Center of Mass

Calculus 2 · Axiom Academy

The single point where an object balances — found by dividing total moment by total mass. The center of mass is the one point where an object's weight is evenly distributed. Support a region there and it rests level; support it anywhere else and it tips. Watch the same lamina balanced on a fulcrum that slides to the correct spot . To pin down that point we use moments . A moment measures turning effect: a mass's moment about a pivot is its mass times its distance from the pivot. At the center of mass the moments on the two sides cancel exactly — that balance is the whole idea. Watch two masses settle level about a pivot: a heavy mass close in balances a lighter mass farther out, because their moments match. For point masses at positions , the balance point is the total moment divided by the total mass: 3. From Sums to Integrals: Laminas A lamina is a thin flat plate. Its mass is spread out continuously, so the sum over discrete masses becomes an integral . Slice the region under y = f(x) into thin vertical strips of width dx : each strip has area and sits at position x , so it contributes a moment . The sweeping strip below builds the balance line as it goes. The numerator adds up every strip's moment; the denominator is the total area (the mass, since is constant). The density appears in both and cancels .

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