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Math Help - Path Independence

  1. #1
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    Path Independence

    The mass of the earth is approximately 6 x 10^27 g and that of the sun is 330,000 times as much. The gravitational constant is 6.7 x 10^-8 m^3/s^2 g. The distance of the earth from the sun is about 1.5 x 10^12 cm. Compute, approximately, the work necessary to increase the distance of the earth from the sun by 1 cm.
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  2. #2
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    Quote Originally Posted by larryboi7 View Post
    The mass of the earth is approximately 6 x 10^27 g and that of the sun is 330,000 times as much. The gravitational constant is 6.7 x 10^-8 m^3/s^2 g. The distance of the earth from the sun is about 1.5 x 10^12 cm. Compute, approximately, the work necessary to increase the distance of the earth from the sun by 1 cm.
    Think about the area under the force versus distance graph for the Earth.
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  3. #3
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    The point of "path independence" here is that because the force is conservative, the integration along a path on which the earth is "moved" one cm more from the sun (which, of course, it does regularly in its orbit) is the same as the "anti-derivative" evaluated at r_0 and r_0+ 1 where r_0 is the initial distance from the sun to the earth in cm.

    Since the force is given by (GmM)r^{-2} (G is the "universal graviational constant", m is the mass of the earth, and M is the mass of the sun), what is the anti-derivative of that?
    Last edited by mr fantastic; August 16th 2010 at 12:20 PM. Reason: Fixed latex tag.
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