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July 2014 Archives

I overheard the following conversation at the best coffee outlet on campus yesterday:

"Well, winter's nearly over. We're past the shortest day so it's getting warmer. And we've had eleven frosts so far this year, and the record for Hamilton is twelve, so there can only be one more to come." - Anonymous

Where do I begin?

Well, first let's point out that the shortest day does not equal the coldest day. Not by a long shot. In fact, I believe that statistically speaking the coldest week of the year for New Zealand is the last week of July (i.e. now). Why the difference? While it's true that it's the sun that provides the heat input to the earth, and that's at a minimum on the shortest day, there's a lot of thermal inertia on the earth, and particularly on the sea. And there is a lot of sea surrounding New Zealand. Temperatures are slow to change. While the sun remains low in the sky, the sea temperatures are slowly cooling, and that is going to influence the temperature in Hamilton. Conversely, the sea temperature in December is still pretty nippy. It's late summer before the sea temperature hits its maximum. Seasonal temperature variation is more about the cumulative heat put in over an extended period of time, as opposed to the heat input from the sun on a particular day.

And then the second point. I've always found it amusing that Hamiltonians count frosts, and think that  minus 4 Celsius (as it has been a couple of mornings recently) is cold. It is only cold because in New Zealand it is (near enough) compulsory to live in poorly heated, uninsulated, single-glazed detached houses. Europeans find this concept laughable, and, I think, Canadians probably sink their heads in their hands in despair.  Anyway, let's leave that aside. So if there have only ever been twelve frosts in a single winter in Hamilton (I doubt this, but don't have statistics on this at hand), and we've had eleven so far, then does that mean there is only at most one more to come?

Um, no. Probability doesn't work like that.  Our weather systems don't have a memory (not in that sense anyway), and they certainly aren't intelligent enough record the number of frosts a particular place has a year and act accordingly in the weeks ahead. I'd say we would be in for a few more frosts yet. That's simply based on the metservice statistics. Go to http://www.metservice.com/towns-cities/hamilton and look at the historical data tab. You'll see that the mean minimum temperature for August is -2 C, and for September it's 0 C, suggesting there can easily be some more negative temperatures coming for 2014. Enjoy. 

I remember several years ago playing a board game with a few friends. We'd had a long run of throws of the dice without seeing a 'six'. One of my friends asked me what the probability was that the next through would be a 'six'.  "One sixth" I answered - "same as for any other throw."  This sparked an intense discussion on whether that was right or not. It is. The dice does not have a memory. It doesn't remember what side it has landed up on in the past. Each throw is equally likely to show 1, 2, 3, 4, 5, or 6.  The probability of a 'six' is one sixth. What was perhaps most interesting is that a friend of mine who was doing a maths degree at the time refused to back me up. 

So is winter nearly over? While it's true that today feels rather spring-like, and the days are now noticeably longer than they were a month ago, winter still has plenty of teeth left.  

 

 

 

 

 

 

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I've been following the weather with interest this week. First of all, I was very glad when the wind and rain disappeared late last weekend. We were at a wedding in Whakatane on Saturday afternoon/evening, and boy, did it rain. With the wedding in a garden in something that was a bit more substantial than a marquee (think marquee with hard walls and floor), with a portaloo outside,  and a four minute walk up a long, dark, mud and puddle infested driveway in a storm separating you from the car, it was certainly a memorable wedding reception. 

Now, with beautiful clear skies, light winds, and frosty mornings, you'd be forgiven for thinking there's a big fat high pressure system sitting over us.  But there isn't.  For the last few days, we (by which I mean at this end of the country) have been in or around a saddle-point, in terms of pressure. There have been lows to the north and south, highs to the east and west, and somewhere in the middle over us. I note today that things have rotated a bit, so the lows now lie east and west, with a high to the north and another approximately south. Here's a picture I've stolen from the metservice website this morning (www.metservice.com, 18 July 2014, 11am); it's the forecaset for noon today. Note how NZ is sandwiched between two lows, but isn't really covered by either. 

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You can see the strength of the wind on this plot by the feathers on the arrow symbols. The more feathers, the stronger the winds. (The arrows point in the direction of the wind). Note how the wind blows clockwise around the low pressures (and anticlockwise, less strongly, around the highs). Have a look just around Cape Reinga (for non-NZ dwellers, and I know there's a few of you out there, that's the northern-most tip of the North Island.) There's a point where the wind (anthropromorphising) doesn't know what to do. It's in what's mathematically termed a saddle point. It's a point where locally there is no gradient in pressure, but is neither a high or a low. Winds are light.  In two dimensions (this is what we have on the earth's surface) with a single variable such as pressure, there are those three possibilities where the gradient of pressure is zero - a the maximum of a high, the minimum of a low, or a saddle. 
 
In terms of terrain, a mountain pass is a saddle point. It's where one goes from valley to valley (low to low), between two mountains. On top of the pass, you are at a point where the gradient is zero. But it's neither a peak or a trough. It's called a 'saddle', because the shape looks rather like a saddle for a horse - which is low on both flanks, but high at the front and back. A marble placed on top of a saddle should, if it were placed exactly at the equilibrium point with no vibrations, stay there. 
 
