Confusing steady states and equilibrium?
Keith S. Taber
"…I am older than I once was
From the lyrics of 'The Boxer' (Simon and Garfunkel song) by Paul Simon
And younger than I'll be
But that's not unusual
No, it isn't strange
After changes upon changes
We are more or less the same
After changes we are more or less the same…"
In a recent post I discussed the treatment of Newtonian forces in a book (Thomson, 2005) about the history of natural theology (a movement which sought to study the natural world as kind of religious observance – seeking to glorify God by the study of His works) and its relationship to the development of evolutionary theory.
The book was written by a prestigious scientist, who had held Professorships at both Yale in the US and at Oxford. Yet the book contained some erroneous physics – 'howlers' of the kind that are sometimes called 'schoolboy errors' (as presumably most schoolgirls would be careful not to make them?)
Read 'Even Oxbridge professors have misconceptions'
'The Watch on the Heath'
by Prof. Keith Thomson
My point is not to imply that this is a poor read – the book has much to commend it, and I certainly thought it was worth my time. I found it an informative read, and I have no reason to assume that the author's scholarship in examining the historical sources was was not of the highest level – even if his understanding of some school physics seemed questionable. I think this highlights two features of science:
- Science is so vast that research scientists setting out to write 'popular' science books for a general readership risk venturing into areas outside their specialist knowledge – areas where they may lack expertise 1
- Some common alternative conceptions ('misconceptions') are so insidious that we confidently feel we understand the science we have been taught whilst continuing to operate with intuitions at odds with the science.
Out of specialism
In relation to the first point, I previously highlighted a reference to "Einstein's relativity theory" being part of quantum physics, and later in the book I found another example of a non-physicist confusing two ideas that may seem similar to the non-specialist but which to a physicist should not be confused:
"In the 1930s, Arthur Holmes worked out the geology of the mechanism [underpinning plate tectonics] and the fact that the earth's inner heat (like that of the sun) comes from atomic fission."
Thomson, 2005: 190
p.190
The earth contains a good deal of radioactive material which, through atomic fission, heats up the earth from within. This activity has contributed to the, initially hot, earth cooling much more slowly than had once been assumed – most notably according to modelling undertaken by Thomson's namesake, Lord Kelvin.2 Kelvin did not know about nuclear fission.
But the sun is heated by a completely different kind of nuclear reaction: fusion. The immense amount of energy 'released' during this process enables stars to burn for billions of years without running out of hydrogen fuel.3
Lord Kelvin did not know about that either, leading to him suggesting
"…on the whole most probable that the sun has not illuminated the earth for 100,000,000 years, and almost certain that he has not done so for 500,000,000 years"
Thomson, 1862
Kelvin suggested this was 'almost' but not 'absolutely' certain – a good scientist should always keep an open mind to the possibility of having missed something (take note, BBC's Nick Robinson).
We now think the sun has been 'illuminating' for about 4 600 000 000 years, almost ten times as long as Kelvin's upper limit. It may seem strange that a serious scientist should refer to the sun as 'he', but this kind of personification was once common in scientific writings.
Read about personification in science
The first atomic weapons were based on fission processes of the kind used in nuclear power stations.
Hydrogen bombs are much more devastating still, making use of fusion as occurs deep in the sun.
(Image by Gerd Altmann from Pixabay)
A non-scientist may feel this conflation of fission and fusion is a minor technical detail. But it is a very significant practical distinction.
For one thing the atomic bombs that were used to devastate Hiroshima and Nagasaki were fission devices. The next generation of atomic weapons, the 'hydrogen bombs' were very much more powerful – to the extent that they used a fission device as a kind of detonator to set off the main bomb! It is these weapons, fusion weapons, which mimic the processes at the centre of stars such as the sun.
