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# Appendix A: The local meter' and second' in the planets of the solar system

The well known small angle formula that gives the period of the simple pendulum (i.e. the elementary text book pendulum) as a function of its length is
 (1)

where is the gravitational acceleration, approximately equal to on Earth. For m we get s. Therefore, each swing takes 1.0035s, that differs from a round second only by a few parts per thousand. Varying by (i.e. from to ), the period changes only by .

In order to understand if there is any physical reason behind this numerical coincidence let us try to understand the property of Earth that mainly influences the period of the pendulum, and if there is any simplification due to the fact that the length of the pendulum is about of the meridian.44 The gross value of depends on mass and radius45 of the Earth with local effects due to not exact sphericity (see Table 4), mass dishomogeneity and above sea level height. Moreover, there is a centrifugal term, null at the pole and maximum at the equator, due to Earth rotation.46In the approximation of a perfect sphere, the gravitational acceleration , i.e. the gravitational force divided by the mass of the pendulum, is given by

 (2)

where kg is the mass of Earth and is the gravitational constant. Expressing the mass in terms of density and volume , we get
 (3)

The gravitational acceleration is then proportional to the planet size and density. Let us now calculate the period of a pendulum whose length is part of a meridian of a spherical planet, i.e. , where is the fixed ratio between this meter' and the planet radius. The period of such a planetary meter' pendulum is
 (4)

and depends only on planet density, and not on planet mass and size separately. In particular, in the inner planets and Earth, for which the density is approximately 5.5 g/cm, such a planetary meter' pendulum would beat approximately the second (see Tab. 6).

 Planet Physical data [48] One meter' pendulum and its period Mass Radius (kg) (km) (g/cm) (m/s) (m) (s) (s) Mercury 2440 5.43 3.70 0.38 1.01 58.6 Venus 6052 5.24 8.89 0.95 1.03 Earth 6378 5.52 9.80 1.00 1.00 1.00 Mars 3397 3.93 3.69 0.53 1.19 1.03 Jupiter 71492 1.33 23.17 11.23 2.19 0.41 Saturn 60268 0.69 8.98 9.47 3.23 0.45 Uranus 25559 1.32 8.71 4.01 2.13 0.72 Neptune 24766 1.64 11.03 3.89 1.87 0.67 Pluto 1137 2.06 0.66 0.19 1.64

However, the half period of this pendulum is approximately equal to the part of the planet rotation only for Earth and Mars, which have approximately equal `days'. For all other planets, the local day can be very different with respect to the Earth one. In fact, the rotation speed is related to the initial angular momentum when the planet was formed and there is no reason why it should come out to be the same in different planets (Venus and Pluto are indeed retrograde, i.e. they rotate East-West).

Next: Appendix B: Tito Livio Up: Why does the meter Previous: Acknowledgements
Giulio D'Agostini 2005-01-25