Many students struggle with Physics calculation questions at KS3 and KS4. Since 40% of the marks on GCSE Physics papers are for maths, this is a real worry for their teachers.
The FIFA system (if that’s not too grandiose a description) provides a minimal and flexible framework that helps students to successfully attempt calculation questions.
Since adopting the system, we encounter far fewer blanks on test and exam scripts where students simply skip over a calculation question. A typical student can gain 10-20 marks.
The FIFA system is outlined here but essentially consists of:
Formula: students write the formula or equation
Insert values: students insert the known data from the question.
Fine-tune: rearrange, convert units, simplify etc.
Answer: students state the final answer.
The “Fine-tune” stage is not — repeat, not — synonymous with re-arranging and is designed to be “creatively ambiguous” and allow space to “do what needs to be done” and can include unit conversion (e.g. kilowatts to watts), algebraic rearrangement and simplification.
Uniquely for Physics, instead of the dreaded “Six Marker” extended writing question, we have the even-more-dreaded “Six Marker” long calculation question. (Actually, they can be awarded anywhere between 4 to 6 marks, but we’ll keep calling them “Six Markers” for convenience.)
The “FIFA-one-two” strategy can help students gain marks in these questions.
Let’s look how it could be applied to a typical “Six mark” long calculation question. We prepare the ground like this:
Since the question mentions the power output of the kettle first, let’s begin by writing down the energy transferred equation.
Next we insert the values. It’s quite helpful to write in any “non standard” units such as kilowatts, minutes etc as a reminder that these need to be converted in the Fine-tune phase.
And so we arrive at the final answer for this first section:
Next we write down the specific heat capacity equation:
And going through the second FIFA operation:
I think every “Six Marker” extended calculation question can be approached in a productive way using the FIFA-One-Two approach.
This means that, even if students can’t reach the final answer, they will pick up some method marks along the way.
I hope you give the FIFA-One-Two method a go with your students.
Why do so many students hold pernicious and persistent misconceptions about forces?
Partly, I think, because of the apparent clash between our intuitive, gut-level knowledge of real world physics. For example, a typical student might find the statement ‘If I push this box, it will stop moving shortly after I stop pushingbecause force is needed to move things‘ entirely unobjectionable; whilst in the theoretical, rarefied world of the physicist the statement ‘The box will keep moving at a constant velocity after I stop pushing it, unless it is acted on by a resultant force such as friction‘ would get a tick whereas the former would get a big angry X and and a darkly muttered comment about ‘bloody Aristotleans.’
After all, ‘pernicious’ is in the eye of the beholder. Physics teachers have to remember that they suffer mightily under the ‘curse of knowledge’ and have forgotten what it’s like to look at the world through anything than the lens of Newtonian mechanics.
We learn about the world through the power of example. Human beings are ‘inference engines’: we strive to make sense of the world by constructing general rules based on the examples presented to us.
Many of the examples of forces in action presented to students are in the form of force diagrams; and in my experience, all too many force diagrams add to students’ confusion.
A bad force diagram
Over the years, I have seen many versions of this diagram. To my own chagrin, I must admit that I, personally, have drawn versions of this diagram in the past. But I now recognise it has one major, irredeemable flaw: the arrows are drawn hanging in mid-air.
OK, let’s address this. Is this better?
No, it isn’t because it is still unclear which forces are acting on which object. Is the blue 75 N arrow the person pushing the cart forward or the cart pulling the person forward? Is the red 75 N arrow the cart pushing back on the person or the person pulling back on the cart?
From both versions of this diagram shown above: we simply cannot tell.
As a consequence, I think the explanatory value of this diagram is limited.
Free Body Diagrams to the Rescue!
A free body diagram is simply one where we consider the forces on each object in the situation in turn.
We begin with a situation diagram. This shows the relationship between the objects we are considering. Next, we draw a free body diagram for each object; that is, we draw each object involved and consider the forces acting on it.
