500 |
Week 12: Gravity |
an infinite number of planets, an infinite number of which have some sort of intelligent or otherwise life may be impossible to verify or refute, but infinite or not the Universe is enormous compared to the scale of the Solar system, which is huge compared to the scale of the Earth, and contains many, many stars with many, many planetary systems.
In fact, the only thing about the Earth that is remarkable may turn out to be – us!
12.4: The Gravitational Field
As noted above, Newton proposed the gravitational force as the cause of the observed orbital motions of the celestial objects. However, this force was action at a distance – it exists between two objects that are not touching and that indeed are separated by nothing: a vacuum! What then, causes the gravitational force itself? Let us suggest that there must be something that is produced by one planet acting as a source that is present at the location of the other planet that is the proximate cause of the force that planet experiences. We define the gravitational field to be this cause of the gravitational force, the thing that is present at all points in space surrounding a mass whether or not some other mass is present there to be acted on!
We define the gravitational field conveniently to be the force per unit mass, a quantity that has the units of acceleration:
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G M |
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~ |
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~g(~r) = − |
rˆ = |
F |
(1050) |
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r2 |
m |
The magnitude of the gravitational field at the surface of the earth is thus:
g = g(RE ) = |
F |
= |
G ME |
(1051) |
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m |
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R2 |
|||
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E |
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and we see that the quantity that we have been calling the gravitational acceleration is in fact more properly called the near-Earth gravitational field.
This is a very useful equation. It can be used to find any one of g, RE , ME , or G, from a knowledge of any of the other three, depending on which ones you think you know best. g is easy; students typically measure g in physics labs at some point or another several di erent ways! RE is actually also easy to measure independently and some classical methods were used to do so long before Columbus.
ME , however is hard! This is because it always appears in the company of G, so that knowing g and RE only gives you their product. This turns out to be the case nearly everywhere – any ordinary measurement you might make turns out to tell you GME together, not either one separately.
What about G?
To measure G in the laboratory, one needs a very sensitive apparatus for measuring forces. Since we know already that G is on the order of 10−10 N-m2/kg2, we can see that gravitational forces between kilogram-scale masses separated by ten centimeters or so are on the order of a few billionths of a Newton.
Henry Cavendish made the first direct measurement of G using a torsional pendulum – basically a barbell suspended by a very thin, strong thread – and some really massive balls whose relative position could be smoothly adjusted to bring them closer to and farter from the barbell balls. As you can imagine, it takes very little torque to twist a long thread from its equilibrium angle to a new one, so this apparatus has – when utilized by someone with a great deal of patience, using a light source and a mirror to further amplify the resolution of the twist angle – proven to be su ciently sensitive to measure the tiny forces required to determine G, even to some reasonable precision.
Using this apparatus, he was able to find G and hence to “weigh the earth” (find ME ). By measuring θ as a function of the distance r measured between the centers of the balls, and calibrating
M m