A lot of huffing and puffing Marcus Wilson Apr 15

While there is some great fiction out there, one really shouldn’t try to learn much physics from it. One case in point, which I am forced to listen to over and over by the youngest member of our house, is the story of the Three Little Pigs. 

I’m not talking here about the relative merits of various building materials for construction of houses. Straw, wood and brick all have their place. I refer to the rather rapid boiling of the pot of water that the Third Little Pig puts on the fire when the wolf comes knocking at the door. 

In the version of the story that we have on CD, thw wolf, fresh from his succesful huffing- and puffing- of the straw and wood houses, arrives at the home of the Third Little Pig, where  the First and Second Little Pigs have taken refuge. A house made of brick. The door is locked in his face. No problem for the wolf – or so he thinks. A little more huff and puff and this one will be blown in to.  But this time he’s mistaken. The house stands still. The angry wolf now resorts to plan B. He puts safety regulations aside and climbs onto the roof of the house, with the intention of gaining ingress via the chimney. 

Time for the third Little Pig to move quickly. He gets a fire going, puts a wolf-sized  pot of water on in, and gets it boiling – just in time, as the wolf drops down the chimney. This version of the story ends with the wolf fleeing in pain (rather than cooked) and the Three Little Pigs jubilant. 

Actually, the story doesn’t end, because Benjamin now pushes the button on the CD player to play it again. And again…

Now, just how much power does the Third Little Pig have at his disposal to boil a wolf-sized pot of water in the time taken for a wolf to climb up onto the roof and head down the chimney. The wolf is in a foul mood, so he’s not going to hang around. Let’s say it’s going to take him  minute for this task. A wolf-sized pot might be around 100 litres in size. If it’s full of water at about room temperature, this 100 litres of water has to gain 75 degrees Celsius in just 60 seconds. 

One litre of water takes 4200 joules of energy to raise its temperature by 1 degree C. That’s called the ‘specific heat capacity’. To raise 100 litres by 75 degrees, we therefore need 4200 times 100 times 75 = 31 500 000 joules. This happens in sixty seconds – thats about half a million joules per second. 

What does that mean? One joule per second is one watt of power. So here we have about 500 kW of power – a kW (kilowatt) being a thousand watts.  

This is something pretty substantial. If you’ve watched the disc on your electricity meter spin around, you’ll know that it’s rotation rate is a measure of your power consumption. Usually 200 revolutions equals 1 kWh of energy. Do the maths and you’ll find that 1 revolution per second (a seriously high domestic consumption) equates to 18 kW of power.  500 kW equates to about 30 revolutions per second. Dizzy stuff.  

If the Little Pigs were relying on electricity they’d be needing to upgrade their mains connection. But they are using wood. Consumer NZ tells me that efficient, domestic wood-pellet fires can produce about 10 kW of power. To hit the 500 kW range, the pigs obviously have a sizeable one indeed. 





How high is a winning cricket score? Marcus Wilson Mar 24

I can’t help thinking that the West Indies team got their run chase strategy wrong on Sunday night. They had a tricky task ahead of them. One might say the problem was one of their own making, judging from the rubbish that they served up to Guptill to hit at the end of tne New Zealand innings, but that was in the past. The fact was they had to chase down 393 in 50 overs. How does a team go about doing that?

I would have thought that the obvious answer was ‘in the same way that the opposition made it’. In other words, keep the scoreboard turning over nicely in the early stages, but without taking excessive risks, and then, with the wickets in hand (especially Gayle’s), hit the accelerator at the end. Rather than taking 50 overs to reach the target, they looked like they were trying to do it in 40. It was never going to work. You can contrast this to the calm manner in which Sri Lanka reached England’s 300+ score in the group stages. Nothing flash, no excessive risks, they just ticked the board over just short of the required run-rate, and then with wickets in hand they pushed on at the end to win easily. No fuss, no stupid shots. They knew exactly what was needed, and they achieved it. 

A school of thought says, other things being equal,  it’s much better to bat second in a limited overs match, because you know exactly what you have to do. It clarifies your batting strategy: Choose whatever strategy maximizes your chances of reaching the target score within 50 overs. Yes, there are other considerations, our friends Duckworth and Lewis being one, but let’s put those aside for the moment.  You have 10 wickets and 50 overs, and a target score. You only need to beat that target by 1 run on the last ball, with one wicket left.  What strategy will maximize your chance of success?

