Environmental Engineering Reference
In-Depth Information
when the ballerina's feet are in contact with the ground and Figure 3.12(b) refers
to the period when the ballerina is in the air. In the former case, the only forces
external to the ballerina's body are gravity and the normal force from the floor.
Thus, motion of her centre of mass upwards is only possible if the magnitude of the
normal force exceeds that of gravity. As soon as the ballerina leaves the floor the
normal force is reduced to zero and the acceleration takes on a value g downwards.
To view the jump in terms of energy we must identify how work is done. During the
upwards acceleration phase the kinetic energy of the ballerina is increasing. Since
the normal force from the floor is responsible for the upwards motion of the centre
of mass it is tempting to suppose that this force does the work that is responsible for
the increase in kinetic energy. This is incorrect: the force coming from the ground
cannot do any work because the point of application does not move. This is not
surprising because we know that the ballerina leaps into the air using her own
internal energy and not energy from the Earth. At first sight, this may seem strange
given that Newton's Third Law requires that all the internal forces sum to zero, i.e.
i,j
F
ij
0
. However, this statement does not imply that
i,j
F
ij
0
. This
is the crux of the matter: not all particles are displaced by the same amount (since
the ballerina constitutes a deformable system). For example, the atoms in the feet
do not move at all. Figure 3.12 shows the essential features of the mechanics. To
simplify matters we model the ballerina as a mass m sitting on a spring, which
initially sits on the floor (i.e. the ballerina's feet are at the base of the spring).
It is a crude model but one that allows us to introduce the internal forces into
the problem. The internal forces that drive the upwards-acceleration phase of the
body are a result of the tension
F
in the compressed spring. This corresponds to
the starting point of the saute, i.e. when the ballerina stands with her legs flexed.
Newton's Third Law dictates that there is a corresponding force acting on the feet,
and since the feet do not move this is also equal to the normal reaction from the
ground, i.e.
N
=
·
d
r
i
=
F
. As the spring extends, the body moves upwards a distance
d
x so
that the work done by the tension in the spring is F
d
x and the net work done on the
body is (F
=
mg)
d
x. Note that this is the result one would obtain
upon considering the free body diagram alone but now it is clear that the work
is done by the spring and not the ground. Notice also that in our model we have
a simple explanation for how the ballerina's feet leave the ground: as the spring
moves from compression to extension, the tension reverses direction and that leads
to a net upwards force on the feet.
−
mg)
d
x
=
(N
−
Conservation of energy in a complex system must therefore take account of the
fact that it is possible for the internal forces to do work. If the internal forces are
non-conservative then some of this work will be dissipated, e.g. it might lead to an
increase in the thermal energy of the system. Of course, external forces may also do
work on the system. Figure 3.13 illustrates how the different categories of force are
able to contribute to the total energy of a system. If we assume that all dissipated
energy becomes thermal energy, then we could write the law of conservation of
energy as
W
ext
=
+
+
K
U
int
E
thermal
,
(3.49)
