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[2] 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 ...
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
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.
Screw theory is the algebraic calculation of pairs of vectors, also known as dual vectors [1] – such as angular and linear velocity, or forces and moments – that arise in the kinematics and dynamics of rigid bodies.
The moment of inertia, denoted by I, measures the extent to which an object resists rotational acceleration about a particular axis; it is the rotational analogue to mass (which determines an object's resistance to linear acceleration). The moments of inertia of a mass have units of dimension ML 2 ([mass] × [length] 2).
=: 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. The equilibrium equations are: [2]
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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 ().