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

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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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

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