Answer: Square Root Calculator

Pick a number. Set the ball at that number on the ramp and let it go. Watch where it falls. The number where it falls is the square root of the number on the ramp.

This works because the velocity of the ball as it leaves the ramp is proportional to the square root of its kinetic energy. Its kinetic energy comes from the potential energy it originally had while sitting on the ramp. That's linear in the height.

Incidentally, not all the ball's initial potential energy gets converted into kinetic energy of translation. Some is also converted into kinetic energy of rotation. However, the fraction of kinetic energy that goes into rotation is constant, so this doesn't affect our ability to build a calculator. It just means the marks on the landing zone have to be closer together than they would for a frictionless, sliding ball.

The thing about the square root calculator is that it uses two linear scales. That's why it's a great machine - it doesn't take any difficult work to decide where the marks go. The laws of physics do the actual computation.

There were a variety of answers to the challenge to create your own analog calculator. Most of them missed the mark of what I was shooting for.

For example, an abacus can do multiplication. It's a physical system, as well. So I suppose it's a physical system that does a useful computation. But it's not interesting from a physics perspective. It's interesting only from a logical perspective. The abacus is probably made of wood, but doesn't depend essentially on the properties of wood to work. I could make the abacus out of stones sitting in from of me, or fancy metal rings, or little holes that I dig or fill in with sand at the beach. The abacus only cares about the positional relationship between its parts, not their physical interactions. I could even, if my powers of imagination were strong enough, picture an abacus with all its beads in my head, move them around mentally, and do abacus-style multiplication completely inside my head. If you imagine a different universe, in which the laws of physics work differently, the abacus wouldn't care. Multiplication would still be the same, and you could still do it with any device whose pieces could move in relation to each other the same way they can in an abacus.

So slide rules are out, too. Electronic computers can certainly compute useful quantities and solve differential equations numerically, and they are physical systems. But again, the magic of the computer is in its logical gates rather than its physics. Some other answers, like tree rings and sundials counting the passage of time, were not really doing a mathematical computation.

If you're good at circuit analysis, you could build an analog computer to calculate all kinds of interesting things. In fact, people used to do this all the time before digital computers got so good. So I won't talk about circuit analysis.

Instead, here's a fairly simple system to do multiplication. It's a tube of dilute gas, say argon. There's a piston allowing you to change the volume to whatever you want. The volume is marked off on the sides like it is in graduated cylinders or Nalgene bottles. You adjust the piston up or down with a stack of weights. If you want the cylinder to have less volume, stack on more weight. By counting the weights you have on there, you can get the pressure. There's a thermometer inside the tube positioned so you can easily read it. Finally, there's a bunsen burner and some cold packs lying around, so you can add or remove heat at will.

To a good approximation, the gas obeys the equation of state PV = nRT. So choose the amount of gas in your cylinder, along with your units pressure, volume, and temperature, so that nR = 1. Then PV = T. You can now do multiplication. Start by setting the weight stack so that the pressure inside is the first number you want to multiply. If the volume is lower than the second number you want to multiply, add heat. If it's greater, suck heat out. Keep adding or subtracting heat until the volume is equal to the second number you want to multiply. Now read off the temperature from the thermometer, and that is the multiple you sought.

The above machine is also a division machine, because you could equally well set the pressure to the denominator, add/subtract heat until the temperature is the numerator, and read off the volume using V = T/P. Similarly the square root calculator is also a square calculator. Take a guess at the square of a number, drop the ball from there and see where it lands. If it lands in front of the number you're squaring, drop it from higher. Repeat until you find the number on the ramp such that when you drop the ball from there, it hits your target number on the landing zone. That's the square.

Any system that satisfies the differential equation df/dx = a*f is a potential logarithm/exponential calculator, since its solution is f = e^ax = (e^a)^x. An example is a system whose friction is proportional to its amplitude. This occurs in the viscous flow of a fluid, for example. A runaway chemical reaction might obey this law, at least over some regime. Population growth of organism does this. The expansion of the universe due to a nonzero vacuum energy does this, since the rate of growth is proportional to the amount of vacuum, but growth is itself the change in vacuum.

Trig functions come from any physical system that obeys d²f/dx² = -a*f, where "a" is positive. A mass on a spring is the canonical example, all sorts of things will exhibit this same behavior. The reason is that particles will minimize their potential energy. When you're near a potential energy minimum, you can expand the potential as a Taylor series, but the constant term is not physically relevant, and the linear term will be zero because you're at a minimum. Then the quadratic term dominates over a certain regime, and in that regime the system obeys the differential equation for trig functions.