Mathematics of Physics and Engineering

(^80) Systems of Point Masses
body occupies the region H in R^3 , then the mass M of the body is given
by the triple (or volume) integral
Af = III p{r)dV.
Without going into the details, let us note that rigid bodies can also be
two-dimensional, for example, a (hard) spherical shell, or one-dimensional,
for example, a piece of hard-to-bend wire. In these cases, we use surface
and line integrals rather then volume integrals. In what follows, we focus
on solid three-dimensional objects.
All the formulas for the motion of a rigid body can be derived from the
corresponding formulas for a finite number of points by replacing rrij with
the mass density function, and summation with integration. For example,
the center of mass of a rigid body is the point with the position vector
VCM = ^ JJJ rp(r)dV. (2.2.36)
n
EXERCISE 2.2.14.C Show that both the mass and the location of the center
of mass of a rigid body are independent of the frame O.
For a rigid body 11 moving in space relative to a frame O, we denote
by lZ(t) the part of the space occupied by the body at time t relative to
that frame O. If the frame O is inertial, then an equation similar to (2.2.5)
connects the trajectory rcM = fCM(t) of the center of mass in the frame
with the external forces per unit mass F^1 -^ = F^E\r) acting on the points
of the body:
MfCM(t) = JJJF^'(r(t)) P(r(t))dV, (2.2.37)
The angular momentum of TZ about O is, by definition,
Lo(t) = III r(t) x r{t) P(r(t))dV. (2.2.38)
n{t)
Similar to (2.2.15), page 71, we have
L 0 (t) = MrCM(t) x rCM(t) + [[[(!) x (*) P(r(t))dV, (2.2.39)