This article is from the In-Depth Report The Science of Pro Football

When Is the Center of Mass Not at the Center of Any Individual Object?

Your body's center of mass is near your belly button, but things change if you're interacting with another person

In the Torque and Center of Mass episode of NBC Learn's "The Science of NFL Football" you see that a linemen lowers his center of mass in order to overpower an opponent.

A human being's center of mass is somewhere around the navel, as explained in the video. If people could stay completely rigid, you could balance them on a single point by placing their center of mass directly above that point.

The situation is not as intuitive when considering a system that involves multiple masses. Take, for instance, Earth and its moon. The two objects are gravitationally bound, and because Earth is so much more massive than its lunar companion, the moon is said to orbit the planet. But in truth, the moon exerts a gravitational pull on Earth as well, nudging our planet around ever so slightly. The two bodies are engaged in a continual orbital dance, each pulling on the other.

So if the mutual attraction between Earth and the moon keeps them both in motion, what are they two objects orbiting around? You guessed it: their center of mass. Earth and its moon are like two children on an invisible seesaw, one much heavier than the other. Their combined center of mass is the point at which a wedge under the seesaw would keep the two children in balance.

With only two objects in a system, it's pretty straightforward to find the system's center of mass. For Earth and the moon (with masses Me and Mm, respectively), say that De is the distance between Earth and the center of mass and Dm is the distance between the moon and the center of mass. The ratio of those distances is the inverse of the ratio of the objects' masses—that is, De/Dm = Mm/Me.

The moon's mass is about 7.35 X 10^22 kilograms, and the mass of Earth is 5.97 X 10^24 kilograms. The ratio Mm/Me, then, is about 0.0123. (In other words, the moon weighs roughly 1.23 percent as much as Earth.) Therefore, the distance from Earth to the system's center of mass is only about 0.0123 times the distance from the moon to the system's center of mass. With a typical distance of 384,400 kilometers separating Earth and the moon's centers, that works out to a center of mass that is roughly 379,700 kilometers from the moon's center and about 4,700 kilometers from Earth's. (Note that 4,700/379,700 = 7.35 X 10^22/5.97 X 10^24, in agreement with the formula De/Dm = Mm/Me.)

Earth's radius is about 6,400 kilometers, so the center of mass for the Earth–moon system lies inside the planet, about 1,700 kilometers below the surface. Both Earth and its lunar companion orbit this subterranean point as they follow their joined path around the sun. For the moon that means making broad laps around our planet; for Earth that means a wobble as it makes its journey around the sun. But for some orbital systems the center of mass is not inside either of the orbiting bodies, so both objects loop around a point in open space.

Pluto and its moon Charon, for instance, are much closer in mass than Earth and its moon. The ratio between Charon's mass and Pluto's mass is about 0.117, and the two bodies are separated by 19,600 kilometers. With those parameters, the system's center of mass is about 17,500 kilometers from Charon and 2,100 kilometers from Pluto. Because Pluto's radius is only 1,150 kilometers, the center of mass of the Pluto–Charon system lies well outside the physical expanse of either body.

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