# The indeterminate cheese flake

A couple of days ago I arrived 'home' to discover our local ant colony at work. There's a nest located somewhere in the bushes at the front of our temporary home, and the occupants have become rather adept at raiding our kitchen. Anything left on the kitchen bench is fair game for the taking. Ants are amazingly strong for their size – any little bit of food like a small oat flake gets picked up and transported back to the nest.

Now, in this case the ants were after the grated cheese. Some sizeable flakes had been left on the bench and these were going to be a feast for an ant colony. By sizeable I mean something you get from a coarse grater – perhaps 3 mm wide and maybe 2 cm long. Now, to get them off the kitchen bench and to the nest, they had to take them across the bench, up the splashback by the sink, across the windowsill, out of the window and then down the wall outside. Getting them across the bench was no problem for a few tough ants – all heave together – and away goes this piece of cheese. Quite amazing to watch them move it so quickly.

But the next bit was tricky. They had to manoeuvre the cheese off the bench and up the wall. They did it by putting one end of the cheese against the wall – then a group of ants on this end slowly lifted it up – while a group on the other end pushed. At least, that's what it looked like they were doing. It seemed to be a touch random, but, on average, that's what was happening. Once the cheese had a bit of an angle to it, it got easier, since it was able to rest there, supported on one end by the friction of the wall, and the other end by the bench. Very soon it was on its way up the wall.

Watching them at work reminded me of the 'ladder on the wall' problem. This is a problem in statics that's often wheeled out to bring the over-enthusiastic "I can solve everything with equations" student back down to the earth with a thud. "A ladder of length 4 m (or whatever) and mass 10 kg (or choose your own) rests at an angle of 80 degrees against a vertical wall with coefficient of static friction 0.8.  The bottom end is resting on the flat ground with coefficient of static friciton of 0.6. (a) Draw a free-body diagram showing the forces acting on the ladder. (b) Evaluate these forces."

The point here is that this problem, as phrased, does not have a unique solution. I'll let you have a go at drawing out the freebody diagram. There's the weight of the ladder (which we know), and we can assume that acts downwards at the centre of mass. That bit is easy. At each of the two ends we have a normal force from the surface and a friction force. That's five forces in total. We know the weight, so it leaves us four to find.  But we can only find three independent equations – we know in equilibrium that (1) the horizontal component of force must be zero, (2) the vertical component of force is zero and (3) moments about the centre of mass are zero. Four unknowns, three equations. We can try to take moments about some other point, such as the ends of the ladder. But that doesn't yield any more information. In equilibrium, the coefficients of friction don't help us find new equations (just inequalities). We are stuck. This is called a 'statically indeterminant problem'.

Now, there are ways of proceeding (the enthusiastic student can read this, for example), but we need to know some more information about the elastic properties of the ladder, the wall and the ground. But my point is that there are some seemingly simple physics problems that just can't be tackled by a naive throw-the-equations-at-it approach. We do need some more careful thinking.

I reckon the ants were not too interested in the finer points of statics when they were manouevring this piece of cheese. In the same way, one doesn't need to solve statically-indeterminant problems in order to safely use a ladder. But, for a physicist, it begs the question "What is the coefficient of friction between a piece of cheese and a tiled wall when the surface is lubricated with ants? And then, "Is there a good way of modelling this?"