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9: Motion in Fields 9.4 Orbital Motion

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Orbital Motion Keplers Third Law: This law relates the time period T of a planets orbit (its year) to the distance r from the star it is attracted to, e.g. for Earth orbiting the Sun. We know that the force between the two bodies is… We also know that the centripetal force acting on a body in circular motion is given by… F = GMm r 2 F = mω 2 r = mv 2 r

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So equating gives... However, the angular speed ω is the angle (in radians) per unit time. So in one orbit, the angle is 2π and the time is the time period T. ω = 2π / T mω 2 r = GMm r 2 Rearranging… 4π 2 = GM T 2 r 3 ω 2 = GM r 3 So… T 2 = 4π 2 r 3 GM

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Clearly the closer the planet to the Star, the shorter the time period. (Kepler discovered his laws using observational data taken by the astronomer Tycho Brahe. A century later Newton derived Keplers laws from his own laws of motion.) Thus for any planet orbiting a star in a circular orbit, T 2 is proportional to r 3. Also the ratio T 2 /r 3 is constant. This is known as Keplers third law. T 2 = 4π 2 r 3 GM

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Kinetic Energy of a Satellite Again by equating the two equations for force acting on an orbiting body, we can now derive a formula for its KE. This time we write the centripetal force formula using v instead of ω: Rearrange and multiply both sides by 1/2 … So, for a satellite… mv 2 = GMm r r 2 ½ mv 2 = GMm 2r KE = GMm 2r

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Potential Energy of a Satellite We already know that the potential energy must be given by… Total Energy of a Satellite Total Energy = KE + PE E p = - GMm r Total Energy = GMm - GMm 2r r Total Energy = - GMm 2r

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Energy Distance r PE KE Total E

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