Mathematics of Physics and Engineering

76 Systems of Point Masses

It is therefore natural to introduce the following notations:

n n n

j=i j=\ j=\

n n

•*xy = -*yx = / j ^'j'^jUji ^xz == *zx = / j TTljXjZj,

•iyz — *zy — / JlmjVizj-

(2.2.30)

With these notations, LCMx = uxIxx - ujyIxy - wzIxz.

EXERCISE 2.2.8.C Verify that

LcMy — —WxIyx + COylyy — U!zIyz, LQMZ — ~^XIZX — UJyIzy + WZJZZ.

EXERCISE 2.2.9.C' Assume that all the point masses are in the (i, j) plane.
Show that Ixz = Iyz = 0 and Ixx + Iyy = Izz.
We can easily rewrite (2.2.28) in the matrix-vector form:

CCM(t) = IcM(t)Ct(t), (2.2.31)

where CcM(t) is the column vector (LCMx{t), LCMv(t), LCMz{t))T, Cl(t)
is the column vector {wx{t), u)y(t), ivz(t))T, and

(2.2.32)

The matrix ICM is called the moment of inertia matrix, or tensor of
inertia, of the system 5 around the center of mass in the basis (i, j, K).
The Latin word tensor means "the one that stretches," and, in mathematics,
refers to abstract objects that change in a certain way from one coordinate
system to another. All matrices are particular cases of tensors. For a
summary of tensors, see page 457 in Appendix.
As much as we would like it, equality (2.2.31) is not the end of our
investigation, and there two main reasons for that:

(1) It is not at all clear how to compute the entries of the matrix ICM-

(2) Because the system S is rotating relative to the frame O, the entries of

the matrix ICM depend on time.