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 (see figure below). 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.

(a) How far from the surface is there a point where the magnitude is 1/2

(b) How far from the surface is there a point where the magnitude is 1/2

FRif we move the apple into the tunnel? (State your answer as a multiple of R.)

Explanation / Answer

a)

here

Fr = (1/2) * Fr = G * M *m /r^2 = (1/2) * G * M * m / R^2

r = sqrt(2) * R

the distance between the apple and the surface is

h = r - R = ( sqrt(2) - 1 ) * R

h = 0.414 R

b)
only the part of the sphere which is inside of the apple exerts force on the apple. The mass of the inside part is

Min = (4/3) * pie * rho * (R/2)^3

Min = (1/8) * (4/3) * pie * rho * (R/2)^2 = M/8

thus the gravitational force is

Fr/2 = G * ( M / 8) * m / (R/2)^2 = 0.5 Fr

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