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

Given,

Radius = R ; F = 0.5 FR

(a)Let this distance be D.

We know that, gravitaional force varies inversely(follows inverse square law) with the square of distance between the objects.

So in order to halve the force we neet to double the square of distance in the denominator.

D = sqrt(2)R = 1.414 R

hence, D = 1.414 R

(b)In this case, the force will be given by:

F = G (4/3 pi r3 x density) Mapple / r2 = 4Gpi r x density x Mapple / 3

So to decrease the force in this case to 0.5F, the apple needs to be moved to D = 1/2R

Hence, D = 1/2 R = 0.5R.

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