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August 2011 Archives

I had a very interesting day yesterday in Auckland, at a NZ Engineering Educator's forum.  Here, there were representatives from across the tertiary sector looking at ways of improving the way that Engineering is done at universities and polytechnics.

The main speaker was Keith Willey, from the University of Technology, Sydney. He gave some great insights into the way that feedback works and how students can use each other to learn (peer learning). He's done a lot of work trying to get these strategies to work really well. He showed a few video clips from his classrooms - the most obvious thing that struck me was just how noisy it was. 

Keith talked a bit about a few strategies he's used - e.g. multiple choice scratch-cards to give the students really instant feedback, but I'll share the most totally outrageous one - allowing students to talk to each other during tests and exams.

It needs a bit of qualifying - students are allowed to talk to each other (but not write anything down) in the first fifteen minutes of the exam. The point is that the exam then becomes a learning experience in its own right - not just a summative exercise. Students can discuss strategies for tackling particular problems before doing them, and learn from each other. 

That, of course, is the point. The idea of an engineering degree is that a student who completes it has 'learned' - has acquired knowledge, skills, abilities etc that are suited for engineering. The role of the teaching staff is to provide them with, and to help them  take, opportunities to learn.  And an exam is one of those opportunities.

Very, very interesting. It would take some nerve to implement it here.

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Lately, I've been doing a bit of reading about the use of group theory in particle physics. I need to do this because I'm meant to be teaching it in the next few weeks. Now, the education research says I can still teach something without being an expert in it -  I just need to be able to inspire my students to learn it themselves. That's re-assuring, because I am certainly no expert in particle physics. Of all the area of physics, it's the one I've always had most trouble grasping. Perhaps it's because you can't neatly summarize it with one equation - I'm not sure.

Anyway, one thing I am sure about is that finding a textbook on group theory in physics (and particle physics in particular) that is accessible to mortals like me is a bit of a mission. The books I've looked at in our library, and there are lots of them, tend to give the impression that the whole thing is insanely complicated. After a fair bit of reading, I'm beginning to get the hang of some of it, but it is really nasty stuff.

As I've already said, group theory is about looking at symmetries. In particle physics, we can see lots of symmetries of various forms, so groups sit naturally here. An example (probably the simplest one) is the symmetry between the neutron and the proton. These two particles make up the nucleus of an atom. At school we're told that the proton has a positive charge, a neutron has no charge, and that the two have near-identical masses. In fact, it's not just (nearly)  the same  mass that the proton and neutron share - they are pretty-well identical, except for their charge.

This leads to the question of whether, in some fundamental way, the proton and neutron are actually  different manifestations of the same thing. Heisenberg developed this idea with his 'isospin' theory. It turns out that there's a clever mathematical way of describing isospin, using what's known as the SU(2) group. This group contains the underlying physics - when we look at its symmetries the neutron and proton states naturally 'drop-out', in the same way that  (some of) the gaits of a quadruped 'drop-out' of the analysis of the symmetries of a rectangle.

But it gets better than that. Not only does this group describe neutrons and protons it describes other, not so well known particles - the pions, as a neat mathematical combination of two nucleons. (By nucleon we mean a neutron or proton).  The whole physics of the way that neutrons and protons interact through exchange of pions is encapsulated in the SU(2) group. Really neat!

Unfortunately, the realm of particle physics is rather bigger than neutrons, protons and pions, and so our SU(2) group doesn't get us terribly far. But it makes a start, and certainly helps us (by which I mean me) to see why particle physicists like to bleat on about symmetry groups.

 

 

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There's a great article in Physics World on crop circles. Not a discussion about man-made / weather-made / UFO-made  - any sensible interpretation would be man-made - but just HOW do you make such intricate and vast patterns so quickly and leave almost no traces behind. Some of the patterns that crop-up (sorry) in crop fields can be fractals, reproduced to an astonishing level of detail.

There's some evidence that the crop-circle makers are really very scientifically based and have moved beyond the rope, peg and stomping board and are armed with magnetrons and other secret techniques by which they carry out their art.

Read the article and see what you think. I love the bit about a couple of the makers being motivated to produce ever more detailed patterns in an attempt to ridicule the scientists who clung to a natural (e.g. whirlwind) view of their formation.

And why are there so few crop circles in NZ?  Are we working our students too hard that they don't have time to get involved in this stuff?

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It's mid-semester break here at Waikato so I have time to breathe and get back to things other than teaching, such as seeing what the PhD students are up to. Yay.