The team batting first has a much less defined problem. They don’t know what a winning score is. True, they’ll have some feel of what one will be, given the ground, the quality of opposition, the state of the pitch and so on, but fundamentally, they don’t know what score is going to be good enough. For example, a team could probably secure a moderate score (let’s say 280) with 80% probability, by batting conservatively. Or they could aim for a larger score (say 350) by taking more risks. They might get there with a probability of 25%,  but then there’s the possibility (maybe also 25%) they crash out on the way and end up with something more like 250. What  should they do? Which strategy is better? 

It’s viewed as a criminal offence if the team batting first fails to bat out its 50 overs. If it’s all out beforehand, it has obviously taken too many risks. But also, one could say its a criminal offence if they end up with just a handul of wickets down. In that case they haven’t been taking enough risk. Balancing all that up, a typical strategy for a team batting first is to go at a moderate rate to start with, and slowly increase the rate as wickets allow. It seems to have stood the test of time. 

It is possible to make this a bit more mathematical. What a team needs to do is to maximize its score, subject to the constraints that a. it only has 50 overs, and b. it only has 10 wickets. Since the rate of fall of wickets is certainly related to the scoring rate (the higher the runs-per-over, the higher the risk, usually, and the higher the wickets-per-over) we can write down some equations for the situation. [You'll be relieved to hear I won't try to do those here].

It turns out to be is an optimization problem, of which there are many in physics. We can tackle many of them with the Euler-Lagrange equation.  For example, what shape does a chain have when it hangs under its own weight? Take a chain of length 2 metres, secure the ends to posts 1.5 metres apart. The chain clearly sags in the middle, but what shape is the resulting curve?  The chain hangs in such a way as to minimize its gravitational potential energy, subject to the constraint that it has a constant length and that it has to start and end at fixed points. One can put this in equation form and solve for the shape. The resulting curve is called a ‘catenary‘. 

Certainly, if one could write down the rate of loss of wickets as a function of scoring rate, wickets lost already and other factors, we should be able to have a go at tackling the cricket scoring rate problem. Given the degree of professionalism of the sport, it wouldn’t surprise me if that has actually been done by someone. I imagine it would be a closely guarded secret.






The difference between a theoretical physicist and a mathematician is… Marcus Wilson Mar 13

A mathematician can say what he likes… A physicist has to be at least partly sane

J. Willard Gibbs 

What is it that makes a physicist sane (if only in part)? Everything has to be related back to the ‘real world’, or the ‘real universe’. That is, a physicist has to talk about how things work in the world or universe in which we live, not some hypothetical universe. That’s how I think of it, and I know, having done a bit of research with some of my students, a lot of them think the same way. That’s not to say mathematicians don’t have a lot to say about this universe too. It’s just that the constraints on them are somewhat less. 

Another way of looking at it is that physicists work with dimensioned quantities. Most things of physical relevance have dimensions. For example, a book has a length, width and thickness. All of these are distances, and can be measured. The unit doesn’t matter; we could use centimetres, inches or light-years – but the physical size of the object is determined by lengths. Also, the book has a mass (one could measure it in kilograms). It might find its way onto my desk at a particular time (measured, for example, in hours, minutes, seconds, millennia or whatever). Perhaps it is falling at a particular velocity – which describes what distance it travels in a particular time. All of these things are physical quantities, and they carry dimensions.

One of my pet hates as a physicist is reading physics material in which the dimensions have been removed. You can do this by writing lengths in terms of a ‘standard’ length, but then only quoting how many of the standard length it is. So we might talk about lengths in terms of the length of a piece of A4 paper (which happens to be 297 mm); a piece of A2 paper has length 2 standard-lengths, and an area of 4 standard-areas. The problem really comes when the discussion drops the ‘standard-length’ or ‘standard-area’ bit and we are left with statements such as a piece of A2 paper has a length of 2 and an area of 4.  It is left to the reader to work out what this actually means in practice. A mathematician can get away with it – she can say what she likes, but not so the physicist. 

