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Tides

Here on the surface of the Earth, we have little occasion to notice that the force of gravity drops off inversely as the square of the distance from the centre of the Earth.^10.12 This is fortunate, since otherwise Galileo would not have been able to do his experiments demonstrating the (approximate) constancy of the acceleration of gravity, g; moreover, scales and other mass-measuring technology based on uniform gravity would not work well enough for commerce of engineering to have evolved as it did. So we don't notice any effects of the inverse square law ``here at home,'' right? Well, let's not be hasty.

The Moon exerts an infinitesimal force on every bit of mass on Earth. At a distance of R_ME = 380,000 km, the Moon's mass of $M_M = 7.4 \times 10^{22}$ kg generates a gravitational acceleration of only $g_ME = 3.42 \times 10^{-5}$ m/s²; in other words, our gravitational attraction to the Moon is $3.5 \times 10^{-6}$ of our Earth weight. Moreover, the Moon's gravitational acceleration changes by only $-1.8 \times 10^{-13}$ m/s² for every metre further away from the Moon we move - a really tiny gravitational gradient. Nevertheless, the fact that the water in the oceans on the side of the Earth facing the Moon is attracted more and that on the side away from the Moon is attracted less leads to a slight bulge of the water on both sides and a concomitant dip around the middle. As the Earth turns under these bulges and dips, we experience (normally) two high tides and two low tides every day.

The consequences of these tides are nontrivial, as we all know. Even though they are the result of an incredibly small gravitational gradient, they represent enormous energies that have been tapped for power generation in a few places like the Bay of Fundy where resonance effects generate huge movements of water. More importantly in the long run (but of negligible concern in times of interest to humans) is the fact that the ``friction'' generated by these tides is gradually sapping the kinetic energy of the Earth's rotation and at the same time causing the Moon to drift slowly further away from the Earth so that in a few billion years the Earth will be ``locked'' as the Moon is now, with its day the same length as a month (which will then be twice as long as it is now) and the same side always facing its partner. ``Sic transit gloria Mundi,'' indeed! Let's enjoy our spin while we can.

A less potent source of tidal forces (gravitational gradients) on Earth is the Sun, with a mass of about $3 \times 10^{40}$ kg at a distance of about 93 million miles or $1.5 \times 10^{11}$ m. You can calculate for yourself the Sun's gravitational acceleration at the Earth: small but not entirely negligible. The Sun's gravitational gradient, on the other hand, is truly miniscule; yet various species of fish seem to have feeding patterns locked to the relative positions of the Sun and the Moon, even at night when the more obvious effects of the Sun are absent. The so-called ``solunar tables'' are an essential aid to the fanatically determined fisherman! Yet, so far as I know, no one has any plausible explanation for how a fish (or a bird or a shellfish, which also seem to know) can detect these minute force gradients.

A more dramatic example of tidal forces is the gravitational field near a neutron star, which has a large enough gradient to dismember travellers passing nearby even though their orbits take them safely past.^10.13 Near a small black hole the tidal forces can literally rip the vacuum apart into matter and antimatter, causing the black hole to explode with unmatched violence; this in fact limits how small black holes can be and still remain stable.^10.14