Mathematics of Physics and Engineering

68 Systems of Point Masses

Typically, each Fj is a sum of an external force F\ ' from outside of S

and an internal system force F], exerted on rrij by the other n — 1 point

masses. Thus, Fj = Ylk^tj Fjk> where FV is the force exerted by rrik
on rrij. Hence,

F

= it

F

J = Y,

F<

J

E)

+ £

F

T =

p{B)

+

F{1)

•

j=l j=l j=l

By Newton's Third Law, F$ = -F$. It follows that F(7) = £"=1 Ff] =

0 and therefore, F = ^)"=1 F(E) = F{E). By (2.2.4), the motion of the

center of mass is then determined by

MrCM = F{E). (2.2.5)

The (linear) momentum PQM °f the center of mass is, by definition,

PCM = MVCM-

With this definition, equation (2.2.5) becomes pCM = F^E\ and if the net

external force F^ ' is zero, then PCM ls constant. By (2.2.2),

n n

PCM = J2 m^ = 12PJ> (^2 -^2 -^6 )

where Pj = rrij Tj is the momentum of rrij. Thus, if F^E' = 0, then the

total linear momentum ps — Yll=i Pj °f ^e system is conserved.

Next, we consider the rotational motion of the system. By definition

(see (2.1.4) on page 40), the angular momentum LQ-J of the point mass

rrij about the reference point O is given by Loj = Tj X rrij Tj = Vj x Pj.

Accordingly, we define the angular momentum Lo of the system S about

O as the sum of the Lo,j'-

n n n

j=i j=i j=\

For the purpose of the definition, it is not necessary to assume that the

frame O is inertial. Note that, unlike the relation (2.2.6) for the linear

momentum, in general LQ ^ rCM x Mr CM-