At the centre of the earth, the force driven by the upper half of the earth is cancelled out due to the force of the lower half of the earth. In this manner, the force caused by any portion of the earth will be nullified at its centre by the force due to the opposite side. Thus, the gravitational force at the centre will be 0. Now, by applying Newton’s Law F= mg. Given that mass of any object cannot be zero, then if F=0, then g has to be 0. So, the value of g is 0 at the centre of the earth.
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