Environmental Engineering Reference
In-Depth Information
B
C
d
r
F
friction
F
friction
A
D
d
r
Figure 3.6
The work done against friction is negative, even for a closed loop.
What if all of the forces acting on a particle are conservative? In that case we are
free to construct a special function of position, which we call the potential energy,
U(
r
)
.Itisdefinedby
U(
r
B
)
−
U(
r
A
)
=−
W
AB
,
(3.15)
which is the negative of the work done by the net force in going from
A
to
B
.
Note that no such special function can be found for non-conservative forces because
the work done in going from
A
to
B
depends on more than just the position of
the end-points. The motivation for taking the negative of the work done in the
definition is because, when combined with Eq. (3.8), we get
U(
r
B
)
−
U(
r
A
)
=
T
A
−
T
B
and therefore
T
A
+
U(
r
A
)
=
T
B
+
U(
r
B
).
(3.16)
Thus for motion of a particle under the influence of conservative forces only we
see that the sum of the kinetic and potential energy is a conserved quantity. We
call this sum the mechanical energy,
E
:
1
2
mv
2
E
=
+
U(
r
).
(3.17)
Note that the opposite sign in Eq. (3.15) would be equally valid, we would just have
to flip the sign of the potential energy in Eq. (3.16) and it would be the difference
rather than the sum of the kinetic and potential energies that would be conserved.
Also note that Eq. (3.15) does not uniquely define the function
U(
r
)
, since it
is possible to add any constant to it and yet still maintain the same difference in
potential energy between any two given points. The mechanical energy of a particle
is therefore not an absolute quantity but rather it is always defined relative to an
arbitrary position of zero potential energy.
