The rope loop - a great activity, but less predictive power than you’d expect

The rope loop analogy has become increasingly popular over the years, and I see teachers (and those who teach teachers) using it more and more. The claim is that it has predictive powers - the problem is that in almost every case it predicts the exact opposite of how real circuits behave.

But it’s so darn convincing as a model - it’s so elegant and tactile that you just will it to do a great job, but there’s no obvious boundary between when it’s applicable and when it isn’t.

At the core of the problem, I think, is that you have to have a thorough understanding of the physics of electric circuits AND a thorough understanding of the physics of the rope loop in order to be able to work out when the rope loop is useful and when it’s not. This is especially difficult if your only mental model for electric circuits is the rope loop!

What you want from a good model is that when you make a change, the model responds in the way the system it’s modelling responds, and that you can interpret that change with minimal translation. What you don’t want is a model that appears to firmly and confidently predict one thing, but the actual system behaves in exactly the opposite way, and for this failure to happen silently.

Because there is a superficial similarity, it’s easy to assume that both physical systems share far more overlap than they actually do, and that is why the rope loop model is so capricious in the hands of the unwary.

It’s not just the rope loop - I think there’s something deep here about mechanical analogues in general. My guess is that there is no mechanical model for electric circuits that makes correct predictions AND is simpler to use than just understanding circuits.

What mechanical analogues find very difficult to do is to model the constant voltage nature of power supplies, and their response to increased resistance - which is to reduce the current they provide and to shift energy slower.

When you increase the ‘resistance’ in any mechanical analogue, whether it’s friction in the rope loop, walking over soft sand, or pushing through a jostling crowd, the response is either to work harder (shift energy faster) or to reduce the movement until the power settles at its starting value. This is not how batteries behave.

Where the rope loop is a great activity to bring to life the internal workings of an electric circuit

The rope loop does a great job at giving a cheap, everyday, interactive simile for the internal workings of a simple electric circuit, which goes a long way to counteracting the most common early misconceptions, and is great at demonstrating the ideas of charge, current, and energy being shifted from store to store.

You can show using the rope loop that:

  • The charges are already there everywhere in the circuit

  • They all start moving everywhere at (almost exactly) the same time

  • The charges don’t get used up as they go round

  • Faster charges means a bigger current (as a first order concept)

  • Energy is shifted from a chemical store (the teacher’s muscles) to a thermal store (in the environment around the student’s hand)

  • The energy is shifted instantaneously (almost) from teacher to student without that energy being carried by parts of the rope (a common criticism of donation models)

  • Changes to the circuit happen instantaneously everywhere even though the charges themselves make very slow progress

The four big problems with the rope loop analogy

1. The frictional force at the student’s hand is NOT an analogue for voltage

This is the first time that the rope loop starts to predict the exact opposite of what actually happens.

While it’s true in a mathematical sense* that frictional force impeding the flow of rope is an exact analogue of voltage, it’s simply not how circuits behave.

There are two major problems with thinking friction at the student’s hand is an analogue of voltage:

  • It predicts that if you increase the frictional force by squeezing the rope harder then the voltage must increase - this is nonsense. (I have seen a physics specialist running subject knowledge CPD for non-specialist science teachers explicitly making this claim). Power supplies are constant voltage providers (if you don’t make them work too hard). If you change the load resistance the p.d. across the load stays the same - only the current changes.

  • It implies that the battery voltage is set by the p.d. across the load, not the other way around. Newton’s 1st Law says that to pull the rope at a constant rate the teacher MUST pull the rope with exactly the same size force as the dynamic frictional force at the student’s hand. By analogy the rope loop predicts that the battery voltage is set by the p.d. across the load - which is to get it exactly the wrong way round.

2. Increasing resistance does NOT make the battery work harder or increase the amount of heating

If you increase the resistance in a simple circuit then the battery responds by providing a smaller current. Energy is shifted slower at the resistance, and the battery works less hard, not harder. Bigger resistances cause less heating.

There is no current that the battery wants to provide - it doesn’t struggle harder when it encounters a bigger resistance. Power supplies are like a lazy teenager - if you make it harder to get out of bed in the morning they don’t work harder - they give up.

However with the rope loop, if you get the student to squeeze the rope harder then the teacher has to pull harder. They have no choice - Newton I demands that the forces must balance for the rope to move at a constant rate.

This is a classic example of where the rope loop makes a prediction - bigger resistance -> more heating, and harder-working battery - that is so convincing even those teaching teachers can be deceived by it.

The real problem is that this argument - that bigger resistances cause more heating - has some truth to it, but the argument is quite subtle. If you have a small resistance (like a plug) in series with a big resistance (like the heater in your washing machine) then energy is shifted quicker in the big resistance because the p.d. across it is bigger, while the current through each is the same.