But, here's a comment about what I was talking about last week with the first year students: conservation of momentum.

If you look in first-year textbooks with regard to conservation of momentum in two dimensions, they tend to be full of examples about colliding billiard balls and car crashes. The former is a rather tedious example of an elastic collision (one in which kinetic energy is conserved) - the latter a nasty example of an inelastic-ish collision (in which the projectiles stick together after collision).  But there are a lot of more interesting examples to be found, and it's always nice when I see a textbook that uses them.

For example, why not talk about the Large Hadron Collider, rather than billiard balls. The LHC collides protons together, and momentum is conserved. True, the products of the collision can be many and varied (maybe even a Higgs Boson - who knows?), and we'll have to use special relativity to analyze them properly, but momentum will be conserved. It's a nice topical example - far more inspiring than billiard balls and car crashes.

Here's another example from the realm of the small - Compton scattering. This happens when a gamma ray or X-ray scatters elastically from an electron. The electron recoils, takes away energy from the gamma ray, which then changes its wavelength. Arthur Compton worked out that there was  a relationship between the observed change in wavelength of the wave and the angle through which the wave is scattered, and this could be explained by a single interaction between the gamma wave and the electron.  To do this he used momentum and energy conservation (it's an elastic collision) - with the complication that it has to be done relativistically. In fact, Compton Scattering can be considered an experimental proof of special relativity and quantum mechanics - the experimental results tie in with the relativistic predictions. We get our third-year students to do this experiment, and it generally works very well. One can even extract the mass of the electron from the results.

Arthur Compton received a Nobel Prize in 1927 for what can be viewed as applying momentum conservation to a simple collision. I think it's well worth talking about in first-year physics - students might struggle with the relativity bit, but the concepts are absolutely easy, and the result is really significant.

Better than car crashes for sure.

 

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There were a couple of moments in the first-year lab yesterday that made me want to despair:

The first one:

Student: My magnetic field doesn't change when I increase the current

Me, seeing what the problem is: How do you connect an ammeter in a circuit?

Student: In series. Um...oh, hang on...we've done parallel, haven't we?

The second one:

Student: We've switched it on, but nothing's happening. 

Me, seeing that the red 'on' light isn't on: Is it plugged in?

At this point I could have despaired - how can we possibly teach students who can't plug something in or put an ammeter in series with what they want to measure.  But, I haven't, because I also teach in the third year labs, where things often are at the other extreme - e.g. students suggesting ways of improving the experiment - and sometimes actually going and doing it. Last week, we were working with a ferrofluid (a magnetic fluid - pretty cool stuff) and a pair of them, while waiting for their experiment to settle, decided that a good way to fill the time would be to build the biggest electromagnet they could out of what they could find in the lab and see how the fluid responded to it.  I think that shows some initiative, some real understanding of physics (since they went on to build the thing) and actual ability to do experimental work for themselves.  The result was a little bit messy but quite interesting.

So we have students who perhaps can't plug in an ammeter properly in year 1 turning into real physicists by the end of year 3, so that's no cause for despair, really.

 

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Those of you who own a four-legs will have noticed that they usually exhibit a range of different gaits depending on the occasion. Taking Mizuna our cat as an example - he walks (back-left, front-left, back-right, front-right, each leg a quarter of a cycle behind the previous), he trots (back-left and front-right together - then back-right and front-left together, each movement half a cycle behind the previous), he runs (back-left, back-right, front-left, front-right, or alternatively back-right, back-left, front-right, front-left).  Mostly he sleeps, but that's not a gait.

There are two other forms of motion he employs occasionally. There's the bound (back legs together, front legs together) which is his gait-of-choice for high-speed ascent of the stairs (he trots on the way down) and the highly amusing pronk, in which he bounces on all four legs simultaneously - this is reserved for moments of great excitement such as being fed. Yes, pronk is a real word!

We bipeds, on the other hand, are rather short of choices when it comes to motion. We can walk (left, right...) or we can jump (legs together) and that is it. I don't count symmetry-breaking gaits here, such as the hop or the skip - they are a bit unusual, just like the quadruped's canter - and I count 'run' as being the same as walk, in that our two legs still move in anti-phase to each other.

The increased range of gaits available to the quadruped can be naively attributed to them having more legs, but perhaps a better description would be that they have increased symmetry. We bipeds have a single mirror plane, left-to-right when it comes to legs, but quadrupeds, with legs arranged in a rectangle, can be thought of as having a front-to-back mirror plane as well. (Yes, I know they don't really have a mirror plane here, but the legs approximately have.)

