Page bee29400_ch14_854-913.indd 905
bee29400_ch14_854-913.indd Page 905
12/13/08
12/13/08
9:31:44 PM user-s172
9:31:44 PM user-s172
/Volumes/204/MHDQ077/work%0/indd%0
/Volumes/204/MHDQ077/work%0/indd%0
REVIEW AND SUMMARY
In this chapter we analyzed the motion of systems of particles, i.e.,
the motion of a large number of particles considered together. In the
first part of the chapter we considered systems consisting of welldefined particles, while in the second part we analyzed systems
which are continually gaining or losing particles, or doing both at the
same time.
We first defined the effective force of a particle Pi of a given system
as the product mi ai of its mass mi and its acceleration ai with respect
to a newtonian frame of reference centered at O [Sec. 14.2]. We
then showed that the system of the external forces acting on the
particles and the system of the effective forces of the particles are
equipollent; i.e., both systems have the same resultant and the same
moment resultant about O:
O
Oma
O (r 3 F ) 5 O (r 3 m a )
n
n
Fi 5
i5 1
ii
(14.4)
i5 1
n
n
i
i
i
i5 1
ii
(14.5)
i5 1
Defining the linear momentum L and the angular momentum HO
about point O of the system of particles [Sec. 14.3] as
O
n
L5
i5 1
m iv i
Effective forces
O (r 3 m v )
Linear and angular momentum
of a system of particles
n
HO 5
i
ii
(14.6, 14.7)
i5 1
we showed that Eqs. (14.4) and (14.5) can be replaced by the
equations
.
.
oM O 5 HO
(14.10, 14.11)
oF 5 L
which express that the resultant and the moment resultant about O
of the external forces are, respectively, equal to the rates of change
of the linear momentum and of the angular momentum about O of
the system of particles.
In Sec. 14.4, we defined the mass center of a system of particles as
the point G whose position vector r satisfies the equation
O mr
Motion of the mass center
of a system of particles
n
mr 5
ii
(14.12)
i5 1
905
bee29400_ch14_854-913.indd Page 906
bee29400_ch14_854-913.indd Page 906
906
12/13/08
12/13/08
9:31:50 PM user-s172
9:31:50 PM user-s172
/Volumes/204/MHDQ077/work%0/indd%0
/Volumes/204/MHDQ077/work%0/indd%0
O m of the particles. Differn
Systems of Particles
where m represents the total mass
i
i5 1
entiating both members of Eq. (14.12) twice with respect to t, we
obtained the relations
L 5 ma
L 5 mv
(14.14, 14.15)
where v and a represent, respectively, the velocity and the accelera.
tion of the mass center G. Substituting for L from (14.15) into
(14.10), we obtained the equation
(14.16)
oF 5 ma
from which we concluded that the mass center of a system of particles moves as if the entire mass of the system and all the external
forces were concentrated at that point [Sample Prob. 14.1].
Angular momentum of a system
of particles about its mass center
y'
miv'i
y
r'i
Pi
G
x'
In Sec. 14.5 we considered the motion of the particles of a system
with respect to a centroidal frame Gx9y9z9 attached to the mass center G of the system and in translation with respect to the newtonian
frame Oxyz (Fig. 14.14). We defined the angular momentum of the
system about its mass center G as the sum of the moments about G
of the momenta miv9 of the particles in their motion relative to the
i
frame Gx9y9z9. We also noted that the same result can be obtained
by considering the moments about G of the momenta mivi of the
particles in their absolute motion. We therefore wrote
O (r 3 m v ) 5 O (r 3 m v)
n
O
x
z'
HG 5
n
i
ii
i5 1
i5 1
i
ii
(14.24)
and derived the relation
z
Fig. 14.14
oMG 5 HG
(14.23)
which expresses that the moment resultant about G of the external
forces is equal to the rate of change of the angular momentum about
G of the system of particles. As will be seen later, this relation is
fundamental to the study of the motion of rigid bodies.
Conservation of momentum
When no external force acts on a system of particles [Sec. 14.6], it
follows from Eqs. (14.10) and (14.11) that the linear momentum L
and the angular momentum HO of the system are conserved [Sample
Probs. 14.2 and 14.3]. In problems involving central forces, the angular momentum of the system about the center of force O will also
be conserved.
