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January 2016 Archives

Alison Campbell alerted me to the following: Physicist Andreas Wahl shoots himself with a gun underwater - and proves a point about drag force. 

For the record - I won't be repeating this. Physics or no physics, the guy is crazy. 

BUT, what I have done, is a quick post-hoc analysis from the safety of my own office. There's a little bit of maths involved, but the gist of it is this. The drag force on an object (in turbulent conditions - which this most certainly is), is given by the equation c rho A v2 where c is the 'drag coefficient', rho is the density of the fluid in which the object moves, A its cross-sectional area and v its speed. If we equate this to the objects mass times acceleration (Newton's second law) we get an expression for the acceleration of the object in terms of some physical parameters. Solving the equation (integrate it!) gives an exponential decay relationship between the velocity of the object and the distance it travels. Thismeans there's a characteristic length-scale, d, given by:

d = rho_bullet x b / (c rho_liquid)  

where b is the length of the bullet.  Broadly speaking, d gives you an indication over what distance the speed will decay over. We can now stuff some numbers in. Let's assume the density of the bullet is about five times the density of the water (note how it's only the ratio of the two that matters) and that the bullet is about 2 cm long. The drag coefficient will be quite low, given its a streamlined object; say about 0.1. That gives a distance scale of around a metre. How far does the bullet travel? From the video, I'd say something around a metre. 

So, what's the difference between water and air? It's the density of the fluid. Air has a density around 1 kg per metre cubed (rather than water's 1000 kg per metre cubed). Fire the bullet in air, and the length scale goes to one thousand metres (1 km). I'm not a gun expert, but that figure seems about right. 

I also think he was very wise to have his head out of the water. Sound travels rather well underwater.

And it's very easy to say all that without a gun pointing at your chest. 


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Don't miss the BBC poll on what is the world's most beautiful equation. Are you a fan of Einstein's field equation, or does the Riemann zeta-function hold you in raptures? There's some great commentary on the twelve candidates here

How did I vote? Well, that would be telling, but the fact that my very first publication is titled Auxiliary-field quantum Monte Carlo calculations for the relativistic electron gas [read it here! - at least if you have access to the Journal of Physics: Condensed Matter] might give the physicist readers a few clues. 


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Going back to my last post, our fancy balance proclaims that it weighs objects from 0 to 200 g with a precision of 0.001 g (that's one milligram).   And it does - put an object on and the balance gives you an attractive-looking number on its prominent display reading 184.139 g, or something similar. It is precise to 1 milligram. It's not reading 184.138 g, neither is it reading 184.140 g, it is reading 184.139 g. 

So does that mean our test object has a mass of 184.139 g? Unfortunately not. Just because the balance gives us that number that precisely, it doesn't mean that it is that accurate. University lecturers always have a good giggle when some poor unsuspecting first year student records an answer to a wildly inappropriate number of significant figures - for example she might measure a speed in the lab of 1.48392348837 m/s.   Precise, yes. Accurate, no. However, when a third year student does the same time (I've usually got the message across by then, aided by deducting marks on assignments for stupid use of significant figures), the humour turns into despair. 

So what does our test object weigh? (What is it's mass? I mean). Well, I can weigh it several times and see how the results are spread. I've done that. It's a few milligrams. On taking the object off, and putting it on again, I don't see the recorded mass change by more than three of four milligrams. If I take a lot of measurements, and work out the mean mass and its standard uncertainty (with a bit of statistics) I can get something that has a random uncertainty of only a milligram or so. 

However, that still doesn't mean our test object has a mass of 184.139 g (or whatever our calculation says). I may have accounted for random uncertainty, by weighing it multiple times, but there's certainly other systematic sources of uncertainty. These are significant. Just look at the graph. This is the mass of the test object (as recorded by the balance) over the period starting Tuesday morning this week. Our object is getting lighter! Quite a lot lighter, too. It's moved about 40 mg over three days! That's about one part in 5000. 


What's happening? One interesting thing to note is that we initially calibrated the balance with a nominal 200.000 g mass (sent with the equipment) and a completely empty pan (0.000 g). I've also been weighing the calibration mass and the empty pan over the course of the three days too. They have shown no drift at all - just a couple of milligrams of random uncertainty, as far as I can see. 

The manual suggests that the equipment is affected by temperature and humidity. Now, Monday was one of those horrible Waikato days with a warm, damp atmosphere and lots of rain. One of those days where, if the humidity got any higher, it would be raining in your office. And there was a lot of rain Monday night, before I made the measurements. On Tuesday morning, everything 'felt' damp - but we've been drying out ever since. Is it a long-term drop in humidity in the lab that's caused the change? And is it because the test object was actually heavier (maybe there was some condensation on it) or is it because the humidity has made the balance 'stick' or affected the electronics in some way. I'm not sure?

But what I am sure about is that saying I have a 184.139 g test mass is, at present, unjustified. 





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It's a New Year and there are lots of things to do at work before the students get back in any numbers. There are still summer students and research students here, and in the last couple of days I've been working with a summer student on getting a new piece of equipment running for our Experimental Physics paper - the Gouy Balance for measuring magnetic susceptibility. 

Magnetic susceptibility is a measure of how magnetically responsive a material is - how much it magnetizes when placed in a magnetic field. Materials can be categorized as diamagnetic, paramagnetic or ferromagnetic. Paramagnetism describes a material that magnetizes with the applied magnetic field - that is, it will be attracted to a region of high magnetic field. A ferromagnetic material goes beyond this - not only is it attracted to a region of high magnetic field it retains its magnetization even after being removed from the field. Iron is the obvious example - once you magnetize it it will stay magnetized. Diamagnetism is the opposite of paramagnetism - a diamagnetic material will magnetize in a direction against the applied magnetic field and therefore be repelled from a region of high magnetic field. Water is a easy-to-get-hold-of example. 

You can demonstrate the diamagnetism of water with some rather simple apparatus. Get a short stick, skewer a couple of nice, ripe tomatoes at each end, and hang the stick by a thread from its centre. Adjust the tomatoes so that the stick is roughly horizontal when you hang it. Wait for it to settle down then take a strong magnet (a rare-earth magnet is best) and move it close to a tomato. The tomato will be repelled.

The Gouy Balance measures susceptibility in a broadly similar way - by measuring the force on a sample of material when it approaches a magnet. With the equipment we have, we actually doing the opposite - we have a magnet on a sensitive balance, and we look at the change in the weight of the magnet as a sample of material is brought towards it. With a paramagnetic material, as we lower the material toward the magnet, the magnet is attracted (slightly) to the material, and the weight recorded on the balance is reduced. The size of the reduction lets us calculate the susceptibility. 

The changes aren't big - with our test sample of titanium powder this morning our magnet's measured mass changed from 184.142 g to 184.014 g as the material approached - a change of 0.128 g, or about 0.07%. One certainly wouldn't feel the difference if one were holding the magnet, but the balance is sensitive enough to pick up the change. 

It's a neat little apparatus and will be fun to play with. And it comes with a demonstration of magnetic levitation with pyrolytic graphite


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