…The rusty wire that holds the cork that keeps the anger in
From the lyrics of 'Two suns in the sunset' (Pink Floyd song) by Roger Waters
Gives way and suddenly it's day again
The sun is in the east
Even though the day is done
Two suns in the sunset, hmph
Could be the human race is run…
In terms of peaceful technologies, fission-based nuclear power stations, whilst not using fossil fuels, have been a major concern because of the highly radioactive waste which will remain a high health risk for many thousands of years, and because of the dangers of radiation leaks – very real risks as shown by the Three Mile Island (USA) and Windscale (England) accidents, and much more seriously at Fukushima (Japan) and, most infamously, Chernobyl (then USSR, now Ukraine). There are also serious health and human rights issues dogging the mining of uranium ore, which is, of course, a declining resource.
For decades scientists have been trying to develop, as an alternative, nuclear fusion based power generation which would be a source of much cleaner and sustainable power supplies. This has proved very challenging because the conditions under which fusion takes place are so much more extreme. Critically, no material can hold the plasma at the extreme temperatures, so it has to be magnetically suspended well away from the containment vessel 'walls'.
The tenacious nature of some misconceptions
My second point, the insidious nature of some common alternative conceptions, is a challenge for science teachers as simply giving clear, accurate presentations with good examples may not be enough to bring about change in well-established and perhaps intuitive ways of thinking, even when students study hard and think they have learnt what has been taught.
I suggested this was reflected in Prof. Thomson's text (Keith, that is, not Sir William) in his use of references to Newton's ideas about force and motion. Prof. Thomson was not as a biologist therefore seeking to avoid referring to physics, but rather actively engaging with Newton's notions of inertia and the action of forces to make his points. Yet, also, seemingly misusing Newtonian mechanics because of a flawed understanding. Likely, as with many students, Prof. Thomson's intuitive physics was so strong that although he had studied Newton's laws, and can state them, when he came to apply them his own 'common-sense' conceptions of force and motion insidiously prevailed.
The point is not that Prof. Thomson has got the physics wrong (as research suggests most people do!) but that he was confident enough in his understanding to highlight Newtonian physics in his writing and, in effect, seek to teach his readers about it.
Newton's laws
What are commonly known as 'Newton' three laws of motion' can be glossed simply as:
N1: When no force is acting, an object does not change its motion: if stationary, it remains stationary; if moving, it carries on moving at the same speed in the same direction.
Indeed, this is also true if forces are acting, but they cancel because they are balanced, i.e.,
N1': When no net (overall, resultant) force is acting, an object does not change its motion: if stationary, it remains stationary; if moving, it carries on moving at the same speed in the same direction.
N2: When a net force is acting on a body it changes its motion in a way determined by the magnitude and direction of the force. (The change in velocity takes place in the direction of the force, and at a rate depending on the magnitude of the force).
So, if the force acts along the direction of motion, then the speed will change but not direction; but if the force acts in any other direction it will lead to a change in direction.
Strictly, the law relates to the 'rate of change of momentum' but assuming the mass of the body is fixed, we can think in terms of changes of velocity. 4
N3: Forces are interactions between two bodies/objects (that attract or repel each other): the same size force acts on both. (This is sometimes unfortunately phrased as 'every action having an equal and opposite reaction') 5.
These (perhaps) seem relatively simple, but there are complications in applying them. Very simply, the first law,when applied to moving bodies does not seem to fit our experience (moving bodies often seem to come to a stop by themselves – due to forces that we do not always notice).
The second law relates an applied force to a process of change, but it is very easy to instead think of the applied force directly leading to an outcome. That is people often equate the change in direction with the final direction. The change occurs in the direction of the force: that does not mean the final direction is the direction of the force.
The third law is commonly misapplied by assuming that if 'forces come in pairs' these will be balanced and cancel out. But they cannot cancel out because they are acting on the two bodies. (If your friend hits you in the eye after one too many pedantic complaints about her science writing you cannot avoid a black eye simply by hitting her back just as hard!)
Often objects are in equilibrium because the forces acting on them are balanced. But they are never in equilibrium just because a force on them is also acting on another body! An apple hangs from a tree because the branch pulls it up the same amount as its weight pulls it down: these are two separate forces, each of which is also acting on the other body involved (the branch, and the earth, respectively).