From version 3 of Force Diagram 1, we can see that it was an attempt to illustrate Newton’s Third Law i.e. that if body A exerts a force on body B then body B exerts an equal and opposite force on body A.
Another bad force diagram
This is a bad force diagram because it is unclear which forces are acting on the cart and which are acting on the person. Apart from a very general ‘Well, 50 N minus 50 N means zero resultant force so zero acceleration’, there is not a lot of information that can be extracted from this diagram.
Also, the most likely mechanism to produce the red retarding force of 50 N is friction between the wheels of the cart and the ground (and note that since the cart is being pushed by an external body and the wheels are not powered like those of a car, the frictional force opposes the motion). Showing this force acting on the handle of the cart is not helpful, in my opinion.
Free body diagrams to the rescue (again)!
The Newton 3 pairs are colour coded. For example, the orange 50 N forward force on the person (object A) is produced as a direct result of Newton’s 3rd Law because the person’s foot is using friction to grip the floor surface (object B) and push backwards on it (the orange arrow in the bottom diagram).
This diagram shows a complete free body diagram body analysis for all three objects (cart, person, floor) involved in this simple interaction.
I’m not suggesting that all three free body diagrams always need to be discussed. For example, at KS3 the discussion might be limited at the teacher’s discretion to the top ‘Forces on Cart’ diagram as an example of Newton’s First Law in action. Or equally, the teacher may wish to extend the analysis to include the second and third diagrams, depending on their own judgement of their students’ understanding. The Key Stage ticks and crosses on the diagram are indicative suggestions only.
At KS3 and KS4, there is not a pressing need to explicitly label this technique as ‘free body force diagrams’. Instead, what I suggest (perhaps after drawing the situation diagram without any force arrows on it) is the simple statement that ‘OK, let’s look at the forces acting on just the cart’ before drawing the top diagram. Further diagrams can be introduced with a similar statements such as ‘Next, let’s look at the forces acting on just the person’ and so on. Linking the diagrams with dotted lines as shown is, I think, useful in not losing sight of the fact that we are dealing piecemeal with a complex and nuanced whole.
The free body force diagram technique (whether or not the teacher decides to explicitly call it that) offers a useful tool that will allow us all to (fingers crossed!) draw better force diagrams.
Draw a situation diagram with NO FORCE ARROWS.
‘Now let’s look at the forces acting on just object 1’ and draw a separate free body diagram (i.e. a diagram showing just object 1 and the forces acting on it)
Repeat step 2 for some or all of the other objects at your discretion.
(Optional) Link all the diagrams with dotted lines to emphasise that they are facets of a more complex, nuanced whole
In the next post, I hope to show how the technique can be used to explain common problems such as how a car tyre interacts with the ground to drive a car forward.
There are three things that everyone should know about simple harmonic motion (SHM).
Firstly, it is simple;
Secondly, it is harmonic;
Thirdly, it is a type of motion.
There, my work here is done. H’mmm — it looks like this physics teaching lark is much easier than is generally acknowledged…
[The above joke courtesy of the excellent Blackadder 2 (1986), of course.]
Misconceptions to the left of us, misconceptions to the right of us…
In my opinion, the misconceptions which hamper students’ attempts to understand simple harmonic motion are:
A shallow understanding of dynamics which does not differentiate between ‘displacement’ ‘velocity’ and ‘acceleration’ but lumps them together as interchangeable flavours of ‘movement’
The idea that ‘acceleration’ invariably leads to an increase in the magnitude of velocity and that only the materially different ‘deceleration’ (which is exclusively produced by resistive forces such as friction or drag) can result in a decrease.
Not understanding the positive and negative direction conventions when analysing motion.
All of these misconceptions can, I believe, be helpfully addressed by using a form of dual coding which I outlined in a previous post.
Top Gear presenters: Assemble!
The discussion context which I present is that of a rather strange episode of the motoring programme Top Gear. You have been given the opportunity to win the car of your dreams if — and only if — you can drive it so that it performs SHM (simple harmonic motion) with a period of 30 seconds and an amplitude of 120 m.
