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 planets 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? (a) Number: L 1 Units: (b) Number: 2 Units: Answer * 1: the tolerance is +/-5% Answer *2: exact number, no tolerance

Explanation / Answer

(a) force at surface = FR = GMm/R2
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force at distance = (0.6)FR = GMm/d2
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We have to find d. So...
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(0.6)GMM/R2 = GMM / d2
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d2 = R2 /0.6 d = 1.2884 R this is the distance from the center of the planet, so...
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distance from surface = d -R = 0.6456 R
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(b) When inside the tunnel, the force decreaseslinearly with distance from the surface. So the force will be (0.6) times the force at the surface when the distanceis (0.6) R.
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This means the distance from the surface must be R - (0.6) R = 0.4R

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