Read about learning difficulties and Newton's third law
Thomson's 'Newtonian Physics'
In the previous posting I noted that Prof. Thomson had written
- "Any trajectory other than a straight line must be the result of multiple forces acting together."
- "the concept of 'a balance of forces' keeping the moon circling the earth and the earth in orbit around the sun…
- "a Newtonian balance of forces… rocks: gradually worn down by erosion, washed into the seas, accumulating as sediments, raised up as new dry land, only to be eroded again"
The first two statements are simply wrong according to conventional physics. Curved paths are often the result of a single force acting. The earth and moon orbit because they are both the subject of unbalanced forces.
Those two statements are contrary to N1 and N2.
The third statement seemed to suggest that a balance of forces was somehow considered to bring about changes. The suggestion appeared to be that a cycle of changes might be due to a balance of forces. But I acknowledged that "this reference to Hutton's ideas seems to preview a more detailed treatment of the new geology in a later chapter in the book (that I have not yet reached), so perhaps as I read on I will find a clearer explanation of what is meant by these changes being based on a theory of balance of forces".
Now I have finished the book, I wanted to address this.
A sort of balance
Prof. Thomson discusses developing ideas in geology about how the surface of the earth came to have its observed form. Today we are familiar with modern ideas about the structure of the earth, and continental drift, and most people have seen this represented in various ways.
However, it was once widely assumed that the earth's surface was fairly static , but had been shaped by violent events in the distant past – a view sometimes called 'catastrophism'. One much referenced catastrophe was the flood associated with the biblical character Noah (of Ark fame) that was sometimes considered to have been world-wide deluge. (Those who considered this were aware that this required a source of water beyond normal rainfall – such as perhaps vast reservoirs of water escaping from underground).
The idea that the earth was continually changing, and that forces that acted continuously over vast periods of time could slowly (much too slowly for us to notice) lead to the formation of, for example, mountain ranges seemed less feasible.
Yet we now understand how the tectonic plates float on a more fluid layer of material and how these plates slowly collide or separate with the formation of new crust where they move apart. Vast forces are at work and change is constant, but there are cyclic processes such that ultimately nothing much changes.
Well, nothing much changes on a broad perspective. Locally of course, changes may be substantial: land may become submerged, or islands appear from the sea; mountains or great valleys may appear – albeit very, very slowly. But crust that is subsumed in one place will be balanced by crust formed elsewhere. And – just as walking from one side of a small boat to another will lead to one side rising out of the water, whilst the opposite side sinks deeper into the water – as land is raised in one place it will sink elsewhere.
This is the kind of model that scientists started to develop, and which Prof. Thomson discusses.
"[Dr John Woodward (1665-1728) produced] "an ingenious theory, parts of it quite modern, parts simply seventeenth century sophistry within a Newtonian metaphor. Woodward's earth, post deluge, is stable, but not in fact unchanging. This is possible because it is in a sort of balance – a dynamic balance between opposing forces."
Thomson, 2005: 156
Plus ça change, plus c'est la même chose
James Hutton (1726 – 1797) was one of the champions of this 'uniformitarianism',
"Hutton's earth is in a constant state of flux due to processes acting over millions of years as mountains are eroded by rain and frost. In turn, the steady raising up of mountains, balances their steady reduction through erosion.
…for Hutton the evidence of the rocks demonstrated a cyclic history powered by Newtonian steady-state dynamics: the more it changed, the more it stayed the same."
Thomson, 2005: 181
p.181
The more it changed, the more it stayed the same: plus ça change, plus c'est la même chose. This, of course, is an idiom that has found resonance with many commentators on the social, as well as the physical, world,
"…A change, it had to come
From the lyrics of 'Won't get fooled again' (The Who song), by Pete Townsend
We knew it all along
We were liberated from the fold, that's all
And the world looks just the same
And history ain't changed
'Cause the banners, they all flown in the last war
…
There's nothing in the street
Looks any different to me
And the slogans are effaced, by-the-bye
And the parting on the left
Is now parting on the right
And the beards have all grown longer overnight…"
Steady states
So, there are vast forces acting, but the net effect is a planet which stays substantially the same over long periods of time. Which might be considered analogous to a body which is subject to very large forces, but in such a configuration that they cancel.
