Environmental Engineering Reference
In-Depth Information
This shows that while work is done by gravity, no work is done by the forces
of constraint. This is a specific example of a more general result, which states
that, for many systems, the work done by the forces of constraint is zero. In the
roller-coaster example the work done depends on the change in height, as it did
with the free projectile. This is an important simplification, which allows us to
calculate the kinetic energy at any point on the path knowing only the height,
despite the fact that we know neither the detailed equation of the path, nor any
details about the forces keeping the roller coaster on the rails.
Power is defined as the rate at which work is done. If a force
F
acts on a
body, which undergoes a displacement d
r
, then the infinitesimal work done is
d
W
d
r
. The instantaneous power
P
is simply this work divided by the time
interval d
t
in which the displacement occurs
=
F
·
d
W
d
t
d
r
d
t
=
P
=
=
F
·
F
·
v
.
(3.13)
In the SI system the unit of power is the watt
3
(W):
1Js
−
1
.
1W
=
3.2
POTENTIAL ENERGY
In the previous section we defined work in terms of the integral over a path
between two points. We also proved that for a uniform gravitational field, the work
done is proportional to the difference in height between the two points, but does
not depend on the path taken between them. Such dependence on the initial and
final positions but not on the path taken is the defining feature of a conservative
field of force.
A field of force
F
(
r
) is conservative if the work done in going between positions
r
A
and
r
B
is independent of the path taken. A field of force that is not conservative
is known as non-conservative or dissipative.
A corollary to the above definition is that if a path ends at the starting point to
form a closed loop the work done is always zero for a conservative force.
Gravity and electrostatic forces provide us with two examples of conservative
forces whilst friction and air-resistance are both non-conservative. It is easy to see
that the work done by friction must always be path-dependent. A longer path will
result in more work being done since the frictional force always acts in opposition
to the motion. For such a situation we obtain
B
µ
k
N
B
A
W
=
F
·
d
r
=−
d
r
=−
µ
k
Nl ,
(3.14)
A
where
l
is the length of the path and
µ
k
is the coefficient of kinetic friction. Notice
that, even for a closed loop like that illustrated in Figure 3.6, the work done by
friction is still negative (which simply means that the kinetic energy reduces).
3
James Watt (1736 - 1819).
