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Gravity and Inertia.

The solar system formed from a cloud of gas and dust which had net angular momentum, that is, it was spinning. As this cloud contracted and eventually condensed into the sun and planets (courtesy of gravity), the conservation of angular momentum (due to rotational inertia) ensured that the final product would also be spinning. Thus, the planets revolve around the sun, the sun and planets all rotate around their axes, and many of the planets have moons which orbit them. Now that the solar system exists, inertia keeps the planets moving and gravity keeps them in orbit. The same general principles apply to the formation of the earth moon system, except that the moon did not form directly from the original cloud, but is believed to have been formed from the earth during a giant collision.

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