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May 2014 Archives

At afternoon tea yesterday we were discussing a problem regarding racing slot-cars (electric toy racing cars).  A very practical problem indeed! Basically, what we want to know is how do we optimize the size of the electric motor and gear-ratio (it only has one gear) in order to achieve the best time over a given distance from a stationary start?

There's lots of issues that come in here. First, let's think about the motors. A more powerful motor gives us more torque (and more force for a given gear ratio), but comes with the cost of more mass. That means more inertia and more friction. But given that the motor is not the total weight of the car, it is logical to think that stuffing in the most powerful motor we can will do the trick. 

Electric motors have an interesting torque against rotation-rate characteristic. They provide maximum torque at zero rotation rate (zero rpm), completely unlike petrol engines. Electric motors give the best acceleration from a standing start - petrol engines need a few thousand rpm to give their best torque. As their rotation rate increases, the torque decreases, roughly linearly, until there reaches a point where they can provide no more torque. For a given gear ratio, the car therefore has a maximum speed - it's impossible to accelerate the car (on a flat surface) beyond this point. 

Now, the gear ratio. A low gear leads to a high torque at the wheels, and therefore a high force on the car and high acceleration. That sounds great, but remember that a low gear ratio means that the engine rotates faster for a given speed of the car. Since the engine has a maximum rotation rate (where torque goes to zero) that means in a low gear the car has good acceleration from a stationary start, but a lower top-speed. Will that win the race? That depends on how long the race is. It's clear (pretty much) that, to win the race over a straight, flat track, one needs the most powerful engine and a low gear (best acceleration, for a short race) or a high gear (best maximum velocity, for a long race). The length of the race matters for choosing the best gear. Think about racing a bicycle. If the race is a short distance (e.g. a BMX track), you want a good acceleration - if it's a long race (a pursuit race at a velodrome), you want to get up to a high speed and hence a huge gear.  

One can throw some equations together, make some assumptions, and analyze this mathematically. It turns out to be quite interesting and not entirely straightforward. We get a second-order differential equation in time with a solution that's quite a complicated function of the gear-ratio. If we maximize to find the 'best' gear, it turns out (from my simple analysis, anyway) that the best gear ratio grows as the square-root of the time of the race. For tiny race times, you want a tiny gear (=massive acceleration), for long race times a high gear.   If one quadruples the time of the race, the optimum gear doubles. Quite interesting, and I'd say not at all obvious. 

The next step is to relax some of the assumptions (like zero air resistance, and a flat surface) and see how that changes things. 

What it means in practice is that when you're designing your car to beat the opposition, you need to think about the time-scales for the track you're racing on. Different tracks will have different optimum gears.

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No, nothing to do with carrots and vitamin A I'm afraid. 

With dark evenings and mornings with us now :(, Benjamin's become interested in the dark. It's dark after he's finished tea, and he likes to be taken outside to see the dark, the moon, and stars, before his bath. "See dark" has become a predictable request after he's finished stuffing himself full of dinner. It's usually accompanied by a hopeful "Moon?"  (pronounced "Moo") to which Daddy has had to tell him that the moon is now a morning moon, and it will be way past his bedtime before it rises. 

I haven't yet explained that his request is an oxymoron. How can one see the dark? Given dark is lack of light, what we are really doing is not seeing. But there's plenty of precedence for attributing lack of something to an entity itself, so 'seeing the dark' is quite a reasonable way of looking at it.  

One can talk about cold, for example. "Feel how cold it is this morning". It is heat, a form of energy, that is the physical entity here. Cold is really the lack of heat, but we're happy to talk about it as if it were a thing in itself. Another example: Paul Dirac in 1928 interpreted the lack of electrons in the negative energy states that arise from his description of relativistic quantum mechanics as being anti-electrons, or positrons. In fact, this was a prediction of the existence of anti-matter - the discovery of the positron didn't come until latter.  

