Is the acceleration due to gravity at the centre of the earth zero??
I understand you reasoning, but it is flawed. The law of universal gravitation treats the masses being studied as points (point masses). When you are ‘inside’ one of the masses, you have to treat it a bit differently. Without going into all of the mathematical explanation (which I can if you want me to), in this sort of situation, one must only consider the mass which is ‘beneath’ you. If you are below the surface of the Earth, not all of the Earth’s mass is beneath you anymore; some is ‘above’ you. If you actually work it out, all of the mass which is now above you cancels itself out due to the symmetry of the sphere, but the mass below you acts normally as a point mass still. An analogy I like to compare this to is an onion. As you travel below the surface of the Earth, the mass which is now located above you is like a layer of the onion which can be ‘pealed’ away, so to speak, without affecting the gravitational pull below you. If you are at the center of the Earth, there is no mass
