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, inertia, arbitrary and/or axis:
“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)
“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)
“Me, whats that after all? An arbitrary limitation of being bounded by the people before and after and on either side. Where they leave off, I begin, and vice versa.”
—Russell Hoban (b. 1925)
“A book is not an autonomous entity: it is a relation, an axis of innumerable relations. One literature differs from another, be it earlier or later, not because of the texts but because of the way they are read: if I could read any page from the present timethis one, for instanceas it will be read in the year 2000, I would know what the literature of the year 2000 would be like.”
—Jorge Luis Borges (18991986)