Graphics Reference
In-Depth Information
35.6.4.2 Buoyancy
Buoyancy arises from pressure within a fluid medium, such as air or water. It is
only observed when the medium and objects within it are exposed to a common
external acceleration, such as that due to gravity.
Let
v
i
be
3
F
b
(
x, i
)
the density (kg/m
3
)ofthe
medium, and
g
the net gravitational acceleration (m/s
2
) at the location of the
object. Assume that
g
is constant in the region of interest. Furthermore, assume
that both the medium and the object are
incompressible,
meaning that their vol-
umes do not change significantly under pressure. Water is incompressible, as are
many of the things that one might expect to find floating in it, such as wood, buoys,
boats, fish, and people.
the volume (in m
3
) of an object
i
,
ρ
y
i
i
r
If object
i
is not in the medium, the buoyancy force that it experiences is obvi-
ously
F
b
(
x
,
i
)=
0 N. If it is submerged (Figure 35.22), then the buoyancy force
is
Figure 35.22: Buoyancy on an
object with volume v
i
submerged
in a fluid with density
ρ
.
F
b
(
x
,
i
)=
−ρ
v
i
g
.
(35.41)
Note that the density of the object does not appear. A dense object will experi-
ence less upward acceleration against gravity because of the high force of gravity
on the object, not because of a reduction in its buoyancy force. Thus, a “buoyant”
object counterintuitively experiences no greater “buoyancy” force than a nonbuoy-
ant one.
The medium of course experiences a force of equal magnitude and opposite
direction. However, the medium is by definition a fluid composed of individual
freely moving particles, so it is internally a very loosely coupled system. In prac-
tice, for animation one often deals with very large bodies of fluid and neglects the
impact of buoyancy on them.
For a sense of scale, the density of liquid water is about 1,000 kg/m
3
at 4
◦
C,
and falls slightly as it is heated or cooled. That this number is exactly a power of
ten is no accident—it is one of the constants around which the metric system was
originally designed.
Note that the mass and density of the submerged object do not appear. So
why do dense objects sink and less dense ones float? Because the
net
force also
depends on gravity:
f
i
=
F
g
(
i
)+
F
b
. If the object is dense, then
F
g
(
i
)
is large
compared to
F
b
and a net downward acceleration is observed.
When the fluid is not at rest, different parts of the fluid exert differing pressure
on itself and the objects within it. This leads to wave and vortex phenomena, both
at the surface and within the fluid.
i
F
s
(
y, i, j
)
F
s
(
y, i, j
)
i
35.6.4.3 Springs
Consider an ideal spring that has no mass of its own and that always returns to
its rest length after being stretched or compressed and then released. Assume that
this spring also only expands and contracts along its “length” axis and is perfectly
rigid along axes orthogonal to the length.
j
j
Compressed
Extended
Let the spring connect objects
i
and
j
(Figure 35.23). Let the spring have
rest length
r
, which means that the spring exerts no force on its ends when
Figure
35.23:
Spring
force
between two bodies.
3. Note that volume
v
i
is distinct from
v
and
x
, the variables used for velocity in this
chapter.