Group theory is a mathematical encapsulation of symmetry. We can use it in physics to simplify problems. A typical example is finding the modes of vibration of a molecule with a particular symmetry. It's often presented as rather abstract mathematics but when applied to physics it becomes beautifully and simply powerful. For example, our pronk, trot, bound, and another gait, the pace (left legs together, right legs together - not sure if any animal does this one), drop out of the analysis for a rectangular arrangement of legs. (The walk and run/gallop are a bit more subtle). Applied to the biped, we simply obtain the walk and the jump. The more symmetry you have, the greater the range of gaits you have.

In the limit of lots of symmetry indeed (the millipede, which approximately has complete translational symmetry along its length) there are a huge number of gait options. We can then start describing these in terms of waves, and, in particular, by wavelengths and frequencies of the action rippling down the body. This then has analogies with other physical systems, such as vibrational modes in solids (phonons), where different frequencies of sound wave travel at different velocities through the solid. Group theory isn't just abstract, as many textbooks would make out, it really is quite practical and fun.

So, next time you come across a four-legs, or a six-legs, have a careful look at how it moves.

P.S. I've drawn a bit from my memory of an Ian Stewart book here, though I can't quite remember which one. I'm not sure I still own it.

 

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Friday morning saw me doing my usual Friday-morning-thing, namely work on my PGCert Tertiary Teaching portfolios. (I've put in a recurring appointment in my calendar every Friday morning for this semester so I actually get down to doing this task.) As part of this, I've been pulling together relevant blog entries on my teaching experiences.

A dangerous move, because I've started reading them again. Some of them are full of good intentions that haven't quite materialized. Here's a quick example. Back in April this year, I wrote about the book 'Assessment for Learning' by Paul Black et al. I highlighted the discovery that giving a (secondary) student a summative mark with a piece of work (e.g. 8 out of 10) completely negates any formative comments you write on the work - i.e. you may as well not have bothered writing any comments. However, if you don't put a mark on the work, the students will take note of your comments and improve. I then said "Worth a shot in one of my papers..."

I now recall that I intended to do this with my experimental physics class. (For this class, it's a fairly easy thing to do because I spend a lot of time with them in the laboratory, and the course is completely under my control - I don't have to fit in with what another lecturer is doing.) So my intention was not to give the students marks on their work, but rather give them feedback and discuss with them where they can improve. However, my intention has not turned into reality. It's halfway there, in that I give them comments, discuss the work with them, and then ask them to give themselves a mark (Phil Race style, www.phil-race.co.uk, see also here), but after they do that I end up giving them a mark (which usually is pretty close to their mark.) Not quite what I had in mind earlier.  There is next year.

Of course, at some point, I would need to get 'summative', because the students need to get a grade for every course they do. But there are probably plenty of options for doing that.

 

Black, P., Harrison, C., Lee, C., Marshall, B. & Wiliam, D. (2003) Assessment for Learning. Maidenhead, U.K.: Open University Press.

 

 

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Going back to my last entry on the sliding car, it's worth commenting a bit more on the nature of friction here. When a car goes round a corner, what prevents it from sliding is the friction between the tyres and the road. Tyres are unsurprisingly designed to be able to give a high frictional force when in contact with the road. If you've got access to a car tyre that's not  attached to a car, try putting it up vertically (as when mounted on a car) and pushing it sideways across the road. Not at all easy, which is quite reassuring, really.

Friction is a complicated beast. We usually separate discussion into 'kinetic' friction and 'static' friction. Kinetic friction is what happens when an object is moving; static when the object is stationary. Kinetic friction can often be described nicely by 'the coefficient of kinetic friction'; in this case the frictional force (which of course acts against the direction of movement) is given by the coefficient of kinetic friction times the normal force that the surface exerts on the object. From Newton's second law, the normal force exerted on an object on a flat surface will equal the object's weight (but that's not true on an inclined surface) and so the heavier an object is, the greater the frictional force on it when it's sliding. That should pretty well tie in with your personal experience, I'm sure. Pushing that filing cabinet is so much easier when you take the files out first.

The coefficient of friction itself depends on the nature of the two surfaces - so rubber on asphalt has a pretty high coefficient of friction, but steel on ice (ice-skate style) is extremely low.

