Assume a planet is a uniform sphere of radius R that (somehow) has a narrow radi

Question

Assume a planet is a uniform sphere of radius R that (somehow) has a narrow radial tunnel through its center. Also assume we can position an apple anywhere along the tunnel or outside the sphere. Let FR be the magnitude of the gravitational force on the apple when it is located at the planet's surface. How far from the surface (what multiple of R) is there a point where the magnitude of the gravitational force on the apple is 0.6 FR if we move the apple (a) away from the planet and (b) into the tunnel?

Explanation / Answer

away from the planet

at a height a of h

gh = g*R^2/(R+h)^2

0.6*g = g*R^2/(R+h)^2

0.6 = R^2/(R+h)^2

(R+h)/R = sqrt(1/0.6) =1.3

1+(h/R) = 1.3

h/R = 0.3

h = 0.3R

b) into the tunnel

at a depth d

gd = g(1-(d/R))

0.6*g = g*(1-(d/R))

0.6 = 1-(d/R)

d/R = 0.4

d = 0.4*R

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