Saddle-points crop up in all kinds of dynamical systems (e.g. brain dynamics) where there's more than one variable involved.  Such a point is termed an unstable equilibrium - any deviation from the equilibrium point will cause the system to move away from it. However, the change may not be terribly rapid. When there are lots of variables involved, such local equilibria may have very complicated dynamics associated with them indeed - the range of possibilities gets very large and dynamics can become very rich indeed. 
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A common technique in physics is 'modelling'. This is about constructing a description of a physical phenomenon in terms of physical principles. Often these can be encapsulated with mathematical equations. For example, it's common to model the suspension system of a car as two masses connected by springs to a much larger mass. Here, the large mass represents the car body, one of the small masses represents the wheel, and the other the tyre. The two springs represent the 'spring' in the suspension system (which on a car is usually a curly spring - though it can take other forms on trucks or motorbikes), and the tyre (which has springyness itself). We can then add in some damping effect (the shock absorber). What we've done is to reduce the actual system into a stylized system that maintains the essential characteristics of the original but is simpler and more suitable for making mathematical calculations. 

That's great. We can now work on the much simpler stylized system, and make predictions on how it behaves. Transferring those predictions to the real situation, that helps us to design suspension systems for real situations. 

There are however, some drawbacks. We have to be sure that our stylized system really does capture the essential features of the actual system. Otherwise we can get predictions completely wrong. On the other hand, we don't want to make our model too complicated, otherwise there is no advantage in using the model. "A model should be as simple as possible, but not simpler" as Einstein might have said

There's another trap for modellers, which is going outside the realm of applicability for the model. What do I mean by that? Well, some simplifications work really well, but only in certain regimes. For example, Newton's laws are a great simplification on relativistic mechanics. They are much easier to work with. However, if you use them when things are moving close to the speed of light, your answers will be incorrect. They may not even be close to what actually happens. We say that Newton's laws apply when velocities are much less than the velocity of light. When that's the case (e.g. traffic going down a road) they work just fine - you'd be silly to use relativity to improve car safety - but when that's not the case (e.g. physics of black holes) you'll get things very wrong indeed. 

A trap for a modeller is to forget where the realm of applicability actually is. In the rush to make approximations and simplifications, just where the boundary is between reasonable and not reasonable can be forgotten. I've been reminded of this this week, while working with some models of the electrical behaviour of the brain. Rather than go into the detail of what that problem was, here's a (rather simpler!) example I can across some time ago now. 

I was puzzling over some predictions made in a scientific paper, using a model. It didn't quite seem right to me, though I struggled for a while to put my finger on exactly what I didn't like about what the authors had done. Then I saw it. There were some complicated equations in the model, and to simplify them, they'd made a common mathematical approximation: 1/(1+x) is approximately equal to 1-x.  That's a pretty reasonable assumption so long as x is a small number (rather less than 1). We can see how large it's allowed to get by looking at the plot here. The continuous blue line shows y = 1/(1+x); the dotted line shows 1-x.  (The insert is the same, at very small x). 

 

 

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We can see for very small x (smaller than 0.1 or so) there's not much difference, but when x gets above 0.5 there's a considerable difference between the two. When x gets larger still (above 1) we have the approximation 1-x going negative, whereas the unapproximated 1/(1+x) stays positive. It's then a completely invalid approximation. 

However, in this paper, the authors had made calculations and predictions using a large x. What they got was just, simply, wrong, because they were using the model outside the region where it was valid. 

This kind of thing can be really quite subtle, particularly when the system being modelled is complicated (e.g. the brain) and we are desperate to make simplifications and approximations. There's a lot we can do that might actually go beyond what is reasonable, and a good modeller has to look out for this. 

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Being a father of an active, outdoor-loving two-year-old, I am well acquainted with the bath. Almost every night: fill with suitable volume of warm water, check water temperature, place two-year-old in it, retreat to safe distance. He's not the only thing that ends up wet as he carries out various vigorous experiments with fluid flow. 

One that he's just caught on to is how the water spirals down the plug-hole. Often the bath is full of little plastic fish (from a magnetic fishing game), and if one of these gets near the plug hole it gets a life of its own. It typically ends up nose-down over the hole, spinning at a great rate as it gets driven round by the exiting water. 

The physics of rotating water is a little tricky. There are two key concepts thrown in together; first the idea of circular motion  - which involves a rotating piece of water having a force on it towards the centre (centripetal force); second is viscosity - in which a piece of water can have a shear force on it due to a velocity gradient in the water. Although viscosity has quite a technical definition, colloquially, one might think of it as 'gloopiness' [Treacle is more viscous than water. The ultimate in viscosity is glass, which is actually a fluid, not a solid - the windows of very old buildings are thicker at the bottom than the top due to the fluid flow over tens or hundreds of years.] In rotational motion there's a subtle interplay between these two forces which can result in the characteristic water-down-plughole motion. 

In terms of mathematics, we can construct some equations to describe what is going on and solve them. We find, for a sample of rotating fluid, that two steady solutions are possible. 

The first solution is what you'd get if all the fluid rotated at the same angular rate - the velocity of the fluid is proportional to the radius. This is what you'd get if you put a cup of water on a turntable and rotated it - all the water rotates as if it were a solid.

The second solution has the velocity inversely proportional to the radius - so the closer the fluid is to the centre, the faster it is moving. This is like the plughole situation where a long way from the plug hole the fluid circulates slowly, but close in it rotates very quickly. Coupled with this is a steep pressure gradient - low pressure on the inside (because the water is disappearing down the hole) but higher pressure out away from the hole. Obviously this solution can't apply arbitrarily close to the rotation axis because then the velocity would be infinite. This is where vortices often occur. (Actually, Wikipedia has a nice entry and animations on this, showing the two forms of flow I've described above). 

A Couette viscometer expoits these effects as a way of measuring the viscosity of a fluid. Two coaxial cylinders are used, with fluid between them. The outer is rotated while the inner one is kept stationary, and the torque required enables us to calculate the viscosity of the liquid.

 

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