This is a fairly reasonable challenge as it would lead to a maximum acceleration of 5.3 m s-2. For reference, a typical production car can go 0-27 m/s in 4.0 s (a = 6.8 m s-2)) but a Tesla Model S can go 0-27 m/s in a scorching 2.28 s (a = 11.8 m s-2). BTW ‘0-27 m/s’ is the SI civilised way of saying 0-60 mph. It can also be an excellent extension activity for students to check the plausibility of this challenge(!)
Timing and the Top Gear SHM Challenge
At what time should the car reach E on its outward journey to ensure we meet the Top Gear SHM Challenge? (15 s since A to E is half of a full oscillation and T should be 30 seconds according to the challenge)
At what time should the car reach C? (7.5 s since this is a quarter of a full oscillation.)
All physics teachers, to a greater or lesser degree, labour under the ‘curse of knowledge’. What we think is ‘obvious’ is not always so obvious to the learner. There is an egregiously underappreciated value in making our implicit assumptions and thinking explicit, and I think diagrams like the above are invaluable in this process.
But what is this SHM (of which you speak of so knowledgeably) anyway?
Simple harmonic motion must fulfil two conditions:
The acceleration must always be directed towards a fixed point.
The magnitude of the acceleration is directly proportional to its displacement from the fixed point.
In other words:
Let’s look at this definition in terms of our fanciful Top Gear challenge. More to the point, let’s look at the situation when t = 0 s:
Questions that could be discussed here:
Why is the displacement at A labelled as ‘+120 m’? (Displacement is a vector and at A it is in the same direction as the [arbitrary] positive direction we have selected and show as the grey arrow labelled +ve.)
The equation suggests that the value of a should be negative when x is positive. Is the diagram consistent with this? (Yes. The acceleration arrow is directed towards the fixed point C and is in the opposite direction to the positive direction indicated by the grey arrow.)
What is the value of v indicated on the diagram? Is this consistent with the terms of the challenge? (Zero. Yes, since 120 m is the required amplitude or maximum displacement so if v was greater than zero at this point the car would go beyond 120 m.)
How could you operate the car controls so as to achieve this part of simple harmonic motion? (You should be depressing the gas pedal to the floor, or ‘pedal to the metal’, to achieve maximum acceleration.)
Model the thinking explicitly
Hands up who thinks the time on the second clock on the diagram above should read 3.75 seconds? It makes sense, doesn’t it? It takes 7.5 s to reach C (one quarter of an oscillation) so the temptation to ‘split the difference’ is nigh on irresistible — except that it would be wrong — and I must confess, it took several revisions of this post before I spotted this error myself (!).
The vehicle is accelerating, so it does not cover equal distances in equal times. It takes longer to travel from A to B than B to C on this part of the journey because the vehicle is gaining speed.
So what is the time when x = 60 m
So we can redraw the diagram as follows:
Some further questions that could be asked are:
Is the acceleration arrow at B smaller or larger than the acceleration arrow at A? Is this consistent with what we know about SHM? (Smaller. Yes, because for SHM, acceleration is proportional to displacement. The displacement at B is +60 m; the acceleration at B is half the value of the acceleration at A because of this. Note that the magnitude of the acceleration is reduced but the direction of a is still negative since the displacement is positive.)
Is the velocity at B positive or negative? (Negative, since it is opposite to the positive direction selected on the diagram and shown by the grey ‘+ve’ arrow.)
Is the magnitude of the velocity at B smaller or larger than at A, and is this consistent with a negative acceleration? (Larger. Yes, since both acceleration and velocity are in the same direction. Note that this is an important point to highlight since many students hold the misconception that a negative acceleration is always a ‘deceleration’.)
How could you operate the car controls so as to achieve this part of simple harmonic motion? (You should have eased off the gas pedal at this point to achieve half the acceleration obtained at A.)
