Let’s say you have a table. This table is better than your average table. It’s perfectly level, absolutely flat to within the thickness of an atom over its entire surface. In fact, this table isn’t even made of atoms. You called up Plato and ordered the platonic ideal of a flat table.
Now you set this table down in your dining room and have Plato’s deliverymen install the table so that it’s perfectly flat with respect to the earth’s surface. Then you take a ping-pong ball and set it down toward the edge of the table. What happens?
It doesn’t stay still. It will roll to the center of the table. The reason is that the table being level doesn’t mean that gravity is pointing locally downward everywhere on the table. A picture is worth a thousand words here, and for clarity let’s make this table very big. Very big.
Here x is how far from the center of the table the ball happens to be at that moment. Re is the radius of the earth. You can see how gravity will be pulling the ball toward the center of the earth, and that means there’s a component of force parallel to the table. We can set this up as a differential equation, with force on the right and mass times acceleration on the left:
Now we see that the mass appears on both sides, so it cancels. We also know that the sine of theta is for small angles approximately equal to the tangent of theta. And tangent of theta is the side opposite over the side adjacent. This puts our differential equation in this form:
Now it’s immediately clear if you’ve been in the physics business a while that this is the differential equation describing simple harmonic motion. It’s the same kind of thing as a pendulum swinging back and forth or a spring oscillating up and down. What’s especially important about simple harmonic motion is that it takes the same amount of time to execute one cycle of motion no matter how large the initial amplitude is. A barely nudged pendulum swinging at one stroke per second will also swing at one stroke per second if the push is larger. It has a longer distance to travel but it’s also moving faster. The two effects cancel out. The angular frequency of an object in simple harmonic motion is the square root of the term in front of the x. But let’s convert that to the period T, which is the time required to make one complete oscillation:
So the ball will roll back and forth on your table once every 84 minutes. You’d only be able to see this motion on a tremendously huge and perfectly flat table, otherwise imperfections in the flatness, local gravity effects, friction, and what have you will ruin the very small forces involved. We’re talking micronewtons at best for actual tabletop displacements.
Connoisseurs of physics problems will probably recognize that 84 minutes number as the period of an object falling through the earth. If you were to drill a tunnel straight down to the other side of the world, neglecting air resistance you’ll fall back and forth in 84 minutes. This means if you decided to step out at the far end of the tunnel your journey would take 42 minutes.
Actually it’s more interesting than that. Turns out that you don’t even have to dig the tunnel straight down. If I gun a tunnel from here to Paris, it would still take 42 minutes. This is not obvious and the math is somewhat trickier than this, but perfectly tractable nonetheless. We’ll do it here eventually. This particular tabletop thought experiment is something of a limiting case of the tunnel problem.
I don’t know if there are any flat surfaces long enough to actually try an experiment like this in real life. The SLAC beam tunnel, perhaps? Now there’s a side project for any clever researchers with some spare time while the computers are crunching data…