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Optimising the acceleration due to gravity on a planet's surface

  • Michael Jewess (a1)
Extract

The Earth (more precisely, the ‘geoid’ thereof) is known to approximate closely to a slightly oblate spheroid whose unique axis coincides with the Earth's axis of rotation [1,2]. (By ‘spheroid’ is meant is an ellipsoid of revolution, i.e. one with two semi-axes equal; a slightly oblate one has these two semi-axes slightly longer than the unique one.) To the nearest km, the diameter of the ‘geoid’ pole-to-pole is 43 km less than the equatorial diameter of 12756 km. There is a reduction of practical significance (0.527%) in the acceleration of free fall" at sea level between the poles and the equator, and therefore in the weight of objects. Of this, 0.345% derives directly from the rotation of the Earth; the balance of 0.182% results from the purely gravitational effect of the Earth's deviation from sphericity.

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1. Cook, A. H., Gravity and the Earth, Wykeham Publications (1969).
2. Cook, A. H., Physics of the Earth and planets, MacMillan (1973).
3. Newton, Isaac, The Principia: mathematical principles of natural philosophy, translated by Cohen, I. Bernard and Whitman, Anne, University of California Press (pbk. 1999), Book 1, Proposition 91, Corollary 2 (pp. 616-617) and Book 3, Propositions 18-20 (pp. 821-832).
4. Courant, R. and Robbins, H., revision by Stewart, I., What is mathematics? (2nd edn.), Oxford University Press (1996), pp.379385.
5. Stephenson, G., Mathematical methods for science students (2nd edn.), Longmans (1973), Chapter 25.
6. Arfken, G. B. and Weber, H. J., Mathematical methods for physicists (6th edn.), Academic Press (2005), Chapter 17.
7. Young, H. D., Friedman, R. A. and (contributing author) Ford, A. L., Sears and Zemansky's university physics (12th edn.) Vol. 1, Addison Wesley (2008), pp. 400403.
8. Råde, L. and Westergren, B., Mathematics handbook for science and engineering (4th edn.), Springer Verlag (1999), indefinite integrals nos 170 and 171, p. 160.
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The Mathematical Gazette
  • ISSN: 0025-5572
  • EISSN: 2056-6328
  • URL: /core/journals/mathematical-gazette
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