Next, we move on to this diagram and ask students to use their knowledge of SHM to decide the values of the question marks on the diagram.
Which hopefully should lead to a diagram like the one below, and realisation that at this point, the driver’s foot should be entirely off the gas pedal.
‘Are we there yet?’
And thence to this:
One of the most salient points to highlight in the above diagram is the question: how could you operate the car controls at this point? The answer is of course, that you would be pressing the foot brake pedal to achieve a medium magnitude deceleration. This is often a point of confusion for students: how can a positive acceleration produce a decrease in the magnitude of the velocity? Hopefully, the dual coding convention suggested in this blog post will make this clearer to students.
‘No, really, ARE WE THERE YET?!!’
Over time, we can build up a picture of a complete cycle of SHM, such as the one show below. This shows the car reversing backwards at t = 25 s while the driver gradually increases the pressure on the brake.
From this, it should be easier to relate the results above to graphs of SHM:
A quick check reveals that the displacement is positive and half its maximum value; the acceleration is negative and half of its maximum magnitude; and the velocity is positive and just below its maximum value (since the average deceleration is smaller between C and B than it will be between B and A) .
I shall leave the final word to the estimable Top Gear team…
There is little doubt that students find understanding how an electric motor works hard.
What follows is an approach that neatly sidesteps the need for applying Fleming’s Left Hand Rule (FLHR) by using the idea of the catapult field.
The catapult field is a neat bit of Physics pedagogy that appears to have fallen out of favour in recent years for some unknown reason. I hope to rehabilitate and publicise this valuable approach so that more teachers may try out this electromagnetic ‘road less travelled’.
(Incidentally, if you are teaching FLHR, the mnemonic shown above is not the best way to remember it: try using this approach instead.)
The magnetic field produced by a long straight conductor
Moving electric charges produce magnetic fields. When a current flows through a conductor, it produces a magnetic field in the form of a series of cylinders centred on the wire. This is usually shown on a diagram like this:
If we imagine looking down from a point directly above the centre of the conductor (as indicated by the disembodied eye), we would see a plan view like this:
We are using the ‘dot and cross‘ convention (where an X represents an arrow heading away from us and a dot represents an arrow heading towards us) to easily render a 3D situation as a 2D diagram.
The direction of the magnetic field lines is found by using the right hand grip rule.
The thumb is pointed in the direction of the current. The field lines ‘point’ in the same direction as the fingers on the right hand curl.
3D to 2D
Now let’s think about the interaction between the magnetic field of a current carrying conductor and the uniform magnetic field produced by a pair of magnets.
In the diagrams below, I have tried to make the transition between a 3D and a 2D representation explicit, something that as science teachers I think we skip over too quickly — another example of the ‘curse of knowledge’, I believe.
Magnetic Field on Magnetic Field
If we place the current carrying conductor inside the magnetic field produced by the permanent magnets, we can show the magnetic fields like this:
Note that, in the area shaded green, the both sets of magnetic field lines are in the same direction. This leads a to stronger magnetic field here. However, the opposite is true in the region shaded pink, which leads to a weaker magnetic field in this region.
The resultant magnetic field produced by the interaction between the two magnetic fields shown above looks like this.
Note that the regions where the magnetic field is strong have the magnetic field lines close together, and the regions where it is weak have the field lines far apart.
The Catapult Field
This arrangement of magnetic field lines shown above is unstable and is called a catapult field.
Essentially, the bunched up field lines will push the conductor out of the permanent magnetic field.
If I may wax poetic for a moment: as an oyster will form a opalescent pearl around an irritant, the permanent magnets form a catapult field to expel the symmetry-destroying current-carrying conductor.
The conductor is pushed in the direction of the weakened magnetic field. In a highly non-rigorous sense, we can think of the conductor being pushed out of the enfeebled ‘crack’ produced in the magnetic field of the permanent magnets by the magnetic field of the current carrying conductor…
Also, the force shown by the green arrow above is in exactly the same direction as the force predicted by Fleming’s Left Hand Rule, but we have established its direction using only the right hand grip rule and a consideration of the interaction between two magnetic field.