If you double the resistance of the (much lower resistance) plug then the p.d. across it will nearly double, but the current through it will stay broadly similar because the proportional increase in the effective resistance of the plug+heater is small. Bigger p.d., similar current, means more heating - which is why having unexpected extra resistance in low-resistance fittings is a fire hazard.

However, if you go on increasing the resistance of the lower resistance, the decreasing current has a bigger and bigger effect, until the heating reaches its maximum when the two resistances are equal. After that the ever-decreasing current more than compensates for the extra share of the voltage the resistance takes, and it produces less and less heating. This is the same argument as the maximum power theorem.

It’s important to remember that, even though increasing the lower resistance initially creates more heating in that resistance, the power supply still works less hard whenever the overall resistance increases, because it provides a smaller current.

3. The rope loop breaks the relationship between p.d., resistance and current

The rope loop cons teachers into believing that they can control something they can’t (the force with which they pull the rope) and that they can’t control something they can (how fast they pull the rope).

The typical suggestion you hear is that the teacher should try and keep the pull on the rope constant when the student changes their squeeze (which is meant to be the analogue for resistance). As we’ve argued above, this is impossible. Newton I says the teacher has to pull as hard as the dynamic frictional force between the rope and the student’s hand for the rope to move at a constant speed.

What the teacher can change is how fast they pull the rope. They can choose to pull it fast or slow - and this is useful for showing the conceptual difference between a big and a small current, but what the rope loop doesn’t do is fix the speed of the rope for a given student squeeze and teacher pull.

The relationship I=V/R says that there is one and only one value of current for a particular resistance with a given p.d. across it. The battery cannot choose what current to provide, unlike the teacher who can choose whatever current they like.

The second implication of the I=V/R relationship is that, regardless of whether R stays constant, you can only change the current through a resistance if you change the p.d. across it.

If, as is claimed, the frictional force at the student’s hand is an exact analogue of p.d., you would expect the frictional force to change if you pulled the rope faster.

This is where the physics of the rope loop diverges yet again. Dynamic frictional forces tend not to change much with speed of sliding, so it’s quite possible to pull the rope faster (not harder) without changing the frictional force at all.

4. The rope loop reinforces the idea that electrical resistance is like mechanical resistance

My view is that the idea of electrical resistance itself is already a mechanical metaphor, and it’s a metapor which brings with it a whole lot of unwanted baggage.

For example our experience of everyday resistances like muddy fields or difficult-to-turn-screws or walking through crowds of people is that we expend more effort when we come across them.

The reasoning goes something like: A and B are connected across the same p.d. The current in B is smaller. I wonder why that is. It’s as if B is resisting the flow of current more.

Electricity is not a mechanical process, and batteries don’t behave like people. There is an explanation using 11-16 concepts that explains why batteries produce a smaller current (and so less heating) in some circuits rather than others. But it’s quite subtle, so as a first order we should just teach the heuristic that they do.

Trying to model resistance as a frictional or other retarding effect will inevitably lead us down the wrong path - but we can still introduce the idea as a way to remember that big resistance means small current.

Conclusions for teaching

The rope loop can be used effectively to visualise how charges flow in simple circuits, and how energy is shifted from a chemical store to a thermal store. It’s good at showing that the charges are already there, that they all start moving everywhere at the same time, and the conceptual difference between a big and a small current.

It’s also good for showing subtle ideas like the fact that the speed at which energy starts being shifted at the ‘resistance’ is much quicker than the rate at which the charges move.

Its elegance is also its biggest weakness. It quickly gets asked to do jobs for which it’s unsuited, and so starts predicting the opposite of how real circuits behave. The fact that it fails silently makes it especially tricksy as a model.

Obviously I think the Electricity Explained simulation is better! But I think the main reason is that it’s not an analogy - it’s a mathematical model with the correct physics hard-wired into the code that has a nice visual output (which looks like a donation model, but isn’t really). It doesn’t require any explanation - it just behaves like a circuit does. Any change you make to the simulation will automatically be correct. You don’t have to translate from an attractive but somewhat deceptive mechanical model, where you can’t tell where the boundaries of applicability are unless you already understand circuit physics.

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*Why voltage (p.d.!) is a mathematical analogue for the dynamic frictional force the student’s hand exerts on the rope.

For an electrical resistance V = P/I

By analogy V = P/I for the rope is represented by P/(length of rope passing per second) = P/(speed of rope) = P/v

Work done per second by friction (i.e. power, P) = Fv

So V analogue for rope = P/v = Fv/v = F.

So frictional force is a mathematical analogue of voltage.