In semiconductor physics, we have 'holes'. These are the lack of electrons in a valence band - a 'band' being a broad region of energy states where electrons can exist. If we take an electron out of the band we leave a 'hole'. This enables nearby electrons to move into the hole, leaving another hole. In this way holes can move through a material. It's rather like one of those slidy puzzles - move the pieces one space at a time to create the picture. Holes are a little bit tricky to teach to start with. Taking an electron out of a material leaves it charged, so we say a hole has a positive charge. That's a bit confusing - some students will usually start of thinking that holes are protons. Holes will accelerate if an electric field is applied (because they have positive charge) and so we can attribute a mass to the hole. That's another conceptual jump. How can the lack of something have a mass? Holes, because they are the lack of an electron, tend to move to the highest available energy states not the lowest energy states. Once the idea is grasped, we can start talking about holes as real things, and that is pretty well what solid-state physics textbooks will do. It works to treat them as positively charged particles. It's easy then to forget that we talking about things that are really the lack of something, rather than something in themselves. 

A more recent example is being developed in relation to mechanics of materials as part of a Marsden-funded project by my colleage Ilanko. He's working with negative masses and stiffnesses on structures - as a way of facilitating the analysis of the vibrational states and resonances of a structure (e.g. a building). By treating the lack of something as a real thing, we often can find our physics comes just a bit easier to work through. 

So seeing the dark is not such a silly request, after all.





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We are three-quarters of the way through semester A. My Friday afternoon :-( tutorial for solid-state physics is still very well attended.  Is this:

A. Because the students are really engaged in this paper, learning a lot, and generally want to be there, or

B. Because they don't have a clue what's happening and are desperate for any hint of help, or

C. Because there's a test next week. 

Answers on a postcard please; the correct answer will be divulged at the end of semester.


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Yesterday I attended a very interesting discussion on the problem of student collusion in assignments. It's a really grey area that is particularly prevelant in the sciences and engineering. This is the kind of thing we mean:

Student A and Student B discuss their assignment one evening. Together, they think through what processes are needed to solve the problems, and how to work through these processes. They then go away and write their answers individually. Unsurprisingly, they end up submitting very similar work.  Their lecturer refers their work to the student discipline committee and the students are very aggrieved because they don't think they've done anything wrong. 

Is this a form of cheating?  As it turns out from our discussion, it depends on who you ask. Some would say no - each student has handed in an individually written assignment - but some would say yes - the ideas behind the assignment have been shared (or copied from one student to another) and that is collusion and unacceptable.

If the teaching staff can't agree on this kind of thing, what message do the students get? We often say things like "We encourage you to discuss the assignment and learn from each other, but what you submit needs to be your own work", but the students are left not knowing what this really means. The complaint is "on the one hand you are telling us to work together, and on the other hand you're telling us not to work together. What do you want?" This is unsurprising since it appears that we (as a group of lecturers) can't agree on what it means. At the discussion yesterday a lecturer from the Faculty of Law told us that Law students are often so paranoid that they might be accused of collusion that they go to great lengths not to talk to each other about assignments. That's probably unhelpful. 

Anyway, is it actually a problem? One could argue that, if students are learning, then that is what we wish to achieve, and the fact that two assignments are rather similar is not a problem.  I think that's a reasonable viewpoint, but then one has to ask the question, "How do you know that the student has actually learnt anything?" Consider the following scenario.

Students A, B, and C meet to work on their assignment. Student A is a strong student. Student B is not so strong, but is keen to learn and do well. Student C is keen to come out of the degree with a bit of paper that says 'BSc' on it. How he gets there and what he learns or doesn't learn on the way is immaterial.  Students A and B have a good conversation about the assignment. Student B learns a lot from student A. Student B asks student A some really tough questions, which gets A thinking too and engaging with the material even more deeply. Student C pretends to listen.  After this, students A, B and C prepare their answers, then have a look at what each other have written. Students A and B are happy to leave their assignments as they are; but student C then goes and changes his work to bring it in line with what the other two have written. 

