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Gravitational potential for various matter configurations

  1. Jun 11, 2017 #1
    1. The problem statement, all variables and given/known data

    Consider the Earth as

    1. with a constant density of matter,
    2. as a thin shell empty sphere and
    3. with a constant linear density of matter ##\rho(r) = \rho_{0}r##.

    In all cases, calculate the gravitational potential and the gravitational field everywhere and make a sketch.

    2. Relevant equations

    3. The attempt at a solution

    1. The gravitational potential outside the Earth is equal to the gravitational potential of a point particle of the mass of the Earth, that is,

    $$V = - \frac{GM}{r}.$$

    Therefore, the gravitational field is simply

    $$g = - \nabla V = - \frac{GM}{r^{2}}.$$

    To compute the gravitational potential within the Earth, do I have to slice up the Earth into thin shells of radius ##dr## and integrate over shells which contribute to the potential?
  2. jcsd
  3. Jun 11, 2017 #2


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