Uniformdensity

Uniform density refers to a material or region in which the density is the same at every point. In classical mechanics, mass density is denoted rho and is defined as mass per unit volume. If rho is constant across a region V, the region is said to have a uniform density. The total mass M is M = rho times V.

Examples include a solid sphere of radius R with uniform density, for which M = (4/3) pi R^3 rho; a right circular cylinder with radius R and height h, for which M = pi R^2 h rho; and a uniform rod of length L and cross-sectional area A, with M = A rho L.

Consequences of uniform density include that the center of mass coincides with the geometric center for symmetric shapes. Moments of inertia have closed forms for standard shapes, such as a solid sphere I = 2/5 M R^2 about any diameter, and a solid cylinder about its central axis I = 1/2 M R^2. More generally, uniform density simplifies integrals needed to compute mass properties.

In practice, uniform density is an idealization used to simplify gravitational and mechanical calculations; real materials exhibit density variations. Uniform-density models are common in applications such as simplifying the interior structure of celestial bodies or geophysical approximations.

In probability, a related notion is a uniform density function, meaning a constant probability density over a region D. The density f(x) is constant, f(x) = 1/Vol(D) for x in D and zero outside, describing a uniform distribution.