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WorkEnergy Principle
Every kinetics problem in CE325 is governed by Newton’s Second Law, i.e.,
X
~
F
=
m~
a,
but sometimes alternate forms of this law is more convenient. One form which is heavily used can
be derived by integrating over distance along the path Newton’s Second Law, i.e.,
Z
2
1
X
~
F
·
d~r
=
Z
2
1
a
·
d~
r.
This will result in the “workenergy formulation” and the above equation is a scalar equation. This
formulation is extremely useful when the path conﬁguration is complicated but the details of how the
particle moved between points 1 and 2 is not important. Clearly, this method destroys information.
Consider now the derivation of the workenergy principle. For the leftside of the equation,
deﬁne
W
as the work done by external forces, i.e.,
W
=
Z
~
r
2
~
r
1
X
~
F
·
r,
and for the rightside of equation, use the fact that
~a
·
=
d~v
dt
·
=
·
dt
=
·
~v
=
·
to evaluate the integral as
Z
~
r
2
~
r
1
m~a
·
=
m
Z
v
2
v
1
·
=
1
2
mv
2
2

1
2
mv
2
1
.
Deﬁne now the kinetic energy
T
as
T
=
1
2
mv
2
,
then the integrated form of Newton’s Second Law can be written as
W
=Δ
T
=
T
2

T
1
.
The term
R
~
r
2
~
r
1
∑
~
F
·
can be simpliﬁed a little further, we can classify the external forces into
(1)
Conservative Forces,
~
F
c
and (2)
Nonconservative Forces,
~
F
n
. Conservative forces include
gravitational forces, elastic spring forces, electromagnetic forces, etc while nonconservative forces
include friction forces and randomly applied forces.
Conservative forces yield an integral that is independent of the path, i.e., the integral
I
~
F
c
·
=0
for any closed path. With this property, the conservative forces can be derived from a potential, i.e.,
–4/1–
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~
F
c
=

dV
d~r
,
where
V
=

Z
X
~
F
c
·
is a scalar potential.
Two types of conservative forces which are most common in dynamics problems are the
gravitational force and the elastic spring force. For their frequent appearances, it is therefore
convenient to write the total external forces as
X
~
F
=
X
~
F
n
+
X
±
~
F
e
+
~
F
g
²
in which
∑
~
F
n
is the sum of all nonconservative forces,
∑
~
F
e
is the sum of all elastic spring
forces and
~
F
g
is the gravitational force on particle
m
. The work term can now be separated as
W
=
Z
~
r
2
~
r
1
X
~
F
n
·
+
Z
~r
2
1
X
~
F
e
·
+
Z
2
1
X
~
F
g
·
=
U

Δ
V
e

Δ
V
g
where
U
is the work done by all nonconservative forces,

Δ
V
e
is the work done by the elastic
forces and

Δ
V
g
is the work done by the gravitational force, all between the interval from
~
r
1
to
~
r
2
. While

Δ
V
is referred to as the work done,
Δ
V
is referred to as the potential energy. With
these new deﬁnitions, the most frequently used workenergy relationship can be written as
U
=Δ
T
+Δ
V
e
V
g
The Derivation of
Δ
V
g
To obtain a more convenient form for the gravitational
potenial energy
Δ
V
g
=

Z
2
1
~
F
g
·
d~
r,
use the cylindrical coordinates. Since the gravita
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 Spring '08
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