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December 2015 Archives

We had our departmental Christmas lunch on Tuesday, outside in the campus grounds. We had some lovely sunshine, but the wind did rather spoil things. I've certainly got used now to living in a very wind-free place - a fresh breeze is something quite unsual here. We were hanging on to our paper plates, but didn't expect to have to hang on to glass drink bottles as well. One particular gust was strong enough to take a newly opened individual-serving-sized glass bottle of lemonade and blow it over. 

So, being a physicist I had a go at estimating just how strong the gust of wind needed to be to push over a lemonade bottle. As the wind hits the bottle it has to change direction, and this causes a change in its momentum. To do that requires a force - the force being the rate of change of momentum of the air. That gives us an estimate of the pushing force in terms of the speed of the wind - specifically the density of air, times the cross section of the bottle to the wind, times the speed of the wind squared. This generates a turning moment about a point on the base - to get this you can multiply the force by the distance of the centre of the bottle from the table. 

The bottle will tip if this force is enough to overcome the turning moment due to gravity the other way. That's simply the weight times the radius of the bottle. Doing the calculation, gave an estimate of about 15 m/s or so, or a bit above 50 km/h. Not particularly high. I was a bit disappointed by the result. 

But then I got thinking about something more interesting. In this case, the bottle tipped. But what determines whether it will tip over or slide along the table? To think about this, we need to work out how strong the wind needs to be in order to slide the bottle. In this case we can equate the sideways force exerted by the wind to the maximum amount of frictional resistance that the bottle-table interface can provide. The latter is simply the bottle's weight multiplied by the coefficient of static friction between the bottle and the table. Doing the maths again, with an estimate of the coefficient of friction of around 0.5, I got something marginally larger - about 60 km/h. 

Now, the curious thing is the ratio of the force needed to slide the bottle to the force needed to tip it. Although each individual force is quite complicated to write down (so I'm not going to), the ratio turns out to be something really quite simple and elegant. Assuming a cylindrical bottle (!) the ratio of the force-to-slide to the force-to-tip is just the square root of the product of aspect ratio (height over diameter) and coefficient of static friction. This means if the coefficient of static friction is larger than the diameter over the height, it will tip rather than slide. If the coefficient of static friction is smaller than the diameter over the height, it will slide rather than tip. 

As an example, if the coefficient of friction is low (e.g. the bottle is on ice) the force to get it to slide is much less than the force to tip it. If the wind blows hard enough, the bottle will slide, not tip. Having a small height compared to diameter also favours sliding rather than tipping - a squat ginger-beer bottle is rather more likely to slide when pushed sideways rather than a slender wine bottle. 

One could potentially use this as an amusing way of measuring the coefficient of static friction. Use differently proportioned cylinders and apply a sideways force to each until they move.  The squatter ones will slide, the more slender ones will tip. Somewhere in the middle will be one that does both at once. The coefficient of static friction is then just the diameter divided by the height of this cylinder. There are simpler ways to do it, such as measuring by the angle of a slope that is just steep enough for an object to slide down. The coefficient of friction is just the tangent of this angle. 

That's it for this year, methinks. Have a Happy Christmas and enjoy the New Year. I think it's time I headed off to Seddon Park to see Mitchell Santner destroy the second-half of the Sri Lankan batting line-up. 

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Yesterday I finally managed to achieve one of the things on my 'to do' list that's been sitting there for about a year - attend a 'Have a Go' session at the velodrome in Cambridge. For those that don't know it, it's New Zealand's new (which means about two years old) world-class velodrome and now 'home' to New Zealand cycling. It's an impressive building - probably more so from the inside than the outside. 

Walking into the central area the first thing that greets you is a terrifying view of the banking on the corners. We were told it's 43.5 degrees. It's the kind of slope you'd find on a steep slide in a playground. And we're meant to cycle on it. 

But our instructor built us up gradually to this. First off, was just getting used to the bikes - large, with a high (fixed) gear, and back-pedal braking. The last of these means no free-wheeling - they need pedalling all the time. There's a nice flat area at the bottom which allowed us to do a couple of laps of the track and get used to starting and stopping, both of which are quite technical affairs. Next, we were up on the blue strip on the inside of the track. That's banked, but not as much as the main track. Once we had this sorted, we were allowed up onto the main track on the straights only, which are less steep. Then the really scary bit. The main track all the way round. 

