Chapter 14 reviev

1
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 S miai
i 1
i 1
n
n
S (ri x Fi ) S (ri x miai)
i 1
i 1
Sharif University-Aerospace Dep. Fall 2004
2
The linear momentum L and the angular momentum Ho
about point O are defined as
n
n
L S mivi
Ho S (ri x mivi)
i 1
i 1
It can be shown that
.
.
S F L 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.
3
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.
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Consider the motion of the particles of a system
with respect to a centroidal frame Gxyz
attached to the mass center G of the system and
in translation with respect to the newtonian
frame Oxyz. The angular momentum of the system
about its mass center G is defined as the
mivi
y
y
ri
Pi
G
x
O
x
z
z
sum of the moments about G of the momenta mivi
of the particles in their motion relative to the
frame Gxyz. The same result is obtained by
considering the moments about G of the momenta
mivi of the particles in their absolute motion.
Therefore
n
n
HG S (ri x mivi) S (ri x mivi)
i 1
i 1
5
n
mivi
y
HG S (ri x mivi)
y
i 1
ri
Pi
G
x
O
We can derive the relation
x
.
z
z
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.
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mivi
The kinetic energy T of a system of particles is
defined as the sum of the kinetic energies of the
particles.
y
y
ri
Pi
n
G
1 2
T S mivi
2
x
O
i 1
x
z
z
Using the centroidal reference frame Gxyz we
note that the
kinetic energy of the system can also be obtained
by adding the kinetic energy mv2 associated
with the motion of the mass center G and the
kinetic energy of the system in its motion
relative to the frame Gxyz
1 2
n
1 2
1 2
T mv 2 S mivi
2
i 1
7
n
mivi
y
1 2
1 2
T mv 2 S mivi
2
y
i 1
ri
Pi
The principle of work and energy can be applied
to a system of particles as well as to individual
particles
G
x
O
x
z
z
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
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(mAvA)1
y
y
y
(mBvB)2
(mAvA)2
(mCvC)2
(mBvB)1
O
x
x
O
x
O
t2
ò
S MOdt
(mCvC)1
t1
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 .
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(mAvA)1
y
y
(mBvB)2
(mAvA)2
(mCvC)2
(mBvB)1
x
O
x
O
(mCvC)1
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.
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(D m)vB
B
B
Smivi
Smivi
S
S
A
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.
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(D m)vB
B
B
Smivi
Smivi
S
S
A
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 dt
SF (vB - vA)
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.
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v
va
S F Dt
m
Dm
S
mv
(Dm) va
S
u va - v
Consider a system of particles gaining mass by
continually absorbing particles or losing mass
by continually expelling 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
(m Dm)
S
(m Dm)(v Dv)
dm dt
P u
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v
va
S F Dt
m
Dm
S
mv
(Dm) va
S
u va - v
(m Dm)
dm dt
P u
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.