Intermediate stages in a process (energy chains and ladders)

This is the first of three posts about situations in which people are tempted to include intermediate stages as part of a chain in an energy analysis. These are sometimes called ‘ladders’ or ‘chains’. In these posts, I will explain why it is better to avoid them and how to do so.

This first post is an overview and I have expanded some of the examples in the following posts on HEP and circuits (like appendices).

Reasons why chains and ladders are unhelpful

When discussing how we represent energy at school, a question often comes up about the role of intermediate processes and whether they need to be included in an energy analysis. It is certainly the case that, historically, we have included them. And therefore, we are used to seeing them. But that does not mean that they are either required (for an energy discussion) or helpful.

In fact, I suggest the reverse is the case. In an energy analysis (for the whole process), intermediate, transient stores serve no useful purpose. In fact, they only complicate and confuse any discussions or activities – especially with younger students. And therefore, it is better to omit them.

Typical situations in which this question comes up are:

1. Lifting a mass with a clockwork motor
2. A hydro-electric power station
3. Lighting a light bulb
4. Car making a journey
5. Heating a room
6. Pedalling a bike
7. A loudspeaker playing music

In teaching schemes and assessments of the recent past (using types and transformations in the way that it had come to be used), an analysis of each of the above would probably have included an intermediate ‘type’. For example:

1. Kinetic energy of the rising mass
2. Kinetic energy of the water, the turbines etc.
3. ‘Electrical energy’ in the circuit and ‘heat’ stored in the lamp’s filament
4. Kinetic energy of the car, its wheels etc
5. ‘Heat’ stored in a radiator
6. Kinetic energy of the chain, the wheels etc
7. ‘Sound energy’

(I have used inverted commas to signify terms that are best avoided).

Including these intermediates as part of a chain is compelling: partly because we are used to them; and partly because they appear to be important (or essential) to the operation and/or fundamental purpose of the device. However, these are not good reasons for doing so.

The role of energy

In all of the examples above, it is certainly the case that, in order to operate, a device has to get up to speed, reach a certain temperature or produce a sound. There are rich and interesting physics explanations about mechanisms of these on-going processes. They will include: moving masses, balanced forces, electric currents, temperature rise in resistors, hot wires radiating across the EM spectrum, vibrating diaphragms producing sound waves and so on.

However, trying to replicate the processes with a labelling exercise relating to energy does not shed any extra light on the phenomena or their processes. Instead, it introduces arbitrariness, confusion and ambiguity. Whilst also masking the underlying physics. Furthermore, it does not achieve any useful learning outcomes in terms of understanding or using energy; i.e. it does not prepare students for calculations or for having meaningful discussions about world energy resources.

In other words, including the intermediate stages is a distraction from the underlying physics without laying any helpful foundations for energy analyses. Therefore, such chains should be avoided.

Let’s see how this applies to the example of the rising mass.

A. The rising mass.

I’ll quickly define the situation. We use a clockwork motor (pre-wound) to lift a 100 g mass through 80 cm from the floor to the bench. The mass is attached to the motor by a light steel wire. Let’s say it takes 16 seconds.

Now, it clearly has to move in order for it to be lifted from the floor to the bench (as it happens, in this example, it moves at an average 5 cm/s).

In an old assessment item, given that it is moving, we might have expected a student to say that the energy story was something like:

elastic energy ⇒ kinetic energy (plus ‘sound energy’) ⇒ gravitational potential energy (plus ‘heat’)

Why is this chain of energy unhelpful?

This emergent chain of energy stages is a tell-tale sign that we are forcing energy into a role for which it is not fit. It is an attempt to replicate or mirror those processes and mechanisms with chains of energy stores (or types). And that is unhelpful. Here are some reasons why.

Including intermediate stores in a chain is:

