centreofmass

The center of mass (centre of mass) of a system of particles is the point where the total mass can be considered to be concentrated for the purpose of analyzing translational motion under external forces. For a discrete system of N point masses m_i located at position vectors r_i, the centre of mass is r_cm = (1/M) ∑ m_i r_i, where M = ∑ m_i. For a continuous body with mass density ρ(r), r_cm = (1/M) ∫ r ρ(r) dV. In a planar lamina, r_cm = (1/M) ∫ r dm.

Under a uniform external force field, such as gravity near Earth’s surface, the centre of mass moves as if all the mass were concentrated at that point; specifically, the acceleration of the centre of mass satisfies M a_cm = F_ext. More generally, r_cm depends on the mass distribution and is the time-dependent average position of the mass.

Relation to center of gravity: in a uniform gravitational field, the centre of mass coincides with the centre of gravity; in nonuniform fields they can differ. The centre of mass is a geometric property of the mass distribution, while the centre of gravity depends on the external field.

Determination and properties: for rigid bodies the centre of mass is fixed relative to the body as it moves without internal deformation. It can be found by balancing torques about various axes or by computing the mass-weighted average position as described above.

Applications: the concept is used in robotics, biomechanics, vehicle safety, sports science, and astronomy. In celestial mechanics, the analogous quantity is the barycenter, the mass-weighted average position of bodies in a system.

Limitations: the centre of mass is a theoretical construct; it may lie outside the physical body for hollow or extended shapes and does not alone determine rotational dynamics, which also depend on the inertia tensor.