Here’s a question which illustrates the point? What is the length of a side of a cube whose volume is equal to its surface area? The over-zealous mathematics student blunders straight in there: Let the length be x. Then volume is x^3, and surface area is 6 x^2 (the area of a face is x^2, and there are six on a cube). So x^3 = 6 x^2 ; cancelling x^2 from both sides, we have x=6.  Six what? centimetres, inches, furlongs, parsecs? The point is that the volume of a cube can never be equal to its surface area. Volume and area are fundamentally different things. 

The Wikipedia page on ‘fundamental units‘ , along with many text books, blunders in this way too. The authors should really know better. (Yes, I should fix it, I know…) For example:

A widely used choice is the so-called Planck units, which are defined by setting ħ = c = G = 1

No, NO, NO!  What is wrong with this? How can the speed of light ‘c’ be EQUAL to Newton’s constant of Gravitation ‘G”. They are fundamentally different things. The speed of light is a speed (distance per unit time), Newton’s constant of gravitation is… well.. it’s a length-cubed per mass per time-squared. It’s certainly not a speed, so it can’t possible be equal to the speed of light. And neither can be equal to 1, which is a dimensionless number. What the statement should say, is that c = 1 length-unit per time-unit; and G = 1 length-unit-cubed per mass-unit per time-unit squared. 

However, doing physics can be more complicated that this. A lot of physics is now done by computer. In writing a computer programme to do a physics calculation, we almost always don’t have explicit record of the units or dimensions in our calculations. Our variables are just numbers. It’s left to us to keep track of what units each of these numbers is in. Strictly speaking, I’d say it’s rather slack. It would be nice to have a physics-programming language that actually keeps track of the units as well. However, I’m not aware of one. (If someone could enlighten me otherwise, that would be fascinating…) Otherwise, I’ll have to have a go at constructing one.

What’s prompted this little piece is that I’ve been reviewing a paper that has been submitted to a physics journal. The authors have standardized the dimensions out of existence, which makes it awfully hard for me to work out what things mean physically. Just how fast is a speed of 1.5? How many centimetres per second is it? While that might be an answer their computer programme spits out, the authors really should have made the effort of turning it back into something that relates to the real world. In a mathematics journal, they might get away with it. But not in a physics journal. At least, not if I’m a reviewer…



Why does time go forwards? Marcus Wilson Mar 10

Further to my last post, here’s a very accessible discussion on some of the physics related to ‘the arrow of time’. Maybe, just maybe, Benjamin has the right idea after all…


The arrow of time Marcus Wilson Mar 02

Benjamin is now two-and-two-thirds, or near enough. As ever, his grasp of physics continues to improve.  In the last few weeks, he has been picking up the idea of time. 

We have a large (more accurately, LARGE) analogue clock on the wall of our lounge. He’s watched me take it off the wall, change the battery and move the hands to a new position when it started to run slow. It’s clear to Benjamin that what the hands do on the clock is related to the time of day, although just how I think is some way off.  Over the weekend he wanted me to read him a book, but didn’t want me to get it for him. He shoved me away, and rather matter-of-factly said “Daddy stay there. Poppet will get a story.” Then he turned to get the book, but quickly came back to me and added “I’ll be back at half-past-four. See you soon.” And off he went to get his favourite book (which, as ever, is about excavators and other large machines). 

That was just one of those amusing moments you get with a young child. But another ‘time’ incident was rather more interesting. Karen was out, and Benjamin was somewhat upset over her absence. I was trying to reassure him that she would be back. “She’ll come home at about half-past-eight”. Benjamin looked at the clock longingly and said “Daddy change the clock so it’s time for Mummy to come home”. 

Now there’s a thought! Wouldn’t it be great if we could just manipulate time by turning the hands of the clock? So, somehow, if we changed what the clock says, then the time of day would actually change. Does Benjamin think that this is how it actually is? Maybe. It would be useful if it were true – extended weekends, short work days; one could cut hours of a plane trip to Europe by taking the clock with you. 

While it might work for science fiction writers, unfortunately that’s not how time works for us. While the clock and time are intimately linked, it is time that controls the clock, not the other way around. We are stuck with progressing through time at the rate of one second every second. 

