Rigid Bodies 81where v(t) = r(t) - rcM(t), and is (2.2.14) replaced by
LCM(t) = llf(t) x «() P(r(t)) dV, (2.2.40)
TC(t)which is the angular momentum relative to the center of mass.
As with finite systems of points, consider the parallel translation of
the frame O to the center of mass of the body. The rotation of the rigid
body relative to this translated frame is described by the rotation vector
u>(t) = u>x{t) i + Wy(t) 3 + ojz(t) k, where (S, j, k) is the cartesian basis in
the translated frame. By analogy with (2.2.30), we write v(t) = x(t)i +
vit)3+ ()&, LCM(t) = LCMx(t)i + LCMy{t)3 + LcMz(t)k, and define
Ixx =
IIIiy2{t) + Z2{t)) P{x{t)) dV
'Iyy =
III{x2{t) + Z2{t))P{x{t)) dV
'
-R(t) n(t)= IJJ(x^2 (t) + y^2 (t))p(x(t))dV,
K{t)•*-xy — -*-yxlyz — J-zyjJjx(t)y{t)p{x{t))dV,
*xz — Izx —JJJx(t)z(t)p(x(t))dV,
Tl(t)= JJJy{t)z{t)p{x{t))dV.
it(t) n(t)W)
(2.2.41)
Then we have relation (2.2.31), page 76, for rigid bodies:
CcM(t)=ICM(t)n(t), (2.2.42)
where CcM{t) is the column vector (LcMx(t), LcMy{t), LcMz(t))T, ft is
the column vector (wx{t), wy(t), uz(t))T, and
(
J-xx ~~±xy *xz \-Iyx Iyy —IyZ I • (2.2.43)
•'zx ~*zy *zz /