fluid_mechanics

68 Chapter 2 ■ Fluid Statics
COMMENT An inspection of this result will show that the line
of action of the resultant force passes through the center of the conduit.
In retrospect, this is not a surprising result since at each point
on the curved surface of the conduit the elemental force due to the
pressure is normal to the surface, and each line of action must pass
through the center of the conduit. It therefore follows that the resultant
of this concurrent force system must also pass through the center
of concurrence of the elemental forces that make up the system.
This same general approach can also be used for determining the force on curved surfaces
of pressurized, closed tanks. If these tanks contain a gas, the weight of the gas is usually negligible
in comparison with the forces developed by the pressure. Thus, the forces 1such as F 1 and F 2
in Fig. 2.23c2 on horizontal and vertical projections of the curved surface of interest can simply
be expressed as the internal pressure times the appropriate projected area.
F l u i d s i n t h e N e w s
Miniature, exploding pressure vessels Our daily lives are safer
because of the effort put forth by engineers to design safe, lightweight
pressure vessels such as boilers, propane tanks, and pop
bottles. Without proper design, the large hydrostatic pressure
forces on the curved surfaces of such containers could cause the
vessel to explode with disastrous consequences. On the other
hand, the world is a more friendly place because of miniature pressure
vessels that are designed to explode under the proper conditions—popcorn
kernels. Each grain of popcorn contains a small
amount of water within the special, impervious hull (pressure vessel)
which, when heated to a proper temperature, turns to steam,
causing the kernel to explode and turn itself inside out. Not all
popcorn kernels have the proper properties to make them pop well.
First, the kernel must be quite close to 13.5% water. With too little
moisture, not enough steam will build up to pop the kernel; too
much moisture causes the kernel to pop into a dense sphere rather
than the light fluffy delicacy expected. Second, to allow the pressure
to build up, the kernels must not be cracked or damaged.
2.11 Buoyancy, Flotation, and Stability
(Photograph courtesy of
Cameron Balloons.)
V2.6 Atmospheric
buoyancy
2.11.1 Archimedes’ Principle
When a stationary body is completely submerged in a fluid 1such as the hot air balloon shown in
the figure in the margin2, or floating so that it is only partially submerged, the resultant fluid force
acting on the body is called the buoyant force. A net upward vertical force results because pressure
increases with depth and the pressure forces acting from below are larger than the pressure
forces acting from above. This force can be determined through an approach similar to that used
in the previous section for forces on curved surfaces. Consider a body of arbitrary shape, having
a volume V, that is immersed in a fluid as illustrated in Fig. 2.24a. We enclose the body in a parallelepiped
and draw a free-body diagram of the parallelepiped with the body removed as shown
in Fig. 2.24b. Note that the forces F 1 , F 2 , F 3 , and F 4 are simply the forces exerted on the plane
surfaces of the parallelepiped 1for simplicity the forces in the x direction are not shown2, w is the
weight of the shaded fluid volume 1parallelepiped minus body2, and F B is the force the body is
exerting on the fluid. The forces on the vertical surfaces, such as F 3 and F 4 , are all equal and cancel,
so the equilibrium equation of interest is in the z direction and can be expressed as
F B F 2 F 1 w
If the specific weight of the fluid is constant, then
F 2 F 1 g1h 2 h 1 2A
(2.21)
where A is the horizontal area of the upper 1or lower2 surface of the parallelepiped, and Eq. 2.21
can be written as
F B g1h 2 h 1 2A g31h 2 h 1 2A V4
Simplifying, we arrive at the desired expression for the buoyant force
F B gV
(2.22)