Things are similar but slightly more complicated when an object doesn't move. We use a coefficient of static friction now, but this time we have to say that the frictional force is less than or equal to the coefficient of static friction times the normal force. (If that force isn't sufficient to hold the object in place, it will start sliding.) So, the larger the coefficient of static friction is, the steeper the ramp you need before an object starts sliding down it.

Now, things often get interesting with friction because the coefficient of kinetic friction can be considerably less that the coefficient of static friction. What this means is that an object can be hard to get moving, but, once it is moving, sliding it becomes much easier. An example is shifting furniture around our new house by pushing it across the carpet. The difficult bit is to get the chest of draws to move to start with - but once it is moving, maintaining its movement isn't so tricky.

A great example of the interplay between kinetic and static friction is with bowing a violin string. The string moves in a 'slip-stick' manner. It will stick to the bow, and move with the bow, until the restoring force on it is large enough to get it to move across the bow (the 'slip') which it will then do very easily, returns towards its original position and overshoots (like simple harmonic motion). Restoring forces then bring down the velocity of the string, and, once the velocity of the string is reduced, it sticks again and the cycle continues. A nice little animation and comprehensive explanation can be found here.

 

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I've recently had a look at the 2010 New Zealand Scholarship Physics exam, for the first time. (This is the exam taken by the top final year school students in physics - the best performers get rewarded with scholarships that will help them financially at university).

The scholarship exams are hard. There's no denying that. For physics, this has often been seen in the way that several concepts can be mixed up in one question. To do well, the students have to be good at identifying what is going on, physically, in a particular situation, and pulling together their knowledge over different areas of physics to make sense of it. The famous 'equation-spotting' method (see what information you are given, see what you are told to find, and find an equation from the list given that contains all those quantities in it) does not work at scholarship level. There's an infamous example from a few years back (the 'phugoid oscillator' - go look up what those are if you're interested) that needed five different equations just to answer one part of the question.

However, in the 2010 scholarship exam I think there was a subtle shift in the type of question asked. Instead of having questions where the student had to identify and pull together the concepts, the questions seemed more focused on individual concepts, but having the student drill down to the core of that concept to see how well they really understood it.

At this point, a big disclaimer is in order: I DO NOT SET SCHOLARSHIP EXAMS; I DO NOT MARK THEM, AND HAVE NOTHING WHATSOEVER TO DO WITH THEIR IMPLEMENTATION; I HAVE NO INSIDER-KNOWLEDGE TO GIVE YOU; what I write here is MY interpretation. I might be completely mistaken.

So, here's an example, straight off the exam (which you can access from www.nzqa.govt.nz - just do a search on 'scholarship physics').

"A television safety advertisement features a car taking corners at dangerously high speeds. The danger is symbolised by land-mines appearing scattered around the corners. As the vehicle approaches a corner, the voice-over says 'There is more force taking you off the road and less force keeping you on it.' The car skids across the road and rolls over an embankment. Discuss the accuracy of the voice-over statement, with reference to centripetal force and friction."

First of all, what a lousy presentation of physics. Why didn't I blog about that when the advert was showing? Anyway, in terms of the question, it gets to the heart of what happens with circular motion. The student really needs to show they have grasped what goes on when a car travels round a curve, that there is centripetal force on the car towards the centre of the circle, and that this force is provided by friction between the tyres and the road.  A common misconception is that centripetal force is somehow generated by an object moving in a circle - that it is an 'extra' force in addition to all the other forces acting on it. It isn't an extra force - it is simply the resultant of summing all the forces acting on the car. The car is accelerating towards the centre of the circle, therefore the net force on it must be towards the centre.

So...the statement is wrong on both counts. First of all, there is no force 'taking you off the road'. Secondly the force keeping you on it actually goes up at high speed - it's the frictional force between the tyres and road. But take the corner at too high a speed and the frictional force is no longer adequate to keep you on it - and you go sliding. 

If you don't really UNDERSTAND centripetal force and circular motion, you don't have the slightest chance of answering this question. By understand, I don't mean being able to say F = m v^2 / r, but I mean knowing physically what causes something to move in a circle. Our centripetal force equation is easy to write, but not so easy to grasp.

 

 

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I've just finished reading a nice little article by Linda Leach, of Massey University, on the engagement (or lack thereof) of tertiary teachers with education theory.  She's interviewed f tertiary teachers and has identified a number of ways that teachers understand 'theory'.

