Massdistribution

Mass distribution refers to how mass is arranged within a physical object or system. It can be described by a density function in continuous media, or by a set of discrete masses at specific locations. The concept is used across physics, engineering, and astronomy to understand how mass influences forces, motion, and stability.

In continuous form, the total mass is M = ∫_V ρ(r) dV, where ρ(r) is the mass density at position r and V is the object’s volume. The center of mass is r_cm = (1/M) ∫_V r ρ(r) dV. For discrete masses, M = Σ_i m_i and r_cm = (1/M) Σ_i m_i r_i. The distribution also determines rotational properties through the moment of inertia. About an axis, I = ∫ r_perp^2 ρ(r) dV, with a more general inertia tensor I_ij = ∫ (r^2 δ_ij − x_i x_j) ρ(r) dV.

Symmetry and density profiles help classify mass distributions. Uniform or radial distributions arise in simple bodies, while more complex systems use profiles like exponential disks, spherically symmetric ρ(r), or halo models in astrophysics. Mass distribution directly affects gravitational fields, gravitational interactions, and rotational dynamics.

Applications span engineering and science. In engineering, the distribution governs balance, loads, and natural frequencies. In astrophysics, estimating the mass distribution of galaxies or clusters from rotation curves and lensing reveals dark matter and formation histories. In practice, mass distribution is inferred from measurements and modeling, since mass is not always directly observable.

Key concepts include center of mass, density functions, total mass, moments of inertia, and how nonuniform distributions influence dynamics.