As a consequence, Students A, B and C hand in very similar work. Students A and B have learned a lot by doing this assignment. Student C has learned nothing. 

The lecturer can't tell, from looking at the assignments, the extent of the student learning. They are all similar. How can the lecturer know then if the students have learned anything? As a result, all three get sent to the student discipline committee. The only way then is to ask the students, which is something that the student discipline committee is having to do very frequently. 

If students aren't learning, the obvious place to look is the assessments. They are probably bad assessments. A good assessment task should drive a student to learn, not drive him or her to copy someone else's.  One might argue that if this scenario is occuring, the lecturer needs to have a careful look at the sort of assessment tasks that are being set. 

There are ways to go forward. There are on-line systems where every student gets a slightly different assignment to do (but addressing exactly the same learning outcomes), that is marked automatically. If student A copies student B, then their answers will be wrong, since student A has a different assessment to student B. (E.g. student A might have to evaluate the centripetal acceleration of an object at 3.5 m/s speed, moving in a circle of radius 2.0 m; student B might have to evaluate the centripetal acceleration when the object moves at 2.5 m/s around a radius of 3.0 m. It's the same learning involved, but the answers are different.) 

Or, if we genuinely want students to collaborate closely, then we should set assignment tasks that demand a close collaboration - e.g. group tasks. The realilty is that almost no scientist or engineer works on their own - projects often have large teams of people on them. Team assignments then are what a student will face when they are in the workplace. So why aren't we giving them more of them as undergraduates?

The issue is a thorny one with no clear-cut answer. We can expect, therefore, a continual plethora of complaints from teaching staff, and counter-complaints of heavy-handedness and double-think from the students. Or, perhaps, we can think about our assessments more carefully. 

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In a long-in-the-past blog entries I commented on how two is a large number and three might be considered a working approximation to infinity. This kind of mathematics suits Benjamin (22 months old now). He's beginning to grasp what counting is about, but there's some way to go. I'm not exactly sure what's going on in his head when he counts things, but the end result is pointing to objects in turn and saying 'two', 'two', 'two', with the occasional 'eight'. A favourite is to count the stairs in our house as he goes up them. There are fourteen (counted conventionally), but, counting Benjamin-style, there are usually two.

In some ways it seems that 'two' is simply a term used to mean 'more than one'.   That will get him so far in life; for example he has two hands, two arms, two legs, and two rabbits. And as I'm discussing with my second-year solid-state physics class, two is quite sufficient when one is counting electrons.

These are the negatively charged particles often associated with atoms. The physics of electrons is extremely important - it is responsible for electricity and electronics doing what they do. The similarity in the names is no co-incidence.  In a crude model, we think of negatively charged electrons in orbit around a positively charged nucleus, but the reality is rather more complicated and very much more bizarre. In the quantum world, electrons can be found in energy levels. Every system (e.g. an atom, or a molecule, or a crystal) has it's own set of energy levels. If we were to give the system a minimum amount of energy, electrons would fill up these energy levels, from the lowest one upwards. One might think of energy levels as rungs on a ladder. But here's the important bit - for electrons an energy level can only hold a maximum of two electrons. The 'two' comes from a property of electrons called 'quantum spin'. A level might carry zero electrons, it might carry one electron. or it might carry two, but it won't have any more. This leads to something called the Fermi-Dirac distribution, which is a rather essential concept for solid-state physics. It tells you that the more electrons you put into a system, the higher the 'rungs' that they must occupy. It will also tell you that the electrons buried on the low rungs aren't going to do anything useful - they can't move because there's no empty spot for them to move to. Only electrons near the top of the occupied section of the ladder (the Fermi energy) can do anything useful and contribute to electronic properties such as electrical and thermal conductivity of materials. 

Benjamin is then ideally placed for studying solid-state physics. All he needs to do is to count up to two, and he can do just that. In fact, as he goes up the stairs ('two', 'two', 'two') he is, perhaps, counting electrons in energy levels...






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