Although scary the first time, it's really not - or at least shouldn't be - for a physicist. The key is simply to be going fast enough. With a bike that highly geared, on a smooth track, with no wind to worry about, that's not difficult. The physics isn't too tricky to do either. Cornering requires a net centripetal force (one towards the centre of the circle) to act - one that's proportional to the square of your speed. The banking ensures that this is more-or-less provided by the normal force acting from the track onto the bicycle. In fact, at the pathetic speeds we were  doing (maybe up to 30 km/h?  my forty-something year old legs didn't seem to work the way they once did...) the banking was providing more than enough centripetal force. This means there's friction acting as well, in this case an upwards force keeping us on the banking. Our bikes were certainly far from perpendicular with the track when we were going round - we were leaning outwards with relative to the track, but a spectator would have seen us leaning inwards relative to the flat.  

Go fast enough (and some quick calculations suggest this is easily in reach of a proper cyclist) and there'll be no friction required at all - and the bike should make a neat 90-degree angle with the track. From the spectator's point of view, the cyclist will be leaning over at 43.5 degrees. Faster still, and the banking won't be enough to provide that centripetal force, and friction will then be required - this time acting down the track. The cyclist will need to lean inwards relative to the track. 

Just as there's a minimum speed required in order to get around the banking (according both the instructor and my own physics calculations about 25 km/h -  and she made sure we were up to it before letting us off the blue strip), there'll also be a maximum speed possible to take the corner safely, even with that extreme banking. My quick estimate of this is around 120 km/h, which I think is beyond what a track cyclist will be able to get to. [Extra note: these calculations make an assumption of a coefficient of static friction of around 0.8. But given that  some sprinters can actually maintain a stationary postion on the banking (so they don't have to lead in the 'sprint') there's clearly more to it than that - possibly the coefficient of static friction is particularly high!]

So, overall, not so scary after all. I'm looking forward to having another go. 


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Last week I attended a seminar by Ian Hawthorn of our Maths Department. He talked about some work which he'd done with a couple of students, Matt Ussher and William Crump. The title is a bit of a mouthful "The physics of sp(2,R)"  (what does that mean?) and I have to say that I didn't follow most of it. But I did follow some of what Ian said, some of the time, in some places. 

Before I comment on what Ian said, first I need to say what dark matter is. Actually, that's quite hard, because we really don't know. It's matter that we think is in the universe, but we can't see. Why do we think it's there? We can observe the structure of galaxies, and how they move and interact, and how they bend light.  On a galactic scale, gravity dominates the other forces, and is what is responsible for galaxies. The gravitational force is generated by mass. So by observing galaxies, we can attempt at calculating the amount of mass in they contain. The problem is, when these calculations are made, there appears to be far more mass than we can account for.

This extra, unobserved mass, is labelled 'dark matter'. It's matter - it has a gravitational effect - but it doesn't interact electromagnetically - that means it does not emit visible, infra-red, radio waves etc. We can't "see" it in any electromagnetic sense. There have been various theories put forward to describe dark matter, but researching it is tricky because we can't actually see any. 

Back to Ian's talk. He used a ten-dimensional algebra to describe the universe. As well as recovering the electromagnetic interaction from it, he recovered Einstein's description of gravity - except with a 'twist'. In Einstein's description, the gravitational effect is seen as a bending of space-time casued by a mass. It's often illustrated by the mass-on-a-membrane analogy. A regular grid on the membrane is distorted by the presence of the mass. Einstein's field equations describe this in a mathematical sense. The description might be complicated but it boils down to this - mass distorts the space-time in which we live, and we perceive this as gravity. 

Now, Ian's result is a bit different. On a 'small' length scale (which actually means something pretty big) everything's the same. But on larger scales the source in Einstein's equation is itself distorted. In other words, what is bending the space-time is not the mass, but something else that is itself caused by the mass. Does this explain dark matter?  It looks as if there must be hidden mass in galaxies, but is this down to the fact that the bending of space-time isn't directly caused by the mass we see, but via some intermediate? Maybe there is no dark matter at all - it's just our description of gravity that needs modifying. 

Ian used the analogy of a car suspension. With no suspension on the car, the car feels all the bumps on the road. But put in a suspension system, and the effect felt by the car is a distorted image of the bumps. The up-and-down movement of the car is still caused by the bumps, but there's an intermediate step (the suspension). The end result is different. So, using this analogy, mass is 'suspended' - the universe feels a distorted version of it. Ian, Matt and Will conclude their commentary on this (which, for those more mathematically-trained than I, you can tackle here), with:

We have explained dark matter by concluding that there is no dark matter as such. There is only gravity behaving as though matter were present where there is none. 

Is this interpretation correct? Is there really no such thing as dark matter? Probably time will tell, but nonetheless it's certainly an interesting possibility. 


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