Mass and Center of Mass

Calculus 3 · Axiom Academy

LESSON Mass and Center of Mass Using a double integral to find the total mass of a variable-density plate and the exact point where it balances. 1. What a Density Function Encodes A density function gives the mass per unit area at each point of a region R — how tightly mass is packed there. Our running plate is on : nearly weightless near the origin, densest at the corner (2,3) . the plate we follow throughout A larger value means more mass packed into that spot. A constant is a uniform plate; a varying is non-uniform. To weigh the plate, integrate the density over the whole region. Each area element dA carries mass ; the double integral sums them all. A moment measures how strongly the mass is thrown to one side of an axis — its tendency to spin the plate about that axis, like torque. Each element is weighted by its distance from the axis before being summed. The center of mass is the balance point — rest the plate on a pin there and it stays level. Divide each moment by the total mass, and mind the cross: comes from M_y , while comes from M_x . The geometric center of R is , but the plate is denser toward (2,3) , so the balance point is pulled up and to the right: 5. Uniform Density: the Centroid When the density is constant , the factor k cancels top and bottom, and the balance point depends only on the shape. That purely geometric point is the centroid . Flatten to a constant and the balance point slides back to the geometric center:

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