S.
Widnall,
J.
Peraire
16.07
Dynamics
Fall
2008
Version
2.0
Lecture
L13
 Conservative
Internal
Forces
and
Potential
Energy
The
forces
internal
to
a
system
are
of
two
types.
Conservative
forces,
such
as
gravity;
and
dissipative
forces
such
as
friction.
Internal
forces
arise
from
the
natural
dynamics
of
the
system
in
contract
to
external
forces
which
are
imposed
from
an
external
source.
We
have
seen
that
the
work
done
by
a
force
F
on
a
particle
is
given
by
dW
=
F
d
r
.
·
If
the
work
done
by
an
internal
forces
F
,
when
the
particle
moves
from
any
position
r
1
to
any
position
r
2
,
can
be
expressed
as
the
di
ff
erence
in
a
scalar
function
of
r
between
the
two
ends
of
the
trajectory,
r
2
W
12
=
F
d
r
=
−
(
V
(
r
2
)
−
V
(
r
1
))
=
V
1
−
V
2
,
(1)
·
r
1
then
we
say
that
the
force
is
conservative
.
In
the
above
expression,
the
scalar
function
V
(
r
)
is
called
the
potential
.
It
is
clear
that
the
potential
satisfies
dV
=
−
F
d
r
(the
minus
sign
is
included
for
convenience).
·
There
are
two
main
consequences
that
follow
from
the
existence
of
a
potential:
i)
the
work
done
by
a
conservative
force
between
points
r
1
and
r
2
is
independent
of
the
path
.
This
follows
from
(1)
since
W
12
only
depends
on
the
initial
and
final
potentials
V
1
and
V
2
(and
not
on
how
we
go
from
r
1
to
r
2
),
and
ii)
the
work
done
by
potential
forces
is
recoverable
.
Consider
the
work
done
in
going
from
point
r
1
to
point
r
2
,
W
12
.
If
we
go,
now,
from
point
r
2
to
r
1
,
we
have
that
W
21
=
−
W
12
since
the
total
work
W
12
+
W
21
= (
V
1
−
V
2
) + (
V
2
−
V
1
)
=
0.
In
one
dimension
any
force
which
is
only
a
function
of
position
is
conservative.
That
is,
if
we
have
a
force,
F
(
x
),
which
is
only
a
function
of
position,
then
F
(
x
)
dx
is
always
a
perfect
di
ff
erential.
This
means
that
we
can
define
a
potential
function
as
x
V
(
x
) =
−
F
(
x
)
dx
,
x
0
where
x
0
is
arbitrary.
In
two
and
three
dimensions,
we
would,
in
principle,
expect
that
any
force
which
depends
only
on
position,
F
(
r
),
to
be
conservative.
However,
it
turns
out
that,
in
general,
this
is
not
su
ﬃ
cient.
In
multiple
dimensions,
1
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the
condition
for
a
force
field
to
be
conservative
is
that
it
can
be
expressed
as
the
gradient
of
a
potential
function.
That
is,
F
C
=
−
∇
V .
This
result
follows
from
the
gradient
theorem,
which
is
often
called
the
fundamental
theorem
of
calculus,
which
states
that
the
integral
r
2
−
∇
V
·
d
r
=
−
(
V
2
−
V
1
)
r
1
is
independent
of
the
path
between
r
1
and
r
2
.
Therefore
the
work
done
by
conservative
forces
depends
only
upon
the
endpoints
r
2
and
r
1
rather
than
the
details
of
the
path
taken
between
them.
r
2
r
2
F
C
d
r
=
−
∇
V
d
r
=
−
(
V
2
−
V
1
)
·
·
r
1
r
1
In
the
general
case,
we
will
deal
with
internal
forces
that
are
a
combination
of
conservative
and
non
conservative
forces.
F
=
F
C
+
F
NC
=
−
∇
V
+
F
NC
.
Note
The
gradient
operator,
∇
The
gradient
operator,
∇
(called
“del”),
in
cartesian
coordinates
is
defined
as
∂
( )
∂
( )
∂
( )
∇
( )
≡
∂
x
i
+
∂
y
j
+
∂
z
k
.
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 Conservation Of Energy, Force, Potential Energy, work and energy

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