Is gravity the same everywhere on Earth?

Answer

The answer is no: at the equator you ‘weigh less’ than at the poles.

This is about the ‘effective gravitational acceleration’. That is a superposition of gravity holding you to the Earth and the acceleration away from the Earth due to the Earth’s rotation. That acceleration is one away from the axis of rotation, not from the center. At the equator that is a vertical acceleration, so upwards; at the pole the vertical component is zero.

Because of the changing (depending on the latitude) influence of the rotation, the Earth is not a sphere, but is somewhat flattened. If the Earth were all sea, its shape would be well approximated by an ellipsoid, whose vertical axis is 14 miles shorter than an axis at the equatorial plane. Mathematically, that shape is given as the surface where the total potential (the sum of gravitational potential plus potential associated with the rotational acceleration) is constant. But that does not mean that the acceleration of gravity on that surface is the same everywhere. At the pole you have less mass under your feet between you and the center of the Earth, but because you are closer to the center, the mass also pulls a little harder.

Historically, the flattening of the Earth has been discovered precisely from the observation that the acceleration of gravity is not the same everywhere. After all, it was established that pendulum clocks made in Europe did not run correctly in Central America. The period during which a pendulum moves depends on the acceleration of gravity, which was the cause of the anomaly.