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The moments of inertia of a mass have units of dimension ML 2 ([mass] × [length] 2). It should not be confused with the second moment of area, which has units of dimension L 4 ([length] 4) and is used in beam calculations. The mass moment of inertia is often also known as the rotational inertia, and sometimes as the angular mass.
The moment of inertia of a flat surface is similar with the mass density being replaced by its areal mass density with the integral evaluated over its area. Note on second moment of area: The moment of inertia of a body moving in a plane and the second moment of area of a beam's cross-section are often confused.
The second moment of area, also known as area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with respect to an arbitrary axis. The unit of dimension of the second moment of area is length to fourth power, L 4, and should not be confused with the mass moment of inertia.
This is useful in calculating moments of inertia or center of mass for a constant density, because the mass of a lamina is proportional to its area. In a case of a variable density, given by some (non-negative) surface density function ρ ( x , y ) , {\displaystyle \rho (x,y),} the mass m {\displaystyle m} of the planar lamina D is a planar ...
An arbitrary shape. ρ is the distance to the element dA, with projections x and y on the x and y axes.. The second moment of area for an arbitrary shape R with respect to an arbitrary axis ′ (′ axis is not drawn in the adjacent image; is an axis coplanar with x and y axes and is perpendicular to the line segment) is defined as ′ = where
English: A diagram showing the elemental area used in calculating the polar moment of inertia of a flat object. ... polar moment of inertia of a flat object.}} ...
Also, the Figure shows I(y)= m * r^2 / 12, which is similar to the formula for "Rod of Length L and mass m" in the Figure immediately below in which the axis of rotation is at the center of mass of the rod (this latter case is derived in Section "Example Calculation of moment of inertia" of article Moment of inertia.
The second polar moment of area, also known (incorrectly, colloquially) as "polar moment of inertia" or even "moment of inertia", is a quantity used to describe resistance to torsional deformation (), in objects (or segments of an object) with an invariant cross-section and no significant warping or out-of-plane deformation. [1]