Center of Mass

The center of mass is the point at which an object's mass distribution balances in every direction — equivalently, the single point an object's whole mass can be treated as concentrated at when analyzing its translational motion, no matter how oddly shaped or non-uniform the object actually is. For a symmetric object of uniform density, it sits at the obvious geometric center; for an irregular or unevenly weighted one, it can sit somewhere that isn't even inside the object's own material, as with a boomerang or a doughnut, where the center of mass lies in the empty hole at the middle.

Center of gravity is a closely related but distinct concept: the point where gravity's net effect can be considered to act. In a uniform gravitational field — true for essentially every terrestrial engineering purpose — the two coincide exactly, which is why the terms get used interchangeably in practice; they diverge only for objects large enough that gravity's strength meaningfully differs across their extent.

Stability against tipping is a center-of-mass question: an object tips when its center of mass moves horizontally past its base of support, which is why a wide stance and a low center of mass both improve stability, and why a Weeble toy's weighted rounded bottom makes it self-right — any tip raises its center of mass, so gravity always pulls it back down to the low-energy upright position.

A tumbling body, no matter how it's thrown, always rotates about its own center of mass — a wrench spinning through the air rotates around that one point even while the wrench as a whole follows a parabolic arc, the two motions (translation of the center of mass, rotation about it) being fully independent — gravity bends the center of mass's path but exerts no torque about it, so the spin carries on unchanged.