fluid_mechanics

2.11 Buoyancy, Flotation, and Stability 69
c
h 2
h 1
A
B
z
F B
Centroid
of displaced
volume
x
y
(d)
D
(a)
C
Centroid
c
y 1
F 1
A
y 2
y c
B
F B
F 3
F 4
FB
D
C
(c)
F 2
(b)
F I G U R E 2.24
Buoyant force on submerged and floating bodies.
V2.7 Cartesian
Diver
Archimedes’ principle
states that the
buoyant force has a
magnitude equal to
the weight of the
fluid displaced by
the body and is
directed vertically
upward.
where g is the specific weight of the fluid and V is the volume of the body. The direction of the
buoyant force, which is the force of the fluid on the body, is opposite to that shown on the freebody
diagram. Therefore, the buoyant force has a magnitude equal to the weight of the fluid displaced
by the body and is directed vertically upward. This result is commonly referred to as
Archimedes’ principle in honor of Archimedes 1287–212 B.C.2, a Greek mechanician and mathematician
who first enunciated the basic ideas associated with hydrostatics.
The location of the line of action of the buoyant force can be determined by summing moments
of the forces shown on the free-body diagram in Fig. 2.24b with respect to some convenient axis. For
example, summing moments about an axis perpendicular to the paper through point D we have
and on substitution for the various forces
F B y c F 2 y 1 F 1 y 1 wy 2
Vy c V T y 1 1V T V2 y 2
(2.23)
where V T is the total volume 1h 2 h 1 2A. The right-hand side of Eq. 2.23 is the first
moment of the displaced volume V with respect to the x–z plane so that y c is equal to the y coordinate
of the centroid of the volume V. In a similar fashion it can be shown that the x coordinate
of the buoyant force coincides with the x coordinate of the centroid. Thus, we conclude that
the buoyant force passes through the centroid of the displaced volume as shown in Fig. 2.24c.
The point through which the buoyant force acts is called the center of buoyancy.
F l u i d s i n t h e N e w s
Concrete canoes A solid block of concrete thrown into a pond or
lake will obviously sink. But, if the concrete is formed into the
shape of a canoe it can be made to float. Of course the reason the
canoe floats is the development of the buoyant force due to the
displaced volume of water. With the proper design, this vertical
force can be made to balance the weight of the canoe plus passengers—the
canoe floats. Each year since 1988 there is a National
Concrete Canoe Competition for university teams. It’s jointly
sponsored by the American Society of Civil Engineers and Master
Builders Inc. The canoes must be 90% concrete and are typically
designed with the aid of a computer by civil engineering students.
Final scoring depends on four components: a design report, an
oral presentation, the final product, and racing. For the 2007 competition
the University of Wisconsin’s team won for its fifth consecutive
national championship with a 179-lb, 19.11-ft canoe.
(See Problem 2.107.)