Measuring Gravity, Baroque Style

I've finally buckled down and started reading Neal Stephenson's Baroque Cycle. Though the author is probably my favorite living fiction writer, the three-volume, ~3000 page monstrosity is just something that's hard for a busy grad student to tackle. I'm only maybe 1/5th through the first volume, but so far so good. Plenty of interesting science as well.

One character (pre-Newton) correctly surmises that the gravitational force of the earth must diminish with distance. To check this, he proposes an experiment that sounds suspiciously Einsteinian but is in fact entirely classical. Put a clock at the bottom of a deep well, and after some time has passed compare it to an identical clock at the top of the well. In doing so, it's possible to compare the gravity at the top and bottom of the well and see if there's any changes. This works in theory not because of any gravitational time dilation, but because the clock is a pendulum clock, and a pendulum being pulled more strongly by gravity will swing faster. This is not so hard to demonstrate - make a pendulum where the weight is a magnet, and then swing it over something magnetic so as to increase the downward force the pendulum feels. It will swing faster.

So how long does it take a pendulum to make a swing? I'll skip the derivation for the moment and just quote the result. The time required for a pendulum to make a complete back-and-forth swing is:

Where L is the length of the pendulum and g is the local acceleration due to gravity. Not knowing a priori how g varied with altitude, our intrepid experimenter would be learning something entirely new by measuring the variation. But we know Newton's law of gravity, and so we can see just how sensitive his measurement would have to be.

Near the earth's surface (and ignoring the shell theorem as a higher-order correction [This is not actually right, though it doesn't greatly affect the subsequent numbers. See comments, and thanks to Mark of the fantastic Arcsecond blog. - Matt], the acceleration due to gravity is:

Where G is the gravitational constant, M is the mass of the earth, and r is the radius of the earth - more precisely, r is the altitude of the observer above the center of the earth. Doing some calculus magic (differentiating and multiplying by the change in altitude), we can see how much g varies with altitude:

Plugging in the appropriate numbers, and using 300 feet as the depth of the well (which is what it was in the book), the change in g is about 0.00014 m/s^2. Compared with the g's actual magnitude of about 9.8 m/s^2, this is pretty small.

Plugging that change in g into the pendulum equation, the period of a pendulum will be reduced by a factor of about 7 parts per million. Obviously it's impossible to measure the swing of a pendulum with microsecond precision, so the experiment is doomed. Right?

Not necessarily. You could just let the pendulum swing for a million seconds, and observe that the clock on the bottom of the well has pulled ahead by about 7 seconds. A million seconds is only a little over 11.5 days, so suddenly this experiment starts to look practical. Indeed pendulum measurements of the local variation of gravity (with latitude, in this case) were first conducted as early as 1672 by Jean Richer. The variation with altitude is considerably smaller, but was successfully observed by Pierre Bouguer in 1737.