That makes time a rather strange concept from a physics perspective. Unlike space, where we are free, more or less, to move to any point in it, we don’t have that option with time. We can only move forward in it, and only move forward at the same rate – one second every second. The past is forever behind us; and the future is always unknown. Physicists call this the ‘Arrow of Time’. It points one way: forward. 

Special Relativity makes it more interesting still. The way time works for you may not be the same as it works for me. If I were to get on a Really Fast Spaceship and travel close to the speed of light for a while, then return to earth, I would be noticeably younger than my identical twin brother. (I actually DO have an identical twin, by the way.)  Not only would the clock in my spaceship be telling me less time had past, I would have actually aged less. From my point of view, I might have been gone for two weeks; from yours, I might have been gone ten years. But even so, each of us will still have perceived time as travelling at one second every second. Forward. 

What about real time-travel – going back in time. Just maybe physics permits this to happen. That’s in the realms of General Relativity and Quantum Gravity and involves some really big masses indeed. Matt Visser’s work at Wellington might give us some pointers here. But the summary of it is: Don’t expect a such a time-machine to be built in Benjamin’s lifetime, even if he does prolong it indefinitely take the battery out of our lounge clock.






First a cricket fan, second a physicist Marcus Wilson Feb 18

I’ve spent most of today thinking Google’s image-of-the-day is a wicket, but have just realized it is in honour of Alessandro Volta.



Static friction is something sticky (as is Scholarship physics) Marcus Wilson Feb 13

In January I had a go at the 2014 Scholarship Physics Exam, as I’ve done for the last couple of years. Sam Hight from the PhysicsLounge came along to help (or was it laugh?) The idea of this collaboration is that I get filmed attempting to do the Scholarship paper for the first time. This means, unlike some of the beautifully explained answers you can find on YouTube, you get my thoughts as I think about the question and how to answer it. Our hope is that this captures some of the underlying thinking behind the answers – e.g. how do you know you’re supposed to start this way rather than that way? What are the key bits of information that I recognize are going to be important – and why do I recognize them as such? So the videos (to be put up on PhysicsLounge) will demonstrate how I go about solving a physics problem (or, in some cases, making a mess of a physics problem), rather than providing model answers, which you can find elsewhere. We hope this is helpful. 

One of the questions for 2014 concerned friction. This is a slippery little concept. Make that a sticky little concept. We all have a good idea of what it is and does, but how do you characterize it? It’s not completely straightforward, but a very common model is captured by the equation f=mu N, where f is the frictional force on an object (e.g. my coffee mug on my desk), N is the normal force on the object due to whatever its resting on, and mu (a greek letter), is a proportionality constant called the coefficient of friction. 

What we see here is that if the normal force increases, so does the frictional force, in proportion to the normal force. In the case of my coffee mug on a flat desk*, that means that if I increase the weight of the mug by putting coffee in it, the normal force of the desk holding it up against gravity will also increase, and so will the frictional force, in proportion.

Or, at least, that’s true if the cup is moving. Here we can be more specific and say that the constant mu is called ‘the coefficient of kinetic friction’: kinetic implying movement.  But what happens when the cup is stationary? Here it gets a bit harder. The equation f=mu N gets modified a bit: f < mu N. In other words, the maximum frictional force on a static object is mu N. Now mu is the ‘coefficient of static friction’. Another way of looking at that is that if the frictional force required to keep an object stationary is bigger than mu N, then the object will not remain stationary. So in a static problem (nothing moving) this equation actually doesn’t help you at all. If I tip my desk up so that it slopes, but not enough for my coffee mug to slide downwards, the magnitude force of friction acting on the mug due to the desk is determined by the component of gravity down the slope. The greater the slope, the greater the frictional force. If I keep tipping up the desk, eventually, the frictional force needed to hold the cup there exceeds mu N, and off slides the cup. 

What this means is that we when faced with friction questions, we do have to think about whether we have a static or kinetic case. Watch the videos (Q4) you’ll see how I forget this fact (I blame it on a poorly written question – that’s my excuse anyway!). 


*N.B. I have just picked up a new pair of glasses, and consequently previously flat surfaces such as my desk have now become curved, and gravity fails to act downwards. I expect this local anomoly to sort itself out over the weekend. 

