The moment of a distribution of mass is the integral that measures the tendency of the mass to rotate about a point or axis. It quantifies how mass is distributed in relation to a particular point.
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The moment about the x-axis is calculated using $M_x = \int_a^b y\rho(x) \, dx$ where $\rho(x)$ is the linear density function.
The moment about the y-axis is calculated using $M_y = \int_a^b x\rho(x) \, dx$ where $\rho(x)$ is the linear density function.
For systems with multiple masses, moments are additive: $M_{total} = M_1 + M_2 + ... + M_n$.
Moments are used to find centers of mass, which are given by $(\overline{x}, \overline{y}) = \left(\frac{M_y}{M}, \frac{M_x}{M}\right)$ where $M$ is the total mass.
Moments of inertia, related but distinct concepts, involve integrals of squared distances weighted by mass.
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Related terms
Center of Mass: A point representing the average position of a system's mass; found using moments.
Moment of Inertia: $I = \int r^2 dm$, a measure of an object's resistance to rotation about an axis.
Density Function: $\rho(x)$, represents how mass is distributed along a line or region.