moments and center of mass

$M_y$ and $M_x$ weight mass by its distance from an axis; $(\bar x,\bar y)=(M_y/m,\ M_x/m)$ is the plate's balance point.

The moment $M_y$ (about the $y$ -axis) weights mass by its $x$ -distance from that axis, and $M_x$ (about the $x$ -axis) weights mass by its $y$ -distance — a point far from an axis contributes disproportionately, the same way a mass far from a seesaw's pivot exerts disproportionate torque. The center of mass $(\bar x,\bar y)=\left(\dfrac{M_y}{m},\dfrac{M_x}{m}\right)$ is the single point where the whole plate balances. When $\rho$ is constant, it factors out entirely and $(\bar x,\bar y)$ depends only on the region's shape — the centroid.

Examples

The triangle bounded by $x=0$ , $y=0$ , $x+y=1$ with constant density has $m=\tfrac12$ , $M_y=M_x=\tfrac16$ , giving centroid $(\bar x,\bar y)=\left(\tfrac13,\tfrac13\right)$ .