The Gravity of the Situation

I see that bailout is working wonderfully. Of course if it hadn't passed and the stock market had behaved identically, we be hearing that we were fools not to pass the bailout. And they would be wrong. Yes, I know God Emperor Paulson hasn't actually used any of his trillion dollar blank check yet. It won't make a difference. Give a cowboy all the expensive equipment in the world and he still won't be able to herd clouds. C'est la vie.

There's nothing to be done about that though so let's talk about gravity. The force due to gravity between objects with masses M and m is given by the old standby equation

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G is a universal gravitational constant and r is the distance between the two objects. So let's say you're standing on a scale. It's measuring the gravitational force between you and the earth. You know your mass and you know (or can at least look up) the value of G, and so from that you can find the mass of the earth. But what if you lived before the value of G was known? You're stuck. You have two missing numbers, the mass of the earth and the value of G. And with only one equation there's no way you can find the value of both at the same time. What to do?

Well, you don't really have any other option. You're going to have to measure the force between two objects whose masses are both already known. This is very difficult - you might have some pretty hefty objects to measure the forces between, but gravity is very weak. Using English units just for the heck of it, if you have two masses of one ton each, separated by one yard, the gravitational attraction between them is a whopping 0.000065 N. That's equivalent to the weight of your average sand grain or thereabouts. It's a tricky measurement, but if you can do it you can get the value of G.

The first guy to do it was Henry Cavendish, in the famous Cavendish experiment. He actually got two large metal spheres of known mass and was able to very delicately measure the gravitational force between them. From there it's easy to calculate the gravitational constant.

Well, that's kind of the "physics legend" version of the story. Cavendish was interested in the mass of the earth, and he really didn't bother with the gravitational constant. Nevertheless this method gave us the value of G.

Today G is still far and away the most badly determined of the fundamental constants of nature. While (for instance) the Planck constant is known to around 8 or so significant figures, G is only known to about 4. Maybe fewer, depending on which experimental results you believe. There's not much immediate prospect of improving that number.

Might be something to work on this weekend!

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Big G is less a problem for its small magnitude than for the impossiblity of gravitational shielding. Quiet EM volumes are easy to create with (super)conductors, inductors, and electrical resistance. If lawn sprinklers activate your G determination sloshes.

The quadrupole torsion pendulum is a tremendous advance in Big G measurement.

Funny you should write this today. I spent my morning trying to figure out how one well-logging (for oil) instrument could determine the change in depth of the instrument based on the change in gravity (using a NEMs accelerometer) over a constant multiplied by the density of the formation, while another similar instrument used the same formulation, but the constant was 4*pi*G, which is a much smaller quantity than the constant in the other formula. Error correction was what I eventually settled upon.

Anyhow, it was messy and there were unit problems everywhere, but I certainly encountered the 'fickle nature' of big G along the way.

Variation of gee with depth is a naughty thing. CRC Handbook of Chemistry and Physics, 88th Ed., section 14, page 13. Fourth column. Somebody didn't read His physics intro text when fabricating the planet.

Just curious, wouldn't it be an easier experiment with cylinders instead of spheres?

I seem to remember vaguely that the off axis force drops off as 1/r instead of 1/r^2 and perfect cylinders should be easier to make than perfect spheres.

Also, check out this experiment for fun.

The hardest part about perfect cylinders is the making them infinitely long.

By Carl Brannen (not verified) on 10 Oct 2008 #permalink