Where Prof. Thomson is in danger of misleading his reader is in confusing a static equilibrium and a macroscopic overall steady state that is the result of many compensating disturbances. This is an important difference when we consider energy and not just the forces acting.
A steady state can be maintained by nothing happening, or by several things happening which effectively compensate.
If we consider a very heavy mass sitting on a very study table, then the mass has a large weight, but does not fall because the table exerts a balancing upward reaction force. Although large forces are acting, nothing happens. In physics terms, no work is done. 6
Now consider a sealed cylinder, perfectly insulted and shielded from its surroundings, containing some water, air and too much salt to fully dissolve. It would reach a stead state where the
- the mass of undissolved salt is constant
- the height of the solution in the tube is constant
On a macroscopic level, nothing then happens – it is all pretty boring (especially as if the cylinder was perfectly insulated we would not be able to monitor it anyway!)
Actually, all the time,
- salt is dissolving
- salt is precipitating
- gases from the air are dissolving in the solution
- gases are leaving the solution
- water is evaporating into the air
- water vapour is condensing
But the rates of 1 and 2 are the same; the rates of 3 and 4 are the same; and the rates of 5 and 6 are the same. In terms of molecules and ions, there is a lot of activity – but in overall terms, nothing changes: we have a steady state, due to the dynamic equilibria between dissolving and precipitating; between dissolving and degassing; and between evaporation and condensation.
This activity is possible because of the inherent energy of the particles. In the various interactions between these particles a molecule is slowed here, an ion is released from electrical bonds – and so. But no energy transfer takes place to or from the system, it is only constantly redistributed among the ensemble of particles. No work is done.
Cycling is hard work
But macroscopic stable states maintained by cyclic processes are not like that. A key difference is that in the geological cycles there are significant frictional effects. In our sealed cylinder, the processes will continue indefinitely as the energy of the system is constant. In the geological systems, change is only maintained because there is source of power – the sun drives the water cycle, radioactive decay in effect drives the rock cycle.
Work is done in forming new crust under the sea between two plates. More work is done pushing one plate beneath another at a plate boundary. It does not matter if the compensating changes were produced by identical magnitude forces pushing in opposite directions – these are not balanced forces in the sense of cancelling out (they act on different masses of material) – if they had been, nothing would have happened.
You cannot move tectonic plates around without doing a great deal of work – just as you cannot cycle effortlessly by using a circular track that brings you back to where you started, even though when cycling in one direction the ground was pushing you one way, and on the way back the ground was pushing you in the opposite direction! (Your tyres pushed on the track, and as Newton's third law suggests, it pushed back on the tyres in the opposite direction – but those equal forces did not cancel as they were acting on different things: or you would not have moved.)
Perhaps Prof. Thomson understands this, but his language is certainly likely to mislead readers:
"Hooke realised that there was a balance of forces: while the geological strata were being formed and mountains were raised up, at the same time the land was constantly being eroded…"
Thomson, 2005: 179
No, there was not a balance of forces.
It could be that Prof. Thomson's use of the phrase 'balance of forces' is only intended as a metaphor or an analogy. 7 However, he also repeats errors he had made earlier in the book
- "the concept of 'a balance of forces' keeping the moon circling the earth and the earth in orbit around the sun"
- "any trajectory other than a straight line must be the result of multiple forces acting together"
which suggests a genuine confusion about how forces act.
One of these mistakes is that planetary orbits (which require a net {unbalanced} force), are due to 'opposing forces',
"…Paley's tortured dancing on the heads of all these metaphysical pins is pre-shadowing of modern ecological thinking and a metaphysical extension of Hooke and Newton's explanation of planetary orbits in terms of opposing forces, or Woodward's theory of matter, or Hutton's geology – it is the living world as a dynamic system of force and counterforce, of checks and balances."