First, it's very clear that 'theory' means different things to different people, and with the article comes a plea for educators to be more precise about what they mean when they use the word. She's considered four different broad interpretations of the word: 1. As the obverse of practice, 2. As a Generalizing or explanatory model, 3. As a body of explanation, and 4. As scientific theory.

What's clear is that many teachers take the view that theory is the opposite of practice. Since, in teaching, it is clearly the practice that matters (since it's what the students experience) this leads to the conclusion that  theory is irrelevant and there is no benefit in engaging with it. Leach says "Few of [those] who avoid theory seem to understand theory as either personal or practical"

The fallacy is the implicit assumption that theory and practice are unconnected. (I mean, you don't have experimental physicists and theoretical physicists working in complete isolation from each other, so why expect that with teaching?) What you believe about student learning will influence the way you teach, whether you formally acknowledge it or not. That's become clear for me as I think about my teaching practice. I have my own 'beliefs', or my own models, call them 'theories', of how students learn, and these influence how I teach. Formally looking at teaching 'theory', though the Postgraduate Certificate of Education, has helped me to identify what my beliefs and philosophies really are, and how these have changed since I've been teaching.

For example, In Pratt's terminology (1998) I take a 'Development Perspective' of teaching - that is, challenging a student's way of thinking about physics - having him or her try to interpret what they are seeing, reading, etc., and build a model of physical understanding that works. That in turn influences the way I set assignments, for example, and how I organize and run lecture and lab sessions.  But this isn't where I started from - in the beginning I was much more towards the 'Transmissive Perspective' - where I held the body of knowledge and it was my job to pass that to the students. A different belief / theory about teaching and learning, which was accompanied by different practice. Identifying what my internal biases are as a teacher (what theories I hold to) is an important step towards improving my practice.

Theory does influence practice, whether you recognize it or not. So it's a good idea to have at least a glance at it, from time to time.

 

Leach, L. (2011). Tertiary teachers and theory avoidance. New Zealand Journal of Teachers' work. 8(1), 78-89.

Pratt, D. D. (1998). Five perspectives on teaching in adult and higher education. Florida: Krieger.

 

 

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One thing we've noticed with our new house is how variable the background noise is. We now live within earshot of State Highway One and we can hear the distant rumble of trucks and other vehicles on it. The noise isn't large - and it's amazing that after just three weeks in the house we no longer can perceive it - we only 'hear' it when we specifically listen for it. But when we do listen for it, just how loud it is depends a lot on wind direction (not neglecting of course, time of day).

If the wind is blowing from the north, not only do we warm up a bit but we get an increase in the noise. Conversely, a southerly wind, though unpleasant in terms of temperature, means the road becomes pretty well inaudible.

The reason is that the sound waves travelling from the road to the house are travelling in the air. Sound needs air (or some other material) to travel in - you may for example have seen the demonstration of an alarm clock in a vacuum jar - you can't hear it because there's no air to carry the sound from it. Sound waves literally get taken along by the wind. If you happen to be downwind, the sound will be a bit louder; if you are upwind, a bit quieter.

Also, but not noticeable, is that the length of time taken for the sound to get from the road to the house will change. If the wave is travelling with the wind, it will travel that bit faster, equal to the speed of sound in still air plus the speed of the wind. If the wave is travelling against the wind, the speed is that bit lower (the speed of sound in still air MINUS the speed of the air).

This phenomenon is used by a sonic anemometer to measure the wind flow. These rather frightening looking things have a number of transducers and sensors that emit and detect sound - by timing just how long sound takes to go from one to another the speed and direction of the wind can be worked out - pretty accurately too. They are pretty common sensors for taking around on field trips where you need to know what the wind is doing.

 N.B. I think they look frightening because of the shiny 'prongs'. It looks like either it's a very painful medical instrument of some description or high voltage arcs are meant to jump across between prongs. Neither is the case.

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I've just received a whole pile of appraisal forms relating to my A-semester papers earlier in the year. As is common in universities, at the end  a paper, I ask my students to fill in a form relating to my teaching (and the content of the paper). It's mostly Likert-scale questions (a statement, which then has the options 'Always', 'Usually',  'Sometimes', 'Seldom', 'Never' - for example "This teacher was enthusiastic"). But also the students have space to record their own comments.

It's usually these comments that provide the most useful information on your teaching. I suspect most lecturers are pretty good at predicting what the scores from the Likert questions will be - I seldom get any surprises in these.  But the free comments can be another story. They can give really useful feedback.

