Apparent Mass

(or associated mass), a quantity having the dimensions of the mass that is added to the mass of a body moving nonuniformly in a fluid medium in order to take into account the action of the medium on the body. For example, if a body with mass m moves with translational motion in an ideal fluid under the influence of the force F, the resistance of the medium is proportional to the acceleration w of the body. In accordance with the fundamental law of dynamics, mw = F – λw or (m + λ)w = F, where the coefficient of proportionality λ is called the apparent mass. Thus, the body moves in the fluid in the same way as it would move in a vacuum if it had the mass m + λ.

The value of the apparent mass depends on the shape of the body, the direction of motion, and the density ρ of the medium. Thus, the apparent mass of a sphere is λ = (2/3)ρπr³, where r is the radius of the sphere. For an elliptic cylinder—that is, a cylinder whose base is an ellipse—moving in a direction perpendicular to one of the axes of the ellipse, λ = ρπa²h, where a is half the length of the axis and h is the altitude of the cylinder.

The determination of apparent mass is of considerable importance in the study of, for example, unsteady motion of bodies fully immersed in water, impact on water, entry of bodies into water, and tossing of ships. In calculations of apparent mass, fluids are assumed to be nonviscous and, usually, of negligible compressibility.