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Applications of Newton - Figure Circular orbit around the...

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Applications of Newton's Law Gravity between planets We can now use Newton's Law to derive some results concerning planets in circular orbits. Although we know from Kepler's Laws that the orbits are not circular, in most cases approximating the orbit by a circle gives satisfactory results. When two massive bodies exert a gravitational force on one another, we shall seethat planets describe circular or elliptical paths around their common center of mass. In the case of a planet orbiting the sun, however, the sun's mass is so much greater than the planets, that the center of mass lies well within the sun, and in fact very close to its center. For this reason it is a good approximation to assume that the sun stays fixed (say at the origin) and the planets move around it. The force is then given by:
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Unformatted text preview: Figure %: Circular orbit around the sun. From the central force acting on the planet is exerting a centripetal force. We know that a centripetal motion has acceleration = and thus = . We can therefore write (note that in what follows r , without the vector arrow denote the magnitude of r--that is r = | | ): = Rearranging we have that: v 2 = Thus we have derived an expression for the speed of the planet orbiting the sun. However, we can also express the speed as the distance around the orbit divided by the time taken T (the period): v = Squaring this and equating this with the result from above: = âá’ T 2 = Thus we have derived Kepler's Third Law for circular orbits from the Universal Law of Gravitation....
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