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So I've followed and understood the proof that inside a thin hollow shell the gravitational potential is constant and hence the force is zero, but when applying it to a problem (namely "how does the force depend on the distance for an object dropped through the Earth in a narrow tunnel") I'm slightly confused:

The solution stated that only the sphere of mass "below" the falling object exerts a gravitational force. Is this because we have broken up the whole planet into 2 parts:

Above the momentary position (that is, all the mass that is farther away from the center of the earth then the object)

Below the momentary position ( The sphere with radius = position from the center)

so the "above" can be thought of as a bunch of thin hollow shells and hence it doesn't exert force, and the below part as regular ball of mass ?

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Yes what you have understood is correct Earth is a solid sphere. You can break a solid sphere into infinite hollow spheres So if you view all the hollow spheres above the momentary position individually, we can conclude force is 0 due to them (as object is inside the hollow sphere) So we just take the mass of the sphere below the position

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