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Would you be weightless at the center of the Earth?
Supposing there is a cavity at the center of the Earth, what is the gravity there? What will be its direction and intensity? Will a body be attracted toward the center of mass of the Earth?
The gravitational field a distance $r$ from the center of a solid sphere with uniformly distributed mass is $$ g\left(r\right) = -\left(\frac{4}{3} \pi G \rho \right) r, $$ where $\rho$ is the mass density of the sphere, and $G$ is the gravitational constant. The value of the field at the center is $$ g\left(0\right) = 0. $$ The actual field at the center of the Earth will not be zero, however, because the Earth is not a perfect sphere, does not have uniform density, and is not the only object in the universe.
