centerofmass

The center of mass is a point that represents the average location of an object's mass. For a system of particles or a rigid body, it is the point about which external forces cause pure translation of the body as if all mass were concentrated there. In a uniform gravitational field the center of mass moves as if the total weight acts at that single point.

Mathematical definitions can be stated for discrete and continuous mass distributions. For N particles with masses m_i and position vectors r_i, the center of mass is R_cm = (1/M) Σ m_i r_i, where M = Σ m_i is the total mass. For a continuous distribution with density ρ(r) over volume V, R_cm = (1/M) ∭ r ρ(r) dV, with M = ∭ ρ(r) dV. The velocity of the center of mass is V_cm = dR_cm/dt = (1/M) Σ m_i v_i, or (1/M) ∭ v ρ dV. The acceleration satisfies a_cm = dV_cm/dt = F_ext/M, where F_ext is the total external force on the system.

Properties and implications: In a uniform gravity field, the total gravitational force is F_g = M g and acts at the center of mass, simplifying dynamics. For a rigid body, motion can be decomposed into translation of the CM plus rotation about the CM. In nonuniform gravitational fields, the center of gravity may differ from the center of mass, since the weight distribution does not coincide with the mass distribution.

Determination and applications: The center of mass can be found by symmetry for simple shapes, by geometry for uniform density, or by integration for complex distributions. It is a fundamental concept in physics and engineering, informing balance and stability in mechanical design, as well as analyses of orbital dynamics, robotics, and computer graphics.