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chapter 4 Systems of Particles
Typology: Lecture notes
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Tenth
Tenth
Edition
Edition
Ferdinand P. Beer
Ferdinand P. Beer
E. Russell Johnston, Jr.
E. Russell Johnston, Jr.
Phillip J. Cornwell
Phillip J. Cornwell
Lecture Notes:
Lecture Notes:
Brian P. Self
Brian P. Self
California Polytechnic State University
California Polytechnic State University
CHAPTER
© 2013 The McGraw-Hill Companies, Inc. All rights rese
Systems of Particles
Vector Mechanics for Engineers: Dynamics
Vector Mechanics for Engineers: Dynamics
14 - 2
Introduction
Application of Newton’s Laws: Eff
ective Forces
Linear and Angular Momentum
Motion of Mass Center of System
of Particles
Angular Momentum About Mass
Center
Conservation of Momentum
Sample Problem 14.
Kinetic Energy
Work-Energy Principle.
Conservation of Energy
Principle of Impulse and Momentum
Sample Problem 14.
Sample Problem 14.
Variable Systems of Particles
Steady Stream of Particles
Steady Stream of Particles.
Applications
Streams Gaining or Losing Mass
Sample Problem 14.
Vector Mechanics for Engineers: Dynamics
Vector Mechanics for Engineers: Dynamics
14 - 4
of particles.
it mass and acceleration. It will be shown that the system of
external forces acting on a system of particles is equipollent
with the system of effective forces of the system.
and its motion described.
impulse-momentum principle to a system of particles will
be described. Result obtained are also applicable to a
system of rigidly connected particles, i.e., a rigid body.
of particles, i.e., systems in which the particles included
in the system change.
Vector Mechanics for Engineers: Dynamics
Vector Mechanics for Engineers: Dynamics
14 - 5
i
in a system of n particles,
effective force
externalforce internal forces
1
1
i i
i ij
i i i
n
j
i i i ij
i i
n
j
i ij
m a
F f
r F r f r m a
F f m a
on a particle is equivalent to the effective
force of the particle.
The system of external and internal forces
acting on the entire system of particles is
equivalent to the system of effective forces.
Vector Mechanics for Engineers: Dynamics
Vector Mechanics for Engineers: Dynamics
14 - 7
particles,
n
i
i i
n
i
i i
n
i
i i
L m v m a
L m v
1 1
1
of system of particles,
n
i
i i i
n
i
i i i
n
i
O i i i
n
i
O i i i
r m a
H r m v r m v
H r m v
1
1 1
1
equal to rate of change of linear
momentum of the system of
particles,
F L
O O
M H
Moment resultant about fixed point O of
the external forces is equal to the rate of
change of angular momentum of the
system of particles,
Vector Mechanics for Engineers: Dynamics
Vector Mechanics for Engineers: Dynamics
14 - 8
by position vector which satisfies
G
r
n
i
G i i
mr m r
1
ma L F
mv m v L
mr m r
G
n
i
G i i
n
i
G i i
1
1
all of the external forces were concentrated at
that point.
Vector Mechanics for Engineers: Dynamics
Vector Mechanics for Engineers: Dynamics
14 - 10
their absolute motion relative to the
Newtonian Oxyz frame of reference.
G G G
n
i
G i i i
n
i
i i
n
i
i i G i
n
i
G i i i
H H M
m r v r m v
r m v v
H r m v
1 1
1
1
the particles in their motion
relative to the centroidal Gx’y’z’
frame of reference,
n
i
G i i i
H r mv
1
i G G
v v v
momenta can be calculated with respect to
either the Newtonian or centroidal frames of
reference.
Vector Mechanics for Engineers: Dynamics
Vector Mechanics for Engineers: Dynamics
14 - 11
particles of a system, then the
linear momentum and angular
momentum about the fixed point O
are conserved.
constant constant
0 0
O
O O
L H
L F H M
problems involving central forces,
constant constant
0 0
O
O O
L H
L F H M
also applies to the analysis of the mass
center motion,
constant constant
constant
0 0
G G
G
G G
v H
L m v
L F H M
Vector Mechanics for Engineers: Dynamics
Vector Mechanics for Engineers: Dynamics
14 - 13
A 20-lb projectile is moving with a
velocity of 100 ft/s when it explodes into
5 and 15-lb fragments. Immediately
after the explosion, the fragments travel
A
= 45
o
B
= 30
o
.
Determine the velocity of each fragment.
SOLUTION:
linear momentum of the system is
conserved.
Write separate component equations
for the conservation of linear
momentum.
for the fragment velocities.
Vector Mechanics for Engineers: Dynamics
Vector Mechanics for Engineers: Dynamics
14 - 14
SOLUTION:
linear momentum of the system is
conserved.
x
y
the conservation of linear momentum.
0
0
5 g v 15 g v 20 g v
m v m v m v
A B
A A B B
x components:
5 cos 45 15 cos 30 20 100
A B
v v
y components:
5 sin 45 15 sin 30 0
A B
v v
fragment velocities.
207 ft s 97. 6 ft s
A B
v v
Vector Mechanics for Engineers: Dynamics
Vector Mechanics for Engineers: Dynamics
14 - 16
(12 ft/s) cos 30 sin 7.4 sin 49.3 (6.29) cos 45
0.12880 0.75813 5.
A B
A B
m mv mv m
v v
(12 ft/s)sin 30 cos 7.4 cos 49.3 (6.29)sin 45
0.99167 0.65210 1.
A B
A B
m mv mv m
v v
0.12880 0.75813 5.
A B
v v
0.99167 0.65210 1.
A B
v v
0.65210 (
)
)
0.83581 5.
A
v
6.05 ft/s
A
v
(1)
(2)
Sub into (1) or (2) to get v
B
6.81 ft/s
B
v
Vector Mechanics for Engineers: Dynamics
Vector Mechanics for Engineers: Dynamics
14 - 17
In a game of pool, ball A is moving with a
velocity v
0
when it strikes balls B and C,
which are at rest and aligned as shown.
v
C
v
A
v
0
v
B
0
Vector Mechanics for Engineers: Dynamics
Vector Mechanics for Engineers: Dynamics
14 - 19
i
,
1 1 2 2
T U T
where represents the work done by the internal forces
and the resultant external force acting on P
i
.
ij
f
i
F
1 2
U
adding the kinetic energies of all particles and considering the work
done by all external and internal forces.
forces will not, in general, cancel out.
ij ji
f f
and
equal to the change in potential energy and
1 1 2 2
T V T V
which expresses the principle of conservation of energy for
the system of particles.
Vector Mechanics for Engineers: Dynamics
Vector Mechanics for Engineers: Dynamics
14 - 20
1 2
2 1
2
1
2
1
L Fdt L
Fdt L L
F L
t
t
t
t
1 2
2 1
2
1
2
1
H M dt H
M dt H H
M H
t
t
O
t
t
O
O O
1
and the impulse of the forces
from t
1
to t
2
form a system of vectors equipollent to the system of
momenta of the particles at time t
2
.
