Environmental Engineering Reference
In-Depth Information
Forces
Internal
External
Conservative
Non-conservative
Potential energy
Dissipation
Work done on system
e.g. thermal energy
Figure 3.13
A scheme showing how different categories of force contribute to energy.
where
W
ext
is the work done on the system,
K
and
U
int
are the changes in kinetic
and internal potential energy, and
E
thermal
is the change in thermal energy. In the
example of the ballerina's saute, the force of gravity can be viewed as an external
force that does negative work on the ballerina (i.e. it takes energy to jump) and we
can therefore write
−
mg
d
x
=
K
−
U
int
(3.50)
as the ballerina increases the position of her centre of mass by a height d
x
. Thus
the kinetic energy can only increase if the work done by internal forces (i.e. the
internal potential energy lost) exceeds the magnitude of the work done by gravity.
Note that since the external force is conservative, we could also think of
W
ext
as a
change in external potential energy, i.e.
W
ext
=−
mg
d
x
.
We are at liberty to choose the boundary of a complex system, and this choice
determines whether forces are to be considered as internal or external. Largely this
is a matter of convenience. In the ballerina example we could have extended the
system boundary to include the Earth, which would have resulted in a description
of the saute purely in terms of internal forces. However, one should be careful
before choosing a system boundary such that there are non-conservative external
forces since the thermal energy generated will probably be distributed on both sides
of the system boundary, i.e. some part of
E
thermal
will be external to the system.
A more detailed study of thermal energy would leads us naturally to the subject of
thermodynamics, but that is outside of the scope of this topic.
U
ext
=−
PROBLEMS 3
3.1 A car of mass 900 kg travelling at 120 km per hour stops in 3.0 seconds.
Calculate the work done on the car by the road. Calculate also the power of
the braking system.
3.2 A force
F
=
3
.
0
i
−
2
.
0
j
(newtons) acts on a particle while it is displaced by
s
=−
5
.
0
i
−
1
.
0
j
(cm). Calculate the work done in units of joules.
3.3 The
“push-me-pull-you”
is
illustrated
in
the
figure.
It
consists
of
two
cars
of
equal
mass
(
m
)
connected
by
a
spring
of
negligible
mass.
