Moment of Inertia Around An Arbitrary Axis
The moment of inertia of a body around an arbitrary axis in space is a scalar that is computed as the sum of the distance squared from the axis to each of the mass elements. This scalar can be computed from the moment inertia matrix of the body using the unit vector along the axis.
Let a rigid assembly of rigid system of N particles, Pi, i=1,...,N, have coordinates ri. Choose R as a reference point and compute the moment of inertia around an axis L defined by the unit vector S through the reference point R. The moment of inertia of the system around this line L=R+tS is computed by determining the perpendicular vector from this axis to the particle Pi given by
where is the identity matrix and is the outer product matrix formed from the unit vector S along the line L.
Introduce the skew-symmetric matrix such that y=S x y, then we have the identity
which relies on the fact that S is a unit vector.
The magnitude squared of the perpendicular vector is
The simplification of this equation uses the identity
where the dot and the cross products have been interchanged. Expand the cross products to compute
where is the skew symmetric matrix obtained from the vector ri-R.
Thus, the moment of inertia around the line L through R in the direction S is given by the scalar
or
where is the moment of inertia matrix of the system relative to the reference point R.
Read more about this topic: Moment Of Inertia
Famous quotes containing the words moment of, moment, inertia, arbitrary and/or axis:
“To the United States the Third World often takes the form of a black woman who has been made pregnant in a moment of passion and who shows up one day in the reception room on the forty-ninth floor threatening to make a scene. The lawyers pay the woman off; sometimes uniformed guards accompany her to the elevators.”
—Lewis H. Lapham (b. 1935)
“Fashion is the most intense expression of the phenomenon of neomania, which has grown ever since the birth of capitalism. Neomania assumes that purchasing the new is the same as acquiring value.... If the purchase of a new garment coincides with the wearing out of an old one, then obviously there is no fashion. If a garment is worn beyond the moment of its natural replacement, there is pauperization. Fashion flourishes on surplus, when someone buys more than he or she needs.”
—Stephen Bayley (b. 1951)
“What is wrong with priests and popes is that instead of being apostles and saints, they are nothing but empirics who say “I know” instead of “I am learning,” and pray for credulity and inertia as wise men pray for scepticism and activity.”
—George Bernard Shaw (1856–1950)
“... no one who has not been an integral part of a slaveholding community, can have any idea of its abominations.... even were slavery no curse to its victims, the exercise of arbitrary power works such fearful ruin upon the hearts of slaveholders, that I should feel impelled to labor and pray for its overthrow with my last energies and latest breath.”
—Angelina Grimké (1805–1879)
“Our security depends on the Allied Powers winning against aggressors. The Axis Powers intend to destroy democracy, it is anathema to them. We cannot provide that aid if the public are against it; therefore, it is our responsibility to persuade the public that aid to the victims of aggression is aid to American security. I expect the members of my administration to take every opportunity to speak to this issue wherever they are invited to address public forums in the weeks ahead.”
—Franklin D. Roosevelt (1882–1945)