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where i represents the i th particle making up the body, ω is the angular velocity of the
body about the axis under consideration, and ( r i × m i ( ω × r i )) is the angular momentum
of the i th particle, which has a magnitude of m i ω r i 2 . For rotation about a given axis, this
equation can be rewritten in the form:
H cg = ∫ ω r 2 dm
Given that the angular velocity is the same for all particles making up the rigid body,
we have:
H cg = ω ∫ r 2 dm
and recalling that moment of inertia, I , equals r 2 dm , we get:
H cg = Iω
Taking the derivative with respect to time, we obtain:
dH cg /dt = d/dt (Iω) = I dω/dt = Iα
where α is the angular acceleration of the body about a given axis.
Finally, we can write:
M cg = I α
As we stated in our discussion on mass moment of inertia, we will have to further
generalize our formulas for moment of inertia and angular moment to account for
rotation about any body axis. Generally, M and α will be vector quantities, while I will
be a tensor 2 since the magnitude of moment of inertia for a body may vary depending
on the axis of rotation (see the sidebar “Tensors” on page 22 ).
A tensor is a mathematical expression that has magnitude and direction, but its mag‐
nitude may not be unique depending on the direction. Tensors are typically used to
represent properties of materials where these properties have different magnitudes in
different directions. Materials with properties that vary depending on direction are
called anisotropic ( isotropic implies the same magnitude in all directions). For example,
consider the elasticity (or strength) of two common materials, a sheet of plain paper
2. In this case, I will be a second-rank tensor, which is essentially a 3×3 matrix. A vector is actually a tensor of
rank one, and a scalar is actually a tensor of rank zero.
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