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
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
where is the moment of inertia matrix of the system relative to the reference point R.
Read more about this topic: Moment Of Inertia
Other articles related to "moment of inertia around an arbitrary axis, moment of inertia, around an arbitrary axis, axis, moment, moment of inertia around an":
... 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 ... 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 ...
Famous quotes containing the words moment of, axis, arbitrary, moment and/or inertia:
“At the thick of the dark
the moment of the colds
deepest plunge we brought branches
cut from the green trees
to fill our need,”
—William Carlos Williams (18831963)
“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 (18821945)
“A state that denies its citizens their basic rights becomes a danger to its neighbors as well: internal arbitrary rule will be reflected in arbitrary external relations. The suppression of public opinion, the abolition of public competition for power and its public exercise opens the way for the state power to arm itself in any way it sees fit.... A state that does not hesitate to lie to its own people will not hesitate to lie to other states.”
—Václav Havel (b. 1936)
“But one sound always rose above the clamor of busy life and, no matter how much of a tintinnabulation, was never confused and, for a moment lifted everything into an ordered sphere: that of the bells.”
—Johan Huizinga (18721945)
“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 (18561950)