movement also had some pernicious negative influences:

- It emphasised the negative contributions of prior knowledge: it almost exclusively characterised prior knowledge as either false or unhelpful which led to “conflict” models of instruction. Ironically, the explicit detailing of “wrong” ideas in order to “overcome” them led to them being strengthened for some students.
- How learning was possible was not a matter that was often discussed in detail. The depth, coherence or strength of particular misconceptions was not always assessed: were they simply isolated beliefs or coherent theories of a similar nature to those held by working scientists? As a result, practical guidance on how to teach particular concepts was not always forthcoming.

As a working Physics teacher, one of the most useful teaching tools that I’ve begun using as a result of becoming aware of diSessa’s work, is that of a *bridging analogy*. This approach was outlined by Hammer 2000: S54-55. For example, how can we successfully introduce the idea of a normal reaction force, say in the context of a book resting on the surface of a table?

Students often invoke the “blocking” p-prim in this context. The table passively “blocks” the action of gravity — and that’s all there is to it.

However, a bridging analogy can be used here. Show an object resting on (and compressing) a spring; identify the forces acting on the object. Because the spring is an “active” component in this situation, students can accept that pushing down on it produces an upward “reaction force”. One can then extend this to (say) a student sitting on a plank (which “bows” slightly with their weight) and then apply it to more stable structure such as a table which exhibits no visible “bowing”.

I have found such approaches to be the most productive: in other words, we aim to work *around* the p-prim rather than attacking the p-prim head on, and along the way we try to get our students to activate more helpful p-prims that have more direct applicability to the context.

As teachers, we only very rarely have the luxury of choosing our students’ starting points. There is no “Well, if you want to get where you’re going, I wouldn’t start from here if I were you.”

We are teachers. Whatever the situation, we start from where our students start. Ladies and gentlemen, we start from *here*.

Carey, S. (1985). Conceptual change in childhood. Cambridge, MA: MIT Press/Bradford Books

diSessa, A. A. (1988). Knowledge in pieces. In G. Forman & P. B. Pufall (Eds.), Constructivism in the computer age (pp. 49-70). Hillsdale, NJ, US: Lawrence Erlbaum Associates, Inc.

diSessa, A. A. (1996). What do” just plain folk” know about physics. The handbook of education and human development: New models of learning, teaching, and schooling, 709-730. [Accessed from http://www.staff.science.uu.nl/~savel101/fi-msecint/literature/disessa1996.pdf on 22/10/18]

DiSessa, A. A. (2014). A history of conceptual change research: Threads and fault lines. [Accessed from https://escholarship.org/uc/item/1271w50q on 22/10/18]

diSessa, A. A. (2018). A Friendly Introduction to “Knowledge in Pieces”: Modeling Types of Knowledge and Their Roles in Learning. In Invited Lectures from the 13th International Congress on Mathematical Education (pp. 65-84). Springer International Publishing. [Accessed from https://link.springer.com/chapter/10.1007/978-3-319-72170-5_5 on 22/10/18]

Hammer, D. (2000). Student resources for learning introductory physics. American Journal of Physics, 68(S1), S52-S59 [Accessed from http://oap.nmsu.edu/JiTT_NMSU_workshop/pdfs/StudentResourcesHammer.pdf on 22/10/18]

McCloskey, M. (1983). Naive theories of motion. In D. Gentner and A. Stevens (Eds.) Mental Models (pp. 299-323). Hillsdale, NJ: Lawrence Erlbaum Associates.

]]>Essentially, the Singapore Bar Model uses pictorial representations (often in the form of a bar or line) to help students bridge the gap between concrete and abstract reasoning. I wrote about one possible application here.

A recent discussion on Twitter started me thinking about if it could be applied to kinetic energy.

For example, how would you explain what happens to the kinetic energy of an object if its velocity is halved?

Many students assume that the KE would halve as well, instead of reducing to a quarter of its original value.

How can we help students grasp this slippery concept without using algebra? Algebra would work fine with your higher sets, of course, but not necessarily for other groups.

This gives a clear visual representation of the fact that the KE *quarters* when the velocity halves. In other words, 0.5 x 0.5 = 0.25.

(Note that I have purposefully used decimals as we know that many students struggle with fractions(!))

Many students found the following question on an AQA paper extremely challenging:

The correct answer is that the power output drops to one eighth of its original value.

Could the Singapore Bar Model helps students to see why this is the case?

I think it could:

]]>