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Chapter 14 SYSTEMS OF PARTICLES
The effective force of a particle Pi of a given system is the
product miai of its mass mi and its acceleration ai with respect
to a newtonian frame of reference centered at O. 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 :
n
n
S Fi =iS=1miai
i =1
n
n
S (ri x Fi ) =iS=1(ri x miai)
i =1
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The linear momentum L and the angular momentum Ho about
point O are defined as
n
L = S mivi
i =1
It can be shown that
.
S F=L
n
Ho = S (ri x mivi)
i =1
.
S Mo = Ho
This expresses 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.
The mass center G of a system of particles is defined by a
position vector r which satisfies the equation
n
mr = S miri
i =1
n
where m represents the total mass S mi. Differentiating both
i =1
members twice with respect to t, we obtain
L = mv
.
L = ma
where v and a are the .velocity and acceleration of the mass
center G. Since S F = L, we obtain
S F = ma
Therefore, 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.
Consider the motion of the particles
of a system with respect to a
y
centroidal frame Gx’y’z’ attached to
r’i
Pi
the mass center G of the system and
G
in translation with respect to the
x’
O
newtonian frame Oxyz. The angular
x
momentum of the system about its
z’
z
mass center G is defined as the
sum of the moments about G of the momenta miv’i of the
particles in their motion relative to the frame Gx’y’z’. The same
result is obtained by considering the moments about G of the
momenta mivi of the particles in their absolute motion. Therefore
y’
miv’i
n
n
i =1
i =1
HG = S (r’i x mivi) = S (r’i x miv’i)
y’
miv’i
y
r’i
G
x’
z
z’
x
HG = S (r’i x mivi)
i =1
n
Pi
O
n
= S (r’i x miv’i)
i =1
We can derive the relation
.
S MG = HG
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.
When no external force acts on a system of particles, the
linear momentum L and the angular momentum Ho of the
system are conserved. In problems involving central forces,
the angular momentum of the system about the center of force
O will also be conserved.
y’
miv’i
y
r’i
G
Pi
x’
O
z’
x
The kinetic energy T of a system of
particles is defined as the sum of
the kinetic energies of the particles.
n
1
2
T = 2 S mivi
i=1
Using the centroidal reference
frame Gx’y’z’ we note that the
kinetic energy of the system can also be obtained by adding the
1
kinetic energy 2 mv2 associated with the motion of the mass
center G and the kinetic energy of the system in its motion
relative to the frame Gx’y’z’ :
z
1
2
T = mv 2 +
n
2
1
miv’i
2 iS
=1
y’
miv’i
y
r’i
G
Pi
x’
O
z
z’
x
n
1
2
2
T = mv + 2 S miv’i
i=1
The principle of work and energy
can be applied to a system of
particles as well as to individual
particles
1
2
T1 + U1
2=
T2
where U1 2 represents the work of all the forces acting on the
particles of the system, internal and external.
If all the forces acting on the particles of the system are
conservative, the principle of conservation of energy can be
applied to the system of particles
T1 + V1 = T2 + V2
y
(mAvA)1
y
t2
S  Fdt
y
(mAvA)2
t1
(mCvC)2
(mBvB)1
O
x
(mCvC)1
(mBvB)2
O
S
t2
t1
x
MOdt
O
x
The principle of impulse and momentum for a system of particles
can be expressed graphically as shown above. 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
(mAvA)1
y
(mBvB)2
(mAvA)2
(mCvC)2
(mBvB)1
O
x
(mCvC)1
O
x
If no external forces act on the system of particles, the systems
of momenta shown above are equipollent and we have
L1 = L2
(HO)1 = (HO)2
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 or
by expressing that the linear momentum, angular momentum,
and energy of the system are conserved.
Smivi
B
S
A
(D m)vB
S F Dt
S
S M Dt
B
Smivi
S
A
(D m)vA
For variable systems of particles, first consider 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. The principle of impulse and
momentum is applied to a system S of particles during a time
interval Dt, including particles which enter the system at A during
that time interval and those (of the same mass Dm) which leave
the system at B. 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.
Smivi
B
S
A
(D m)vB
S F Dt
S
S M Dt
B
Smivi
S
A
(D m)vA
Equating 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. From this result, we can derive the expression
dm
SF =
(vB - vA)
dt
where vB - vA represents the difference between the vectors vB
and vA and where dm/dt is the mass rate of flow of the stream.
v
va
m
mv
S F Dt
Dm
S
S (Dm) va
u = va - v
Consider a system of particles gaining
(m + Dm)
mass by continually absorbing particles
S
or losing mass by continually expelling
(m + Dm)(v + Dv)
particles (as in the case of a rocket).
Applying the principle of impulse and
momentum to the system during a time interval Dt, we take care
to include particles gained or lost during the time interval. The
action on a system S of the particles being absorbed by S is
equivalent to a thrust
dm
P = dt u
v
va
m
mv
S F Dt
Dm
S
S (Dm) va
u = va - v
dm
P = dt u
(m + Dm)
S
(m + Dm)(v + Dv)
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 P is
in a direction opposite to that in which particles are being
expelled.