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    Earth as a rigid axisymmetric body

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    The earth maybe considered as a rigid axisymmetric body with a small quadrupole deformation. (There are two problems (a) and (b))

    (a) If the exterior gravitational potential is written as:

    V(r)=-M_e*G*1/r*[1-J(R_e/r)^2* P_2(costheta)]

    Here, M_e is the mass of the earth, R_e is the equatorial radius and theta the colatitude, show that J=(I_3-I_1)/M_e*R_e^2)
    P_2(costheta)=3costheta^2-1 as easily found in any quantum book.

    ----------> I have tried to take the derivative of this potential with r to find the "external force" in order to use the equation
    (external Torque)=(dL/dt)_body+wXr

    I'm not sure how to take the next step.
    Could you please show me the full process until I can arrive to the correct solution?
    Another related problem(b) is attached as file.

    © BrainMass Inc. brainmass.com October 10, 2019, 6:28 am ad1c9bdddf
    https://brainmass.com/physics/torques/earth-as-a-rigid-axisymmetric-body-545349

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    The earth may be considered a rigid axisymmetric body with a small quadrupole deformation.

    (a) If the exterior gravitational potential is written ...

    Solution Summary

    We derive an approximate formula for the gravitational potential outside the Earth in terms of its quadrupole moment about its axis of symmetry.

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