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When we find the gravitational field on an object that lies inside Earth we simplify the problem by considering the sphere whose radius is the distance of the object from the center of Earth. We use mass of this sphere to find gravitational pull. How do we come to know this?

The way I visualise this is, divide the sphere by a plane* passing through the point object dividing the sphere into two portions. The portion above the object will try to pull it upwards while the portion below it will try to pull downwards. What is wrong with this approach?

*(plane passes such that volume of one portion is minimum whereas volume of the other portion is maximum, unless object is at center then both have equal volumes)

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That an object inside earth (assumed spherical) experiences only the gravitational pull of the sphere corresponding to the distance of the object from the earth's center is due to Newton's Shell Theorem, which consists of two parts: 1. The gravitational force exerted by a spherically symmetric body correspond to the force exerted by a mass point of equal mass located at the center. 2. The gravitational force of a spherical shell on an object inside the shell is zero. From this follows that an object inside earth doesn't experience a net force of the spherical earth mass shell above it. It only feels the force of the spherical mass below it. This theorem has already been proven mathematically by Newton. The derivation can also found at the linked Wikipedia article.

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