I think I'm of the opinion that there is no such thing as 'positive' and 'negative' feedback. All feedback is positive feedback if you choose to pay attention to it. (I'm ignoring the abusive kind of feedback - fortunately I've only ever had one comment that I would put into that category). However, it can be pretty sobering to read comments sometimes.

So, this latest bunch has a couple of surprises in it. I won't write out specific comments (the students of course make them in confidence - I'm sure some of them won't want their writing plastered over a blog).

One surprise was from the mechanical engineering class, which is that I put too many symbols into my examples and not enough numbers. So, for example, I would use 'F' for force, and carry that through my calculations, rather than substituting a value in, say  53 newtons. Now, to me, the physicist, using F is the way to go - for one thing it's shorter to write, moreover, you end up with a more general expression - one that applies no matter how big the force is. But to the engineer, that's not necessarily so. Put numbers in where you have them, muliply them out to a single number, then it's simpler. This is a physics versus engineering thing, I feel. My physics classes don't complain about it. It's something I clearly need to be conscious of, however, when I teach engineering classes as opposed to physics classes. This will be particularly difficult in first year, for example, where physics students and engineering students are in the same class. How do I get around that one? Perhaps mix up the methods a bit.

A second surprise was from another third year class (electronics and physics) who suggested that I keep my alternative teaching style.  Alternative style?  This has me a bit worried and slightly perplexed. What exactly do they mean?  I don't think anything I do is desperately alternative - most is pretty well straight out of the teaching literature. Or maybe that's what they think IS alternative - being taught according to science/physics pedagogy.  I hope not.

Which, incidently, leads to a final comment which is addressed at any students who are reading this - a lot of the comments are awfully hard to decipher. So be specific about what you write - if it's vague and I mis-interpret it, it won't have achieved its purpose of providing useful feedback, and that means next year's class won't get the benefit from it.

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Physics is all about describing physical quantities. Whether it's length, velocity, force, electric current or heat flux, it takes physics to describe what it is and what it does. Central to this is our system of units. The three really common base units (in the S.I. system) are the metre (unit of length), the second (unit of time) and kilogram (unit of mass). With these three, we can construct a whole range of units for other quantities - e.g. the newton for force, pascal for pressure, joule for energy and so on.

The 'metre', 'second', and 'kilogram', have a long history behind them. And so they are not necessarily the 'easiest' or 'best' units for describing certain processes. When we get into particle physics, for example, and describe things of tiny mass (e.g. an electron is about ten to the power of minus 30 kilograms) we end up using rather large multipliers on our units. That makes them a bit awkward.

However, we are not confined to using a 'metre', 'second' and 'kilogram'. Any length, time and mass unit could be chosen. One choice that's popular with particle physicists is to make as many fundamental constants as possible equal to 1 something - e.g. choose a natural length unit and a natural time unit such that the speed of light is equal to 1 natural length per natural time.

With three units to play with, we can choose three things to 'set to 1'. A common choice is the speed of light, Planck's constant divided by 2 pi (called h-bar), and the rest mass of the proton. Particles people will claim this makes things easier, because you don't have to bother writing down the symbols 'c', 'h-bar', and 'm_0' for the speed of light, h-bar and the mass of the proton respectively whenever you have an equation.

However, I'm not so sure it simplifies things - at least not in understanding what's going on. When a constant 'disappears' from an expression, it's hard to see just what that expression entails. For example, the rest mass energy of an electron in natural units becomes m_e;   (m_e being mass of an electron); it comes from the famous E equals m c squared (mass times speed of light squared), but since the speed of light in these units is one, and the mass we have is the mass of the electron, then it's just E equals  m_e.

Now, do you agree that this is a touch confusing?  If we have E = m_e, it looks like we have an energy on the left hand side of the equals sign, and a mass on the right hand side. That can't work in physics. An energy equals an energy; a mass equals a mass.  It's because our particle physicists take a few liberties with the units when they say c = 1, h-bar = 1, m_0 = 1. (Or worse still, c = h-bar = m_0 = 1).  And I see plenty of textbooks that are written like this. Writing this is WRONG; what should be written is c = 1 natural length unit per natural time unit; h-bar = 1 natural mass unit natural length unit squared per natural time unit, and m_0 = 1 natural mass unit.  The units are not dispensable. Take them out and you start losing the physics. Saying c = h-bar = m_0 = 1 is claiming that the speed of light equals the mass of the proton. The two are utterly different entities.

Science papers using natural units therefore bug me somewhat, and students won't catch me using them in lectures.

 

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