Kinetic energy of a system
of particles
The kinetic energy T of a system of particles was defined as the sum
of the kinetic energies of the particles [Sec. 14.7]:
T5
1
2
Omv
n
i5 1
2
ii
(14.28)
bee29400_ch14_854-913.indd Page 907
bee29400_ch14_854-913.indd Page 907
12/13/08
12/13/08
9:31:54 PM user-s172
9:31:54 PM user-s172
/Volumes/204/MHDQ077/work%0/indd%0
/Volumes/204/MHDQ077/work%0/indd%0
Review Summary
Using and the centroidal frame of reference Gx9y9z9 of Fig. 14.14, we
noted that the kinetic energy of the system can also be obtained by adding the kinetic energy 1 m v 2 associated with the motion of the mass
2
center G and the kinetic energy of the system in its motion relative
to the frame Gx9y9z9:
T 5 1 mv 2 1
2
1
2
O m v
n
ii
2
907
(14.29)
i5 1
The principle of work and energy can be applied to a system of
particles as well as to individual particles [Sec. 14.8]. We wrote
Principle of work and energy
(14.30)
T1 1 U1 y 2 5 T2
and noted that U1 y 2 represents the work of all the forces acting on
the particles of the system, internal as well as external.
If all the forces acting on the particles of the system are conservative,
we can determine the potential energy V of the system and write
Conservation of energy
(14.31)
T1 1 V1 5 T2 1 V2
which expresses the principle of conservation of energy for a system
of particles.
We saw in Sec. 14.9 that the principle of impulse and momentum for
a system of particles can be expressed graphically as shown in
Fig. 14.15. It states that the momenta of the particles at time t1 and the
impulses of the external forces from t1 to t2 form a system of vectors
equipollent to the system of the momenta of the particles at time t2.
y
y
(mAvA )1
(mBvB)1
x
O
y
t
+
t2
t
t2
1
O
(m CvC)1
F dt
=
x
1
(mBvB)2
(mAvA )2
x
O
(m CvC)2
MO d t
(b)
(a)
Principle of impulse and momentum
(c)
Fig. 14.15
If no external force acts on the particles of the system, the
systems of momenta shown in parts a and c of Fig. 14.15 are equipollent and we have
L1 5 L2
(HO)1 5 (HO)2
(14.36, 14.37)
Many problems involving the motion of systems of particles can
be solved by applying simultaneously the principle of impulse and
momentum and the principle of conservation of energy [Sample
Prob. 14.4] or by expressing that the linear momentum, angular
momentum, and energy of the system are conserved [Sample
Prob. 14.5].
Use of conservation principles in
the solution of problems involving
systems of particles
bee29400_ch14_854-913.indd Page 908
bee29400_ch14_854-913.indd Page 908
908
12/13/08
12/13/08
Systems of Particles
Variable
Variable systems of particles
Steady stream of particles
9:31:57 PM user-s172
9:31:57 PM user-s172
/Volumes/204/MHDQ077/work%0/indd%0
/Volumes/204/MHDQ077/work%0/indd%0
In the second part of the chapter, we considered variable systems of
particles. First we considered a steady stream of particles, such as a
stream of water diverted by a fixed vane or the flow of air through a
jet engine [Sec. 14.11]. Applying the principle of impulse and momentum to a system S of particles during a time interval Dt, and including
the particles which enter the system at A during that time interval
and those (of the same mass Dm) which leave the system at B, we
concluded that the system formed by the momentum (Dm)vA of the
particles entering S in the time Dt and the impulses of the forces
exerted on S during that time is equipollent to the momentum (Dm)vB
of the particles leaving S in the same time Dt (Fig. 14.16). Equating
( m) vB
m ivi B
m ivi B
F t
+
S
S
=
M t
S
A
A
( m) vA
(a)
(b)
(c)
Fig. 14.16
the x components, y components, and moments about a fixed point
of the vectors involved, we could obtain as many as three equations,
which could be solved for the desired unknowns [Sample Probs. 14.6
and 14.7]. From this result, we could also derive the following expression for the resultant oF of the forces exerted on S,
oF 5
dm
( vB 2 vA )
dt
(14.39)
where vB 2 vA represents the difference between the vectors vB and
vA and where dm/dt is the mass rate of flow of the stream (see footnote, page 886).
Systems gaining or losing mass
Considering next a system of particles gaining mass by continually
absorbing particles or losing mass by continually expelling particles
[Sec. 14.12], as in the case of a rocket, we applied the principle of
impulse and momentum to the system during a time interval Dt, being
careful to include the particles gained or lost during that time interval
[Sample Prob. 14.8]. We also noted that the action on a system S of
the particles being absorbed by S was equivalent to a thrust
P5
dm
u
dt
(14.44)
where dm/dt is the rate at which mass is being absorbed, and u is the
velocity of the particles relative to S. In the case of particles being
expelled by S, the rate dm/dt is negative and the thrust P is exerted in
a direction opposite to that in which the particles are being expelled.