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

a) Given FR be the magnitude of the gravitational force on the apple when it is located at the planet's surface

Then FR = GMm/R^2

where G =gravitational constant

M = mass of the planet

m = mass of apple

R = radius of the planet

Let r be the point away from the planet where the magnitude of the gravitational force on the apple is 0.6 FR

Then GMm/r^2 = 0.6GMm/R^2

r = R/sqrt(0.6) = 1.29R

b) Into the tunnel

Let r be the point inside the tunnel where the magnitude of the gravitational force on the apple is 0.6 FR

Density of the planet = M/[4/3*Pi*R^3] = Constant

Mass of the planet with radius r

Mr = M*[(4/3*Pir^3)/(4/3*PiR^3)]

Mr = M*[r^3/R^3]

GMr*m/r^2 = 0.6GMm/R^2

M*[r^3/R^3]/r^2 = 0.6M/R^2

r/R^3 = 0.6/R^2

=> r = 0.6R

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