Great balls of fire!

Jonathan Dawes

Fellows of the Royal Society have from time to time been associated with hula dancers, fire-eaters, strongmen, clowns, bareback horseriders and tightrope walkers....


....but not at the Convocation this time, unfortunately.



It has been pointed out to me that the first few lines of each blog are trailed on the main blog page. Therefore it is essential that these lines provide a decent enough 'hook' to get you to click on the link to the main post. I suspect this has worked since you're now reading this. But, judging by the absence of comments (thanks, flo from Cardiff!), you're probably in quite select company.

Anyway, today's topics are bouncing and colliding, and then Brazil Nuts.


1. Bouncing and colliding.

Bouncing is easy, right !? Newton's law of restitution asserts that in a collision (i) total momentum is conserved (there are no external forces, you see) and (ii) the relative velocity of the 2 objects is reduced by a factor called the 'coefficient of restitution' r. This is all we need to work out the speeds after the collision. Done. And dusted.

So if you think that's the answer, clearly, because you're a modern scientist and you've read Karl Popper, you're going to test the law of restitution experimentally. So, first pick your balls. Then measure r. Unfortunately, r isn't as constant as you hoped. It turns out that the value you get for r,as well as depending on the material of the balls (which is what we would expect), also depends hugely on the actual relative velocity of the objects involved. When that relative velocity is small, r is always close to 1, i.e. very little energy is lost in low-impact collisions. This is clearly a key observation for the dynamics of 2 balls, or even of one ball bouncing repeatedly on the ground.

But for a large number of balls together, this observation becomes even more important - it determines how a large number of balls, or granules in a granular medium, settle down after being all shook up. This overall settling time depends on the details of the velocity-dependent coefficient of restitution but only at the low-velocity end of the graph - the high-velocity part doesn't actually play a role because if we're interested in the final settling time, by then most of the impacts will be very low-velocity, and so these will have the larger influence. This is a topic of substantial current research interest (and another example of  carrying out some kind of averaging process to get a macroscopic answer from microscopic details). I heard a talk on it in Barcelona only last month - the speed at which the granules settle down at long times is known as Haff's Law.

2. Brazil Nuts.

Tasty. And always at the top of your muesli, curiously - not evenly mixed in. This separation of a granular mixture into the big stuff coming to the top and the small powdery bits remaining underneath is a graphic example of the difference between granular materials and liquids. When you shake up a liquid it mixes. When you shake a granular mixture it separates out. This separation is so reliable that, for example, it can be used in the field by geographers to establish past avalanche activity - the interactions between rocks that are driven by gravity in the avalanche cause the big rocks to come up and smaller ones to fall down between them as the avalanche progresses. The resulting cross-section shows what is referred to as inverse grading.

The explanation for the Brazil Nut Effect, broadly, is that as the muesli, or sand and a shell as in our hand-held demo, is shaken up and down, gaps open up around the grains and beneath the larger particles. Small grains tumble down into these gaps where large grains can't fit. So the small grains rachet the larger particles up to the top. Because it's so simple, it's pretty universal! So that's the physics. The mathematical challenge is to take the 'limit of large numbers of grains', i.e. to take the description of what individual grains experience and turn it into an averaged equation for the collective motion of the grains together. That would be a simpler problem to solve, either analytically (i.e. with pen and paper) or by computer, so it would be great to understand how to do this, and guarantee that the simpler model problem made the right predictions. We're not there yet, though.