The riddle part is that polar bears live at the North Pole, not the South Pole. Furthermore, a polar bear at the equator wouldn’t be a polar bear, would he? He’d be an equatorial bear.
But let’s take the question at face value; namely, will a bear, or anything else, for that matter, weigh less at the equator than it does at either pole? All other things being equal (and, of course, they never are), the answer would be yes. Slightly.
First of all, because Earth bulges out somewhat around the equator, the bear will be a bit farther from the center of Earth and gravity’s pull will therefore be a bit weaker.
But what your daughter undoubtedly had in mind was the effect of our planet’s rotation, which is one complete turn every twenty-four hours. (Isn’t that a neat coincidence? Of course not. That’s how we humans defined twenty-four hours in the first place.) At the equator, which is 24,900 miles (40,070 kilometers) around, that works out to a ground speed, palm trees, bears and all, of 1,040 miles per hour (1,670 kilometers per hour). Back home at the exact North Pole, however, the bear wasn’t traveling around at all; he was just rotating in place, at the center of the merry-go-round.
Because of the Earth’s rapid rotational speed, bears (and everything else) are subjected to an outward centrifugal force tending to fling them off the planet, just as a dog flings water off his back by rotating himself rapidly after a bath. But the reason that the space around Earth isn’t filled with flying bears is that the planet’s much stronger gravitational force holds them firmly to the ground.
Nevertheless, the outward-flinging centrifugal force detracts slightly from the Earth-holding gravitational force, so that an equatorial bear’s weight is slightly diminished, by a little more than three-tenths of a percent. An 800-pound (360-kilogram) bear would weigh about 3 pounds (1.4 kilograms) less at the equator than he does at the North Pole. In human terms, a 150-pound (68-kilogram) person would weigh half a pound (200 grams) less at the equator than at the North Pole.
Of course, these are the extremes. Anywhere in between the equator and the poles, the rotational speed of the planet is somewhere between zero and the equatorial speed, because the distance around is shorter.
So there is a gradual loss of weight as one moves toward the equator from anywhere in the northern or southern hemisphere. If you weigh 150 pounds (68 kilograms) at the latitude of Washington, D.C., and Madrid, Spain, for example, you’d weigh about 5 ounces (14 grams) less at the equator.
That’s not a very effective weight-loss strategy, however, unless you get there by walking.