The Catapult Field for an electric motor
First, let’s make sure that students can relate the 3D arrangement for an electric motor to a 2D diagram.
The pink highlighted regions show where the field lines due to the current in the conductor (red) are in the opposite direction to the field line produced by the permanent magnet (purple). These regions are where the purple field lines will be weakened, and the clear inference is that the left hand side of the coil will experience an upward force and the right hand side of the coil will experience a downward force. As suggested (perhaps a little fancifully) above, the conductors are being forced into the weakened ‘cracks’ produced in the purple field lines.
The catapult field for the electric motor would look, perhaps, like this:
On a practical teaching note, I wouldn’t advise dispensing with Fleming’s Left Hand Rule altogether, but hopefully the idea of a catapult field adds another string to your pedagogical bow as far as teaching electric motors is concerned (!)
I have certainly found it useful when teaching students who struggle with applying Fleming’s Left Hand Rule, and it is also useful when introducing the Rule to supply an understandable justification why a force is generated by a current in a magnetic field in the first place.
The catapult field is a ‘road less travelled’ in terms of teaching electromagnetism, but I would urge you to try it nonetheless. It may — just may — make all the difference.
Student: Did you know FIFA is also the name of a video game, Sir?
Student: Yeah. It’s part of a series. I just got FIFA 20. It’s one of my favourite games ever.
Me: Goodness me. I had no idea. I just chose the letters ‘FIFA’ completely and utterly at random!
The FIFA method is an AQA mark scheme-friendly* way of approaching GCSE Physics calculation questions. (It is also useful for some Y12 Physics students.)
I mentioned it in a previous blog and @PedagogueSci was kind enough to give it a boost here, so I thought I’d explain the method in a separate blog post. (Update: you can also watch my talk at ChatPhysics Live 2021 here.)
The FIFA method:
Avoids the use of formula triangles
Minimises the cognitive load on students when approaching calculations.
Why we shouldn’t use formula triangles
Formula triangles are bad news. They are a cognitive dead end.
During a university admissions interview for veterinary medicine, I asked a prospective student to explain how they would make up a solution for infusion into a dog. Part of the answer required them to work out the volume required for a given amount and concentration. The candidate started off by drawing a triangle, then hesitated, eventually giving up in despair. […]
They are a trick that hides the maths: students don’t apply the skills they have previously learned. This means students don’t realise how important maths is for science.
I’m also concerned that if students can’t rearrange simple equations like the one above, they really can’t manage when equations become more complex.
[Update: this 2016 article from Ed Southall also makes a very persuasive case against formula triangle.]
I believe the use of formula triangle also increases (rather than decreases) the cognitive load on students when carrying out calculations. For example, if the concentration c is 0.5 mol dm-3 and the number of moles n required is 0.01 mol, then in order to calculate the volume V they need to:
recall the relevant equation and what each symbol means and hold it in working memory
recall the layout of symbols within the formula triangle and either (a) write it down or (b) hold it in working memory
recall that n and c are known values and that V is the unknown value and hold this information in working memory when applying the formula triangle to the problem
The FIFA method in use (part 1)
The FIFA acronym stands for:
FINE TUNE (this often, but not always, equates to rearranging the formula)
Lets look at applying it for a typical higher level GCSE Physics calculation question
We add the FIFA rubric:
Students have to recall the relevant equation as it is not given on the Data and Formula Sheet. They write it down. This is an important step as once it is written down they no longer have to hold it in their working memory.
Note that this is less cognitively demanding on the student’s working memory as they only have to recall the formula on its own; they do not have to recall the formula triangle associated with it.
Students find it encouraging that on many mark schemes, the selection of the correct equation may gain a mark, even if no further steps are taken.
