I teach 8th graders that the closer you are to the center of gravity the more you weigh. Therefore, you would weigh more at the poles then at the equator, since the earth is not round, but flattened at the poles. Students seem to reason that if you go into a cave you should weigh more because you are getting closer to the center of the earth. On a TV special about errors in textbooks, a scientist said that this is not so. Please clear up the confusion.
Good question. We need to remember that gravity is the attraction of mass for other mass, so that when we're in a cave the mass of the rocks ABOVE us is pulling UP on us. So that even if we're closer to the center of mass of the earth, gravity will not go up much. If you are at the center of the earth, gravity is zero because all the mass around you is pulling "up" (every direction there is up!).
A few of other points concerning gravity:
If you are outside a spherical hollow "Death Star" its gravitational
attraction is the same as if all its mass were concentrated at its center,
but as soon as you go through the surface into the hollow interior,
gravity is ZERO at all places inside the sphere.
If you are near a flat plate that stretches to infinity in all directions, its gravitational attraction is a constant and doesn't change with distance from the plate.
There are three things that cause gravity to be different between the poles of the earth and the equator. You got one of them - the fact that gravity increases as you get closer to the center of mass, because the earth is flattened at the poles. Another is the spin of the earth. The farther you are from the spin axis, the lower "gravity" is, because another force (centrifugal, or centripetal) caused by the spin pulls you in the opposite direction. If the earth were about 36,000 km in diameter with the same mass and length-of-day then the gravity at the equator would be zero. This is the altitude of geostationary orbits. The last is the fact that there is more material between you and the center of the earth when you are near the equator because of the flattening. Since there is more mass, gravity increases some. The net effect of these three factors is an increase in gravity near the poles. You'd never notice it, but its very important for space flight, since its much cheaper to launch near the equator than near the poles for most orbits.
Dr. Fred Duennebier, Professor
Department of Geology and Geophysics
University of Hawaii, Honolulu, HI 96822