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Weighing the Earth

Suppose you know your own mass m, determined not from your weight but from experiments in which you are accelerated horizontally by known forces. Then from your weight W you can calculate the mass of the Earth, M_E, if only you know G, the universal gravitational constant, and R_E, the radius of the Earth. The trouble is, you cannot use the same measurement (or any other combination of measurements of the weights of objects) to determine G. So how do we know G? If we can measure G then we can use our own weight-to-mass ratio (i.e. the acceleration of gravity, g) with the known value of $R_E = 6.37 \times 10^6$ m to determine M_E. How do we do it?

The trick is to measure the gravitational attraction between two masses m₁ and m₂ that are both known. This seems simple enough in principle; the problem is that the attractive force between two "laboratory-sized" masses is incedibly tiny.^10.9 Cavendish devised a clever method of measuring such tiny forces: He hung a "dumbbell" arrangement (two large spherical masses on opposite ends of a bar) from the ceiling by a long thin wire and let the system come completely to rest. Then he brought another large spherical mass up close to each of the end masses so that the gravitational attraction acted to twist the wire. By careful tests on shorter sections of the same wire he was able to determine the torsional spring constant of the wire and thus translate the angle of twist into a torque, which in turn he divided by the moment arm (half the length of the dumbbell) to obtain the force of gravity F between the two laboratory masses M₁ and m₂ for a given separation r between them. From this he determined G and from that, using

$$g = {G \, M_E \over R_E^2}$$ (10.5)

he determined   $M_E = 5.965 \times 10^{24}$ kg  for the first time. We now know G a bit better (see above) but it is a hard thing to measure accurately!