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

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
radius^{45} 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.^{46}In the
approximation of a perfect sphere, the gravitational acceleration
, i.e. the gravitational force
divided by the mass of the pendulum, is given by

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

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

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).