• Obscuring: it operationalises the energy analysis and hides all the interesting underlying processes and mechanisms – i.e. the physics.
• Misleading: The chain implies that, somehow, there is a kinetic store that is emptied in order to fill a gravitational store. And that this is happening throughout the upward journey – i.e. that the gravitational store can only fill if energy has passed through a kinetic store. i.e. that energy has somehow been carried by the moving masses. This picture does not reflect the physical reality. A more helpful representation is to refer to the clockwork motor exerting a force on the mass and, because that force is moving, the system is working (mechanically) to lift the mass against gravity.
• Arbitrary: if we were to include kinetic energy, it would be an arbitrary choice. There are other ways in which energy is temporarily stored but we have chosen to ignore them. For example, the supporting wire will be in tension. An elastic store fills (a little) at the start, stays constant and empties at the end. Omitting the elastic store and including the kinetic store would be entirely arbitrary. If we (quite rightly) ignore the minimal amount of energy stored elastically, then the consistent approach is also to omit the energy stored kinetically as it ascends.
• Unenlightening: it does not reveal anything interesting about the way in which the device operates. Nor does it achieve anything positive relating to learning about energy. It does not provide any tools to help students think about, explain or analyse the situation. Specifically, it is...
• ...not relevant for any calculation: If we wanted to calculate the speed of the masses or the time it takes to rise, we would not use an energy analysis to do so (at no point is the amount kinetic energy gained equal to the amount that the energy stored elastically has decreased). Similarly, there is no (useful) calculation in which we would think of the kinetic store emptying and a gravitational store filling. One way of thinking about this is that, the amount of energy stored due to its movement will not affect the final outcome (please note, I am aware that its speed will affect the final outcome – because of increased friction – but that is different).
• Minimal: not a reason on its own to exclude kinetic energy; but worth noting. It accounts for about 0.01% of the increase of energy stored gravitationally.
• Ambiguous and confusing: once it is included, there are many other stores of similar magnitude that could be included. And many ways of including them. There is no unambiguous solution to the energy analysis. This is more obvious in an example of, say, a hydro-electric power station. For a student who is given a basic set of tools to think with (whether it is stores or types), it wouldn’t be possible to reason their way to an unambiguous solution for the chain of stores that arises at a HEP station (or any new situation). And, given that such is the case, it’s not much like physics – it is simply a labelling exercise with ad-hoc, imposed rules.
• It has led to spurious forms of energy: the terms ‘sound energy’, ‘light energy’, ‘heat energy’ and ‘electrical energy’ have been introduced in order to try to mirror physical processes in the labelling exercise. None of these is helpful, some of them are misleading and some have no physical referent.

A more helpful approach

These concerns have arisen by trying to describe an on-going process with an energy story that mirrors the description based on physical processes and mechanisms. And in doing so masks that physics. A more helpful approach is to analyse the journey in terms of forces and motion. And reserve the energy analysis for determining the change in the system between the beginning and end of the on-going process. i.e. to use start and end points – chosen in a way that keeps the discussion simple.

First let’s look at the mechanisms and processes for the rising mass:

Mechanisms and processes

In brief, the coiled spring in the motor exerts a force on the mass (bigger than the gravitational force). The unbalanced force gets the mass moving but, within a very short time, friction in the system balances the net upwards force and the mass reaches terminal velocity. It continues to be dragged up at a constant speed. All the while, the frictional forces cause the temperature of the gears and axle to rise a little and they, in turn, raise the temperature of the surroundings.

This discussion can be expanded, reduced, embellished, challenged and recast until there is an agreed version. Because it is based on physics rather than on different interpretations of labelling rules.

Trying for a more helpful and elegant energy story?

The first step is to choose start and end points. Let’s start at the point where the motor is fully wound and the mass is on the floor; and choose an end point when it has reached the top. This makes the energy story simpler, less confusing and (I hope) easier to reach agreement.

Between the start and end points:

There is a reduction in the energy stored elastically and an increase in both the energy stored gravitationally and the energy stored thermally. During the process (of rising), the clockwork motor is working (mechanically) to haul the masses up against gravity. The temperature of the gears rises (through working) and the hotter gears rise the temperature of the surroundings (through heating).

If you are using the language of stores, then the phrasing becomes simpler again:

An elastic store has emptied (a bit) and, through mechanical working, the levels have increased in a gravitational store, a thermal store (associated with the mechanism) and a thermal store associated with the surroundings.

It might look like this in a diagram.

Summary of suggestions

• Avoid chains of intermediate stores. To do so will lead to making arbitrary choices or devising arbitrary (unphysical) rules.
• Introduce the idea of start and end points and choosing (or allowing students to choose) them carefully.
• Beware of trying to replicate the physics story with a chain of energy stores and how they are linked up.
• As ever, I suggest separating the physical description from the energy analysis: 
  • talk about processes and mechanisms to tell the physics story – there will be a lot in it and it will always be worthy of discussion. This should always be the first step – so as not to lose the physics by over-operationalising the task of energy analysis.
  • introduce energy as an analysis tool. Defining a start and end point is helpful. It allows us to have an unambiguous and agreeable description of how the energy is stored before and after an event.

This approach does not require a change in the way that you represent energy (i.e. it is not imperative to talk about stores and pathways). However, it works well if you do so. And then choose start and end points that make the story simple and allow you to make the point(s) that you would like to make without getting bogged down in filling in gaps.

I have gone into some of the examples (HEP and circuits) in more detail in the following posts.

Also it is always worth checking out the SPT resources on energy.