Next, we insert the values. I find it useful to provide a framework for this such as:
We can ask general questions such as: “What data are in the question?” or more focused questions such as “Yes or no: are we told what the kinetic energy store is?” and follow up questions such as “What is the kinetic energy? What units do we use for that?” and so on.
Note that since we are considering each item of data individually and in a sequence determined by the written formula, this is much less cognitively demanding in terms of what needs to be held in the student’s working memory than the formula triangle method.
Note also that on many mark schemes, a mark is available for the correct substitution of values. Even if they were not able to proceed any further, they would still gain 2/5 marks. For many students, the notion of incremental gain in calculation questions needs to be pushed really hard otherwise they will not attempt these “scary” calculation questions.
Next we are going to “fine tune” what we have written down in order to calculate the final answer. In this instance, the “fine tuning” process equates to a simple algebraic rearrangement. However, it is useful to leave room for some “creative ambiguity” here as we can also use the “fine tuning” process to resolve difficulties with units. Tempting though it may seem, DON’T change FIFA to FIRA.
We fine tune in three distinct steps (see addendum):
Finally, we input the values on a calculator to give a final answer. Note that since AQA have declined to provide a unit on the final answer line, a mark is available for writing “kg” in the relevant space — a fact which students find surprising but strangely encouraging.
The key idea here is to be as positive and encouraging as possible. Even if all they can do is recall the formula and remember that mass is measured in kg, there is an incremental gain. A mark or two here is always better than zero marks.
The FIFA method in use (part 2)
In this example, we are using the creative ambiguity inherent in the term “fine tune” rather than “rearrange” to resolve a possible difficulty with unit conversion.
In this example, we resolve another potential difficulty with unit conversion during the our creatively ambiguous “fine tune” stage:
The emphasis, as always, is to resolve issues sequentially and individually in order to minimise cognitive overload.
The FIFA method and low demand Foundation tier calculation questions
I teach the FIFA method to all students, but it’s essential to show how the method can be adapted for low demand Foundation tier questions. (Note: improving student performance on these questions is probably a more significant and quicker and easier win than working on their “extended answer” skills).
For the treatment below, the assumption is that students have already been taught the FIFA method in a number of contexts and that we are teaching them how to apply it to the calculation questions on the foundation tier paper, perhaps as part of an examination skills session.
For the majority of low demand questions, the required formula will be supplied so students will not need to recall it. What they will need, however, is support in inserting the values correctly. Providing a framework as shown below can be very helpful:
Also, clearly indicating where the data came from is useful.
The fine tune stage is not needed, so we can move straight to the answer.
Note also that the FIFA method can be applied to all calculation questions, not just the ones that could be answered using formula triangle methods, as in part (c) of the question above.
I believe that using FIFA helps to make our thinking and methods in Physics calculations more explicit and clearer for students.
My hope is that science teachers reading this will give it a go.
*Disclaimer: AQA has not endorsed the FIFA method. I describe it as “AQA mark scheme-friendly” using my professional own judgment and interpretation of published AQA mark schemes.
I am embarrassed to admit that this was the original version published. Somehow I missed the more straightforward way of “fine tuning” by squaring the 30 and multiplying by 0.5 and somehow moved straight to the cross multiplication — D’oh!
My thanks to @BenyohaiPhysics and @AdamWteach for pointing it out to me.
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 involves first differentiating the concepts of current, voltage and energy before relating 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, energy lost per coulomb per amp.
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
Because the two resistors are identical, the 3 V supply is shared equally across both resistors. That is to say, there is a potential difference of 1.5 V across each resistor. But let’s check this by applying V = IR (eq. 18). The total potential difference is 3 V and the total resistance is 1 ohm + 1 ohm = 2 ohms.
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.
Driver, R., Squires, A., Rushworth, P., & Wood-Robinson, V. (1994). Making sense of secondary science: Research into children’s ideas. Routledge.
The AQA GCSE Science specification calls for students to understand and apply the concepts of not only thermal energy stores but also internal energy. What follows is my understanding of the distinction between the two, which I hope will be of use to all science teachers.