Thomson, 2005: 242 (my emphasis)
p.242
The other was that a single force cannot lead to a curved path,
"…the philosophical concept of reduction, namely that any complex system can be reduced to the operation of simple causes. Thus the parabolic trajectory of a projectile is the product of two straight-line forces acting on each other [sic];…"
Thomson, 2005: 264 (my emphasis)
p.264
Forces are interactions between bodies, they are abstractions and do not act on each other. The parabolic path is due to a single constant force acting on a body that is already moving (but not in the direction of the applied force). It can be seen as the result of the combination of a force (acting according to N2) and the body's existing inertia (i.e., N1). Prof. Thomson seems to be thinking of the motion itself as corresponding to a force, where Newton suggested that it is only a change of motion that corresponds to a force.
However, whilst Prof. Thomson is wrong, he is in good company – as one of the most common alternative conceptions reported is assuming that a moving body must be subject to a force. Which, as I pointed out last time, is not so daft as in everyday experience cars and boats and planes only keep on moving as long as their propulsion systems function (to balance resistive forces); and footballs and cricket balls and javelins that do not have a source of motive power (to overcome resistive forces) soon fall to earth. So, these are understandable and, in one sense, very forgiveable slips. It is just unfortunate they appear in an otherwise informative book about science.
Sources cited:
- Thomson, K. (2005). The Watch on the Heath: Science and religion before Darwin. HarperCollins.
- Thomson, W. (1862). On the Age of the Sun's Heat. Macmillan's Magazine, 5, 388-393.
- Thorn, C. E., & Welford, M. R. (1994). The Equilibrium Concept in Geomorphology. Annals of the Association of American Geographers, 84(4), 666-696. http://www.jstor.org/stable/2564149
Notes
1 Although there are plenty of 'academic' books in many fields of scholarship (usually highly focused so the author is writing about their specialist work), the natural sciences tend to be communicated and debated in research journals. Most books written by scientists tend to be for a more general audience – and publishers expect popular science books to appeal to a wide readership, so these books are likely to have a much broader scope than academic monographs.
2 When he was ennobled, William Thomson chose to be called Baron Kelvin – after his local river, the river Kelvin. So the SI unit of temperature is named, indirectly, after a Scottish River.
Kelvin's reputation was such that when he modelled the cooling earth and suggested the planet was less that a 100 000 000 years old, this caused considerable concerns given that geologists were suggesting that much longer had been needed for it to have reached its present state.
3 For a brief discussion regarding energy changes during processes of this kind, see 'How much damage can eight neutrons do?'
4 The rate of change of momentum is proportional to the magnitude of the applied force and takes place in the direction of the applied force.
As momentum is mv, and as mass is usually assumed fixed (if the motion is well below light speeds) 'the rate of change of momentum' is the mass times the rate of change of the velocity – or ma. (F=ma.)
The key point about direction is that it is not that the body moves in the direction of the force, but the change of momentum (so change of velocity) is in the direction or the force.
As the body's momentum is a vector, and the change in momentum is a vector, the new momentum is the vector sum of these two vectors: new momentum = old momentum + change in momentum.
The object's new direction after being deflected by a force is in the direction of the new momentum
5 When there is force between two bodies (let's call them A, B) the force acting on body B is the same size as the force acting on body A, but is anti-parallel in direction.
6 This is an ideal case.
A real table would not be perfectly rigid. A real table would initially distort ever so slightly with the area under the mass being ever so slightly compressed, and the weight dropping to an ever so slightly lower level. The very slight lowering of the weight does a tiny amount of work compressing the table surface.
Then, nothing more happens, and no more work is done.
7 Thorn and Welford (1994) have referred to "the fuzzy and frequently erroneous use of the term…equilibrium in geomorphology" (p.861), and how an 1876 introduction of the "concept of dynamic equilibrium resembles the balance-of-forces equilibrium that appears in dynamics, but by analogy rather than formal derivation" (p.862).