Abstract
The phenomena of the ocean tides is one of the most visible and dramatic manifestations of gravity. It is a vivid demonstration of the “action at a distance” which characterizes its deep mystery. We know that the force of gravity is determined by two things: the existence of a mass and its distance from us. Any particle of mass will attract every other particle of mass, and particles twice as far away will exert their attraction one fourth as strongly. Understanding this inverse square relationship is the key to understanding the tides. As Rachel Carson put it in her classic, The Sea Around Us,
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Notes
- 1.
Carson [1].
- 2.
Sir Isaac Newton, Mathematical Principles of Natural Philosophy, Book III, Proposition XXIV, Theorem XIX, pp 395–337 (Translation by Andrew Motte, 1729, revised by F. Cajori). University of California Press, Berkeley, 1949. (Short title) Principia.
- 3.
Principia, Book III, Proposition XXXVI, Corollary, 478.
- 4.
We have avoided the terminology of “pseudo forces” and “fictitious forces” which may be confusing to the student. These are forces that emerge in accelerating reference frames and are not physically real in the sense of the four fundamental forces of nature. For example, a ball placed on a moving merry-go-round will move to the outer rim. But no “force” has pushed or pulled it – it is the rotation of the reference frame itself that has displaced its location due to the ball’s inertia, its tendency not to move with the rotating frame. If we say that the gravitational force (or acceleration) of the Moon acting at the center of the Earth is exactly balanced by the fictitious force of inertia F = − mf (or acceleration, F/m = − f) acting in the opposite direction (a so-called “centrifugal force”), we get exactly the same result as described in the text. For a useful discussion of frames of reference and the confusion they can cause when explaining tides, see P. Sirtoli, Tides and Centrifugal Forces (2005), at http://www.vialattea.net/maree/eng/index.htm. An interesting but considerably more technical treatment is given by E.I. Butikov, A Dynamical Picture of the Oceanic Tides (1989), http://faculty.ifmo.ru/butikov/Planets/Tides.pdf. The image of the spaceship Earth was drawn from this paper.
- 5.
Using Newton’s Second Law, F = mf. See Eric M. Rodgers, Physics for the Inquiring Mind, p. 328 (Princeton, 1960) for a similar treatment of this problem which inspired the approach.
- 6.
A succinct way of looking at it is this: if R is the radius of the Earth, the gravitational potential on the surface is U = −GM 1 m/R. Since g = GM 1 m/R 2, then GM 1 m = R 2 g. Substituting this last expression into the first equation yields U = − gR. We can ignore the sign if we choose in this context, since the difference in U is what will concern us.
- 7.
We again emphasize that the actual tides and their timing are as noted earlier affected of course by many factors, such as the eccentricity of the lunar and solar orbits, inclination of the orbits, friction, latitude on the Earth, ocean depth, contours of the land, currents, etc. The point to remember is that water flows and the oceans tend to move and hump up under the areas of least earthly gravitational potential, along the Earth – Moon line. Owing to these differences, tides can be many meters in some places on Earth and barely detectable in others.
- 8.
I credit Moaz [2] for this problem. This difference in weight is measurable and it would be an enjoyable experiment to try over the course of a lunar month.
References
Carson R (1961) The sea around us, Rev edn. New American Library, New York, pp 142–143
Moaz D (2007) Astrophysics in a nutshell. Princeton University Press, Princeton, p 112
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MacDougal, D.W. (2012). What Causes the Tides?. In: Newton's Gravity. Undergraduate Lecture Notes in Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5444-1_17
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