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.
Thermal Energy Through the Macroscopic Lens
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 Through the Microscopic Lens
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 part (but not the whole) of the internal energy of the system.
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!
Nave, R. (2000). HyperPhysics. Georgia State University, Department of Physics and Astronomy.
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.
Part the First: How to get the *wrong* answer
Imagine pushing an object with a mass m with a constant force Fso that it accelerates with a constant acceleration aso that covers a distance s in a time t. The object was initially at rest and ends up moving at velocity v.
(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: Wd = F x s
But remember s = v x t so:
Step 3: Wd = F x vt
And also remember that F = m x a so:
Step 4: Wd = ma x vt
Also remember that a = change in velocity / time, so a = (v – 0) / t = v / t.
Step 5: Wd= m (v / t) x vt
The ts cancel so:
Step 6: Wd = mv2
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: Ek = mv2.
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: Ek=mv2!?!?
That’s just wrong: where did the half go?
Houston, we have a problem.
Part the Second: how to get the *right* answer
The problem lies with Step 3 above. We wronglyassumed 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 s.
So Step 3 should read:
But remember that s = (average velocity) x t.
Because the object is accelerating at a constant rate, the average velocity is (v + u) / 2 and since u = 0 then average velocity is v / 2.
Step 3: Wd= F x (v / 2) t
And also remember that F = m x a so:
Step 4: Wd= ma x (v / 2) t
Also remember that a = change in velocity / time, so a = (v – 0) / t = v / t.
Step 5: Wd= m (v / t) x (v / 2) t
The ts cancel so:
Step 6: Wd= ½mv2
Based on this, of course, Ek = ½mv2
(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 v.
Another way of representing the kinetic energy as a solid prism is shown below.
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 s than if it was moving at a constant velocity v from the very beginning.
So there we have it, Ek = ½mv2 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.
The theory of dual coding holds that the formation of mental images, in tandem with verbal processing, is often very helpful for learners. In other words, if we support verbal reasoning with visual representations, then better learning happens.
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.
SUVAT: “Made darker by definition”?
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. Cow is 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).
Visual Symbols for the Dual-coding of SUVAT problems
1.1 Analysing a simple SUVAT problem using dual coding
Problem: a motorcycle accelerates from rest at 0.8 m/s2 for a time of 6.0 seconds. Calculate (a) the distance travelled; and (b) the final velocity.
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.
1.2a Analysing a more complex SUVAT problem using dual coding (Up is positive convention)
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?
1.2b Analysing a more complex SUVAT problem using dual coding (Down is positive convention)
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.
1.3 Analysing a projectile problem using dual coding
Let’s look at this typical problem from AQA.
We could annotate the diagram like this:
Guiding our students through the calculation:
Just Show ‘Em!
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 Unreasonable Effectiveness of Mathematics in the Natural Sciences
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.
This Is The Root You’re Looking For
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.
The first rule of IoP Energy Club is: you do not talk about energy . . .
. . . unless you’re gonna do a calculation.
— with apologies to Brad Pitt and Chuck Palahniuk
In the UK, the IoP (Institute of Physics) has developed a model of energy stores and energy pathways that has been adopted by all the exam boards. Although answers couched in terms of the old “forms of energy” model currently get full credit, this will almost certainly change over time (gradually or otherwise).
This post is intended to be a “one stop” resource for busy teachers, with suggestions for further reading.
Please note that I have no expertise or authority on the new model beyond that of a working teacher who has spent a fair amount of time researching, thinking about and discussing the issues. What follows is essentially my own take, “supplemented by the accounts of their friends and the learning of the Wise” (if I may borrow from Frodo Baggins!).
Part the First: “Why? For the love of God, why!?!”
The old forms of energy model was familiar and popular with students and teachers. It is still used by many textbooks and online resources. However, researchers have suggested that there are significant problems with this approach:
Students just learn a set of labels which adds little to their understanding (see Millar 2014 p.6).
