Weight of a body at the centre of earth is

Inside a uniformly dense sphere, only the mass enclosed within a sphere of radius r (measured from the centre) pulls on a point at that radius — by the shell theorem, the material outside that inner sphere forms concentric shells whose net gravitational force on an interior point is zero. This makes the acceleration due to gravity g′ at a depth vary linearly with the distance r from the centre: g′ = g × (r/R), where R is Earth's radius and g is the surface value.

1. At the centre of the Earth, the distance from the centre is r = 0.

2. Substituting r = 0 into g′ = g × (r/R) gives g′ = 0.

3. Weight is W = m × g′; with g′ = 0, the weight of any body at the centre is exactly zero, regardless of its mass.

This is different from the region above the surface, where g decreases as the inverse square of distance (g′ = g×R2/r2) and never actually reaches zero. The small weight difference between the poles and the equator is a separate, surface-only effect caused by Earth's rotation and its equatorial bulge — it does not apply at the centre, where the pull from every direction cancels out completely by symmetry.