66 Systems of Point Masses
rotation of the frame 0\ relative to this translated frame O'. We can now
combine relation (2.1.44) for rotation with relation (2.1.12) on page 47 for
parallel translation to get
ro(*) = roi(*)+ri(t)
+ 2u(t) x fi(t) +w(f) x (u(t) x ri(t)) +u(J) x n(t).EXERCISE 2.1.21? Verify (2.146). Hint: Apply (2.1.45) ton, replacing frame
O with O'.
Suppose that the frame O is inertial, and a force F is acting on the
point mass m. Then, by Newton's Second Law, mro{t) = F; to simplify
the notations we will no longer write the time dependence explicitly. By
(2.1.46),
mf\ = F - mr-Qi - 2mw x r\ - mu x (u x n) -mCo x T\. (2.1.47)As before in (2.1.13), page 47, and in (2.1.24), page 53, we have several
corrections to Newton's Second Law in the non-inertial frame 0. These
corrections are the translational acceleration force Fta = —in^oi,
the Coriolis force Fcor = —2mu> x ri, the centrifugal force Fc =
—rawx (wxri), and the angular acceleration forceFaa = -mwxri.
2.2 Systems of Point Masses
The motion of a system of point masses can be decomposed into the motion
of one point, the center of mass, and the rotational motion of the system
around the center of mass. In what follows, we study this decomposition,
first for a finite collection of point masses, and then for certain infinite
collections, namely, rigid bodies.
2.2.1 Non-Rigid Systems of Points
n
Let 5 be a system of n point masses, mi,..., mn, and M — J^ rrij, the total
mass of S. We assume that the system is non-rigid, that is, the distances
between the points can change. Denote by rj the position vector of rrij in
some frame 0. By definition, the center of mass (CM) of S is the point