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This is also called the polar moment of the area, and is the sum of the second moments about the - and -axes. [24] The stresses in a beam are calculated using the second moment of the cross-sectional area around either the x {\displaystyle x} -axis or y {\displaystyle y} -axis depending on the load.
The left hand side of the equation—which includes the sum of external forces, and the sum of external moments about P—describes a spatial wrench, see screw theory. The inertial terms are contained in the spatial inertia matrix
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.
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
The moment of force, or torque, is a first moment: =, or, more generally, .; Similarly, angular momentum is the 1st moment of momentum: =.Momentum itself is not a moment.; The electric dipole moment is also a 1st moment: = for two opposite point charges or () for a distributed charge with charge density ().
The parallel axis theorem, also known as Huygens–Steiner theorem, or just as Steiner's theorem, [1] named after Christiaan Huygens and Jakob Steiner, can be used to determine the moment of inertia or the second moment of area of a rigid body about any axis, given the body's moment of inertia about a parallel axis through the object's center of gravity and the perpendicular distance between ...
=: the sum of the moments (about an arbitrary point) of all forces equals zero. Free body diagram of a statically indeterminate beam In the beam construction on the right, the four unknown reactions are V A , V B , V C , and H A .
Felix Klein saw screw theory as an application of elliptic geometry and his Erlangen Program. [11] He also worked out elliptic geometry, and a fresh view of Euclidean geometry, with the Cayley–Klein metric. The use of a symmetric matrix for a von Staudt conic and metric, applied to screws, has been described by Harvey Lipkin. [12]