Massajakauma

Massajakauma refers to how mass is distributed within a physical object or system. It describes the spatial arrangement of mass and determines key mechanical and physical properties such as the center of mass and the moment of inertia.

In continuous bodies, mass density ρ(x) (or ρ(r)) defines the differential mass dm = ρ dV, and the total mass is M = ∫ ρ dV over the body. The center of mass is r_cm = (1/M) ∫ r ρ dV, or, for discrete masses, r_cm = (1/M) Σ m_i r_i. The moment of inertia about an axis is I = ∫ r^2 dm; for a planar lamina with surface density σ(x,y), M = ∫∫ σ dA and I about a perpendicular axis is I = ∫∫ (x^2 + y^2) σ dA.

Intuitively, uniform density yields a center of mass at the geometric center of symmetric objects, while nonuniform density shifts the center toward heavier regions. Mass distribution strongly influences mechanical behavior, including balance, vibrations, natural frequencies, and stability. In engineering, accurate knowledge of mass distribution is essential for design, balancing rotating parts, and predicting dynamic responses. In physics and astrophysics, it affects gravitational fields, orbital dynamics, and the mass profiles of astronomical objects.

Examples illustrate the concept: a uniform rod has its center of mass at its midpoint; a solid sphere's center of mass lies at its geometric center; a hollow cylinder’s mass distribution determines its polar moment of inertia.