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 0.5 FR if we move the apple (a) away from the planet and (b) into the tunnel?

Explanation / Answer

Let r be the position of the apple measured from the center of the sphere. Since r>R, the sphere exerts force on the apple as if all of its mass were concentrated at the center. We have

Fr = (1/2)FR

GMm/r2 = (1/2)GMm/R2

r = (2)R

the distance between the apple and the surface is

d = r – R

= ((2)-1)R

= 0.414R

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The mass of the part of the sphere which is inside of the apple is

Min = (4/3)r3 × = (4/3) r3 × M/((4/3)R3)) = M(r/R)3

Fr = (1/2)FR

GM(r/R)3m/r2 = (1/2)GMm/R2

r = (1/2)R

Measuring the position from the surface, we obtain

d= R – r

= [1 – (1/2)]R

= 0.500R

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