Do I weigh less on the equator than at the North Pole?
Category: Earth Science Published: January 7, 2014
Yes, you weigh less on the equator than at the North or South Pole, but the difference is small. Note that your body itself does not change. Rather it is the force of gravity and other forces that change as you approach the poles. These forces change right back when you return to your original latitude. In short, a trip to the equator is not a viable long-term weight-loss program.
Your weight is the combination of all the large-scale, long-term forces on your body. While the earth's gravity is by far the strongest large-scale force, it is not the only one. What you experience as "something pulling you down" is actually the total of all the forces and not just gravity. The four dominant large-scale, long-term forces are:
- The earth's gravity
- The sun's gravity
- The moon's gravity
- The earth's centrifugal force
Note that although earth's Coriolis force plays a major role in shaping hurricanes and ocean currents, since it is not a static force, it does not contribute to your overall weight. Also, additional forces appear when you ride a roller-coaster, an elevator, a swing, or another vehicle, but these forces are transient, so they do not contribute to your overall, long-term weight. Finally, electromagnetic and nuclear forces are either too small-scale, or too short-term to contribute to your overall weight.
For our purposes, we want to consider the forces that differ significantly at the equator versus the poles. While the sun's gravity is strong enough to keep us and the earth in orbit, the sun's position relative to a spot on the equator versus the poles is constantly changing. As a result, averaged over a few days, the gravitational force of the sun on a spot on the equator is the same as the gravitational force of the sun on a spot on the poles. The same situation applies to the moon. This leaves only earth's gravity and earth's centrifugal force as the two forces that contribute to your weight differing at the equator versus the poles.
As we learned in high school, earth's gravity is approximately constant all over the surface of the earth. But this is only an approximation. If the earth were perfectly spherical and if its density were perfectly uniform, then the strength of earth's gravity would be exactly constant at all points on its surface. But it's not. There are three major complications to earth's gravitational field. First the earth is not a sphere. The earth is spinning, causing it to slightly flatten like a pizza crust thrown and spun in the air. As a result, the earth is an oblate spheroid and not a perfect sphere. If you stand at sea level on the equator, you are 6378 km from the center of the earth. In contrast, at each pole, you are only 6357 km from the center of the earth. Since the strength of gravity weakens as you get farther away from a gravitational body, the points on the equator are farther and have weaker gravity than the poles. The other two complications to earth's gravitational field; non-uniform internal density and local surface mass variations such as mountains; are small enough factors that we will neglect them here. Therefore, assuming the entire mass of the earth is located at its center, we can calculate the force of earth's gravity at the equator and at the poles. Using Newton's law of gravity, we find that the force of earth's gravity on your body at the equator is 9.798 m/s2 times the mass of your body, whereas at the poles it is 9.863 m/s2 times the mass of your body.
The earth's centrifugal force also varies with latitude. The centrifugal force is the outward force felt whenever you are in a rotating reference frame. While the centrifugal force is a non-fundamental force caused ultimately by the inertia of bodies, it is very real for the body in a rotating reference frame, such as your body on the surface of the rotating earth. The centrifugal force is proportional to the tangential speed of the rotating reference frame. The equator is moving quickly as the earth's spins, so it has a lot of centrifugal force. In contrast, the poles are not spinning at all, so they have zero centrifugal force. Since centrifugal force points outwards from the center of rotation, it tends to cancel out a little bit of earth's gravity. If the earth were not spinning, you would be heavier as you would feel the full force of gravity. Since there is more centrifugal force at the equator to cancel gravity, your overall weight at the equator versus at the poles is even less. The centrifugal force on your body at the equator is 0.034 m/s2 times the mass of your body. The centrifugal force at the poles is zero.
Your total weight at sea level at the equator (gravity minus centrifugal force) is therefore 9.764 m/s2 times your mass, whereas your weight is 9.863 m/s2 times your mass at the poles. If we use a more accurate model (such as taking into account the shape of the continents) these numbers will be slightly different, but the overall point will be the same: you weigh about 1% less at the equator than at the poles. If you weigh 200 pounds at the North Pole, you will weigh 198 pounds at the equator. Note that we have focused on the equator and the poles as the extremes, but the same effect applies to all latitudes. You weigh slightly less in Mexico City than in New York City, as Mexico City is closer to the equator.