The “forms of energy” approach focuses attention in the wrong place: it highlights the label, rather than the physical process. There is no difference between chemical energy and kinetic energy except the label, just as there is no difference between water stored in a cylindrical tank and a rectangular tank. (See Boohan 2014 p.12)
The new IoP Stores and Pathways model attempts to address these issues by limiting discussions of energy to situations where we might want to do calculations.
Essentially, the IoP wanted to simplify “energy-talk” and make it a better approximation of the way that professional scientists (especially physicists) actually use energy-concepts. The trick is to get away from the old and nebulous “naming of parts” approach to a newer, more streamlined version that is fit for purpose.
Part the Second: How many energy stores?
The second rule of IoP Energy Club is: youdo not talk about energy . . . . . . unless you’re gonna do a calculation.
— with apologies to Brad Pitt and Chuck Palahniuk
The IoP suggests eight named energy stores (listed below with the ones likely to be needed early in the teaching sequence listed first).
Many will be surprised to see that electrical energy, light energy and sound energy are not on this list: more on that later.
There are, I think, two very important points:
All of these energy stores represent quantities that are routinely measured in joules.
All of the energy stores represent a system where energy can be stored for an appreciable period of time.
For example, a rattling washing machine is not a good example of a vibration energy store as it does not persist over an extended period of time: as soon as the motor stops, the machine stops rattling. On the other hand, a struck tuning fork, a plucked guitar string or a bell hit with a hammer are good examples of vibration energy stores.
Similarly, a hot object is not a vibration energy store: it is better described as a thermal energy store. Thermal energy stores are useful when there is a change in temperature or a change in state.
Likewise, a lit up filament bulb is not a good example of a thermal energy store because it does not persist over an extended period of time; switch off the current, and the bulb filament would rapidly cool.
Note also that the electric-magnetic energy store applies to situations involving magnets and static electric charges. It is not equivalent to the old “electrical energy”.
The thread linking all the above examples is we limit discussions of energy to situations where we could perform calculations.
Thermal energy store is an appropriate concept for (say) the water in a kettle because we can calculate the change in the thermal energy store of the water and the result is useful in a wide range of situations. However the same is not true of a hot bulb filament as the change in the thermal energy store of the filament is not a useful quantity to calculate (at least in most circumstances). For further discussion, see this blog post and also this section of the IoP Supporting Physics website.
Part the third: How many energy pathways?
The third rule of IoP Energy Club is: there ain’t no such thing as ‘light energy’ (or ‘sound energy’ or ‘electrical energy’).
— with apologies to Brad Pitt and Chuck Palahniuk
In the new IoP Energy model, there is no such thing as a “light energy store”. Instead, we talk about energy pathways.
Energy pathways describe dynamic quantities that are routinely measured in watts. That is to say, they are dynamic or temporal in the sense that their measurement depends on time (watts = joules per second); energy stores are static or atemporal over a given period of time.
It is not useful to talk about a “light energy store” because it does not persist over time: the visible light emitted by (say) a street lamp is not static — it is not helpful to think of it as a static “box of joules”. Instead it is a dynamic “flow” of joules which means its most convenient unit of measurement is the watt.
As an analogy, think of an energy store as a container or tank; in contrast, think of a pathway as a channel or tap that allows energy to move from one store to another. )
You can read more on the “tanks and taps” analogy here.
The cautious reader should note that the IoP describe slightly different pathways which you can read about here. (Mechanical and Electrical Working are in, but the IoP talk about “Heating by particles” and “Heating by radiation”; on this categorisation, sound would fit into the “Mechanical Working” category!)
The fourth rule of IoP Energy Club is: I don’t care what you call it, if it’s measured in watts, it’s a pathway not an energy store, OK?
— with apologies to Brad Pitt and Chuck Palahniuk
You can look forward to more ‘IoP Energy Club Rules’, as and when I make them up.
Important note: all of the above content is the personal opinion of a private individual. It has not been approved or endorsed by the IoP.