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    Gravity Work required to move a mass radially outward from the c.m. of a planet

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    See attachment #1 for diagram showing parameters.

    A uniform spherical planet has mass M= 8.5 E 25 kg and radius R= 7.5 E 6 m.
    A hole is drilled from the surface to mass m= 360 kg at an initial position r1= 2.5 E 6 m from the center.

    PART a. Find the initial gravity force F that the planet exerts on mass m.
    PART b. Find the work required to bring mass m to the surface at a constant speed.
    PART c. Find the work required to move mass m at constant speed from initial distance r2= 8.2 E 6 m to distance r3= 9.4 E 6 m, from the center of the planet.

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    Solution Preview

    PART a.
    The gravity field g at a point is defined as the force per unit mass at that point. In this case, the distance r1 is less than the radius R so the mass m is interior to the planet. For a field within a planet, universal gravitation law;
    (1) F= G M m /r^2 will not hold. The field of a planet at interior points is given (as developed in a separate posting in the Solutions Library #6055) by the linear function:
    (2) g = (G M/R^3)(r1)
    From the definition of g, we can write:
    (3) g= F/m.
    Equating (2) and (3) and solving for F, we get:
    (4) F= (G M ...

    Solution Summary

    The solution is very detailed in its explanation of the problem and the answers. The initial positions from the center of a spherical planets are given.