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.
I loved those books.
Please make sure you write a similar review once you get to the bit about the boat carrying a large cargo consisting of flasks semi-filled with Mercury.
I don't think the shell theorem is "higher order".
g = -GM/r^2
dg/dr = -G/r^2 * dM/dr + GM/r^3
which term is bigger? multiplying all by r^2/G, we're now comparing
dM/dr and M/r
plugging in to google search bar, i get that the shell effect is roughly 50% bigger than the altitude effect, so the pendulum actually runs slower at the bottom of the well, and the difference is only half as large as calculated for the altitude effect alone
spoke too soon. it looks like i forgot a factor of two in one of my derivatives. I still think it's worth point out that the effects are actually comparable in size, not one dwarfed by the other.
also, note that to do the calculation, i used the density of the crust in finding dM/dr, rather than the density of the earth as a whole. on wikipedia i found roughly 3g/cc for the crust and 5g/cc for the earth on average.
You are entirely correct, both in reasoning and numbers. Maybe it's better experimentally to dodge the issue by using a tower rather than a well!
Funny thing, I bought the series a couple of weeks ago and, as a fellow grad student, I was planning to start them over break too.
Don't you have to account for the fact that a piece of the earth is also above the clock when it's inside the well? Especially when the well gets deeper. Or is that negligible at 300 feet?
Oh, I just read that bit about "shell theorem" and see that you already thought of that! Sorry, I spoke too soon!
Use a tall tower *and* a deep hole. Mind the temperatures and air viscosities. Gee increases with depth to 50% Earth radius.
CRC Handbook Chem. Phys., 88th Ed., p. "14-13"
The Baroque Cycle is well worth the time and attention. It keeps on giving. If, after the first volume, you thought you spent too much time in what seemed tedious stretches (and it does have some), take heart; in the subsequent volumes they are much shorter. We might like to consider a second edition with those stretches reduced, but would we trade away Anathem to get that?
The practical pendulum dampens quite a bit in two weeks. Take two similar pendulums. Describe the the motion with linear constant coefficient differential equation with dampening (just an another model to take the dampening into account somehow, disregard everything else). Does the pendulum in lower g stop first? How much sooner, if we have some idea about the magnitude of the dampening?
@7: It was Newton who thought of that, and some believe that he invented the calculus to solve that particular problem.
@10: You would do the experiment with a pendulum clock, and the variation of gravity with latitude is MUCH bigger than what was estimated here so the effect is minutes per day.
I have added something on Richer's discovery here