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Seventh Edition
CHAPTER VECTOR MECHANICS FOR ENGINEERS:
14DYNAMICS
Ferdinand P. Beer Systems of Particles
E. Russell Johnston, Jr.
Lecture Notes:
J. Walt Oler
Texas Tech University
© 2003 The McGraw-Hill Companies, Inc. All rights reserved.
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h Introduction
In the current chapter, you will study the motion of systems
of particles.
The effective force of a particle is defined as the product of
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.
The mass center of a system of particles will be defined
and its motion described.
Application of the work-energy principle and the
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.
Analysis methods will be presented for variable systems
of particles, i.e., systems in which the particles included
in the system change.
© 2003 The McGraw-Hill Companies, Inc. All rights reserved. 14 - 3
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h Application of Newton’s Laws. Effective Forces
Newton’s second law for each particle P
in a system of n particles, i
r n r r
F + ∑ f =ma
i j=1 ij i i
r r n r r r r
()
r ×F + ∑ r × f =r ×ma
i i j=1 i ij i i i
r r
F = external force f = internal forces
i ij
r
ma = effective force
i i
The system of external and internal forces 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.
© 2003 The McGraw-Hill Companies, Inc. All rights reserved. 14 - 4
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h Application of Newton’s Laws. Effective Forces
Summing over all the elements,
n r n n r n r
∑F +∑∑f =∑ma
i ij i i
i i j i
=1 ==11 =1
n r r n n r r n r r
() ()
()
∑r×F +∑∑r×f =∑ r×ma
i i i ij i i i
i=1 i==11j i=1
Since the internal forces occur in equal
and opposite collinear pairs, the resultant
force and couple due to the internal
forces are zero,
r r
∑F =∑ma
i r i i
r r r
()()
∑r×F =∑ r ×ma
i i i i i
The system of external forces and the
system of effective forces are
equipollent by not equivalent.
© 2003 The McGraw-Hill Companies, Inc. All rights reserved. 14 - 5
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h Linear & Angular Momentum
Linear momentum of the system of Angular momentum about fixed point O
particles, of system of particles,
r n r r n r r
()
L=∑mv H =∑ r ×mv
i i O i 1 i i i
i=1 =
r n r n r r n r r n r r
& & & & &
()()
L=∑mv =∑ma H =∑r×mv +∑ r×mv
i=1 i i i=1 i i O i=1 i i i i=1 i i i
n r r
()
= ∑ r ×m a
Resultant of the external forces is i=1 i i i
equal to rate of change of linear
momentum of the system of Moment resultant about fixed point O of
particles, the external forces is equal to the rate of
r r
& change of angular momentum of the
∑F=L
system of particles,
r r
&
∑MO=HO
© 2003 The McGraw-Hill Companies, Inc. All rights reserved. 14 - 6
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h Motion of the Mass Center of a System of Particles
Mass center G of system of particles is defined
r
r
by position vector which satisfies
n G
r r
mr =∑mr
G i=1 i i
Differentiating twice,
r n r
& &
mr =∑mr
G i=1 i i
r n r r
mv =∑mv =L
G i=1 i i
r r r
&
maG = L = ∑F
The mass center moves as if the entire mass and
all of the external forces were concentrated at
that point.
© 2003 The McGraw-Hill Companies, Inc. All rights reserved. 14 - 7
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h Angular Momentum About the Mass Center
The angular momentum of the system of
particles about the mass center,
r n r r
′ ′ ′
()
H =∑ r ×mv
G i=1 i i i
r n r r n r r r
& ′ ′ ′ ′
()()()
H =∑ r ×ma =∑ r ×m a −a
G i=1 i i i i=1 i i i G
r r r′ n r r ⎛ n r ⎞ r
a =a +a
i G i ′ ⎜ ′⎟
()
= ∑ r ×m a − ∑m r ×a
i i i ⎜ i ⎟ G
i=1 ⎝i=1 ⎠
Consider the centroidal frame n r r n r r
′ ′
()()
of reference Gx’y’z’, which = ∑ r ×m a = ∑ r ×F
i=1 i i i i=1 i i
translates with respect to the r
Newtonian frame Oxyz. =∑MG
The centroidal frame is not, The moment resultant about G of the external
in general, a Newtonian forces is equal to the rate of change of angular
frame. momentum about Gof the system of particles.
© 2003 The McGraw-Hill Companies, Inc. All rights reserved. 14 - 8
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h Angular Momentum About the Mass Center
Angular momentum about G of particles in
their absolute motion relative to the
Newtonian Oxyz frame of reference.
r n r′ r
()
H =∑ r ×mv
G =1 i i i
i
n r′ r r′
()()
= ∑ r ×m v +v
i=1 i i G i
r r r′
v =v +v
i G G ⎛ n r ⎞ r n r r
′ ′
⎜ ⎟ ()
= ∑mr ×v +∑ r ×mv
Angular momentum about G of ⎜ i i ⎟ G i i i
⎝i=1 ⎠ i=1
the particles in their motion r r′ r
relative to the centroidal Gx’y’z’ HG =HG =∑MG
frame of reference, Angular momentum about G of the particle
r n r r momenta can be calculated with respect to
′ ′ ′
()
H =∑r×mv
G i=1 i i i either the Newtonian or centroidal frames of
reference.
© 2003 The McGraw-Hill Companies, Inc. All rights reserved. 14 - 9
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