P.S. 17 February 2015. Sam now has the videos uploaded on physicslounge  

Conduction in semiconductors – the tennis ball model Marcus Wilson Feb 04

Not so long ago, a tennis ball appeared in our garden. It’s a rather distinctive red one. It doesn’t belong to us. It was lying close by to the (low) fence between us and our neighbour, so I just chucked it back. 

Next morning, it was there again. I threw it back.

And, more or less immediately, it was back with us. Evidently, it didn’t belong to next door. They were working on the assumption it belonged to us. The next-likely suspect was the house at the back of us, which has some rather energetic children. Over went the ball into their garden. 

Next day it was back with us. Not their ball, either. Suddenly, this ball has become highly mobile. It flits from garden to garden, and doesn’t appear to be finding a home anywhere? Where did it come from?

I can’t help thinking that this is a good analogy with conduction of electrons in n-type semiconductors. Although silicon underlies so much of modern electronics, it comes as a real surprise to many students to learn that silicon is really quite a lousy electrical conductor. That’s unsurprising when you look at its structure – the silicon atoms are locked in a lattice, with each atom bonded by strong covalent bonds to four other atoms. There are no free electrons – all the outermost electrons that would contribute to conduction are tightly bound in chemical bonds. Without free, or losely bound, electrons, there’s not going to be much electrical conduction. 

So how come silicon devices are at the heart of modern electronics? The key here (in the case of n-type silicon) is that extra electrons have been put into the lattice. This is done by adding impurity atoms with five, not four, electrons in their outer shell (e.g. phosphorus). These electrons aren’t involved with bonding, and become extremely mobile, because none of the silicon atoms finds it favourable to take them on. They flit from atom to atom, finding a natural home nowhere, as does our tennis ball. Unlike a tennis ball, however, electrons are charged particles. Apply an electric field, and they have a purpose, and we suddenly have movement of electrical charge (which is simply what an electrical current is).

There’s a second way to make silicon conduct, and that’s the reverse. Rather than adding in electrons, we take them away. How does that work? Introduce now an atom into the lattice that only has three outer-shell electrons (e.g. boron). It is likely to grab one from a neighbour, to allow itself to make four covalent bonds. But now its neighbour is devoid of an electron. It will grab one from one of its neighbours. And so on. Now the ‘lack of an electron’, or ‘hole’, as its known in semiconductor physics, is what is mobile. Since electrons are positively charged, the lack of an electron (i.e., a hole) is positively charged. Apply an electric field and the hole moves – and we have electrical current again. This is ‘p-type’ silicon (‘p’ for positive, since conduction is through positively charged holes; contrast ‘n’ for negative, where conduction is through movement of negatively charged electrons). 

In our tennis ball analogy, the p-type lattice corresponds to a less desirable neighbourhood – someone on discovering that one of their tennis balls is missing makes up a complete set by sneaking round into the next-door garden to steal one, thus transferring the problem elsewhere. 

Seeing spots before my eyes Marcus Wilson Jan 23

“Doctor, Doctor, I keep seeing spots before my eyes”

“Have you ever seen an optician?”

“No, just spots”.

The concept of seeing an optician floating across my field of view is a scary one indeed. However, the concept of seeing spots doing the same is one I’m coming to terms with. 

I had a talk to an opthamologist about this last week, as part of an eye check-up. He was very good, I have to say, and we discussed in detail some optical physics, particularly with regard to the astigmatism in my right eye (and why no pair of glasses ever seems quite right).  He also reassured me that seeing floaters is nothing, in itself, to be worried about. It’s basically a sign of getting old. How nice. He did though talk about signs of a detached retina to look out for (pun intented) – and did some more extensive than usual examination. 

So what are those floaty things I see? To use a technical biological phrase, they are small lumps of rubbish that are floating around in the vitreous humour of the eye. They are real things – not an illusion – although I don’t ‘see’ them in the conventional manner that I would see other objects. 

The eye is there to look at things outside it. Its lens focuses light from objects onto the retina, where light sensitive cells convert the image to electrical signals that are interpreted by the brain. But given that the floaters are actually between the lens and the eye, how am I seeing them?

