(^82) Systems of Point Masses
tion p. The principal axes frame, with center at the center of mass and
basis vectors i, j, k, is attached to the body and rotates with it. If
u>(t) = u i + LJJ + w k, then the Euler equations (2.2.35) describe
the rotation of the body about the center of mass. Equations (2.2.37) and
(2.2.35) provide a complete description of the motion of a rigid body.
As an example, consider the DISTRIBUTED RIGID PENDULUM. Recall
that in the simple rigid pendulum (page 41), a point mass is attached to the
end of a weightless rod. In the distributed rigid pendulum, a uniform rod
of mass M and length I is suspended by one end with a pin joint (Figure
2.2.1).
--3
Cross-section of the rod
Fig. 2.2.1 Distributed Pendulum
We assume that the cross-section of the rod is a square with side a. Then
the volume of the rod is la^2 and the density is constant: p(r) = M/(£a^2 ).
EXERCISE 2.2.15.c Verify that the center of mass CM of the rod is the
mid-points of the axis of the rod.
Consider the cartesian coordinates (z, j, k) with the origin at the
center of mass (Figure 2.2.1). As usual, k = i x j.
Let us compute I*z, as this is the only entry of the matrix 7£M we will
need:
- y^2 )p(r)dV
M
la?
/ a/2 a/2 1/2 e/2 a/2 a/2 \
dy I dz I x^2 dx + I dx I dz I y^2 dy
-a/2 -a/2 -1/2 -1/2 -a/2 -a/2
= (pa^2 £^3 /12) + (plo^jYL) = (M/12){1^2 + a^2 ).