There are a couple of phenomena going on. First of all, a floater can cast a small shadow onto the retina. You can see this effect by using a lens to put an image of something (e.g. the scene outside) onto a piece of card, and then put something between the lens and the image. Some of the light can’t get to the card, and so part of the image is shadowed. The appearence of the shadow depends on how close the object is to the card – if its right by the lens there will be very little effect – but if close to the card there’ll be a tight, well-defiined shadow. My experience is that these spots are definitely most noticable in bright conditions – presumably because the shadows on the retina then appear in much greater contrast than under dull conditions. 

Secondly, however, they can bend the light. Their refractive index will be different from that of the vitreous humour, and therefore when a light ray hits a floater it will bend, a little. The consequence is a defocusing of a little bit of the image, which wil be visible. If the floater stayed still, it would probably barely be noticable, but when it moves, the little bit of bluriness moves with it, and the brain picks up the movement rather effectively. 

The most interesting thing to me is that it just isn’t possible to look at these things. When I try, my eyes move, and consequently these bits of rubbish flit out of view. Rather like quantum phenomena, you can’t observe them without changing where they are and where they are moving to.  




Modes of a square plate Marcus Wilson Jan 15

Alison has drawn my attention to this video. It demonstrates vibrational modes of a square plate by using sand. At certain frequencies, there are well defined modes of oscillation, in which parts of the plate ‘nodal lines’ are stationary. The sand will find its way to these parts and trace out some lovely pictures. 

Vibrational modes are often illustrated through waves on a guitar string. Here, the string is held stationary at both ends, but is free to vibrate elsewhere. There is a fundamental frequency of oscillation, where the distance between the ends of the string is half of a wavelength (this ensures the displacement of both ends of the string is zero since they are clamped).  Since wavelength is related to frequency (frequency = speed/wavelength) that means if the wavelength is 2 L where L is the distance between the ends of the string, we have frequency = speed/2L.  

But that’s not the only possible mode. Another one would have L equal to a whole wavelength (equals two half wavelengths). Or one-and-a-half wavelengths (equals three half-wavelengths.) This gives us, rather neatly, frequency = n speed/2L, where n is an integer. We see that our ‘harmonics’ are just integer multiples of the fundamental frequency. Rather neat.

However, if you look at the frequencies given in the video, they appear to be all over the place. I challenge you to pull out the relationships between these (I’ve tried). There are a few reasons why the case shown on the video is considerably more complicated than the waves on the string. 

1. The boundary conditions. The edges of the plate aren’t clamped in place. This makes it less straightforward to define the modes geometrically. 

2. The plate is square, giving rise to ‘degeneracy’ in the modes. This term refers to two or more distinct modes having the same frequency. You can see it rather well with the 4129 Hz mode. Basically, there are horizontal stripes shown. But equally, with the same frequency, you could get vertical stripes. Why don’t the two occur together? They do. You can see the effect of having a little bit of vertical stripe most clearly at the far end of the plate, where the pattern becomes more square-like. Also, with a square, you can get two completely different types of mode with the same frequency. This occurs because what matters are the sums squares of pairs of integers. Broadly speaking (at least for a square clamped on the edges, which I must point out this ISN”T), our modes follow the relationship:

f = C sqrt(n^2 + m^2)

where C is a constant, ‘sqrt’ means square-root, and n^2 is n-squared. So, for example, not only is 50 equal to 5-squared plus 5-squared, it is also equal to 1-squared plus 7-squared (or 7-squared plus 1-squared). This gives us three  modes all competing to appear at exactly the same time. What happens then isn’t easy to tell. 

3. Non-linear effects. This a physicist’s code-word for ‘it’s all too difficult’. That’s not quite true – arguably most of the interesting physics research happening in the world is looking at non-linear effects. What this really means is that, if A and B are both solutions of a problem, then some combination of A and B is NOT a solution. A lot of physics IS linear – Maxwell’s equations in a vacuum is a good example – but a whole lot isn’t. With waves, the speed of the wave usually depends on frequency (i.e. is not constant) which means we lose the nice, integer-multiple relationship of our waves-on-a-string mode.

So, enjoy the video for what it is, and don’t try to analyze it TOO closely. 



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