2.2 Water at Rest: Hydrostatics
21
Substituting Eq. (2.5) and using the fact that z = -h, we find that the piezometric head, H, is a constant throughout the fluid, i. e.:
H = Pa .
(2.7)
Pwg
If water is kept in containers being filled to the same height, the pressure on
the base is the same and independent on shape of the container (Fig. 2.2c).
The pressure at the container's base is controlled by the depth, h, not by the
weight of water in container as given by Eq. (2.4). The resultant force acting
on the container's base is a product of the corresponding pressure, P, and base
surface A:
F =pA.
(2.8)
Sometimes the pressure at a given point in the water column is calculated
relative to the atmospheric pressure. Therefore, pressure at point P simply is
p = Pwg h .
From Eq. (2.4) it can be deduced that the pressure difference between two
points of different submergence, hI and h2 (hI> h2)' is:
(2.9)
2.2.2 Buoyancy
The forces on submerged objects determine whether they sink, float or remain
where they are. These three stages are usually categorized as negative, positive
or neutral buoyancy. To examine the nature of the buoyancy force, let us
consider a vertical submerged cylinder situated in a large volume of water
(Fig. 2.3). Using Eq. (2.4), and assuming that Pa = 0, the vertical forces on
cylinder top and bottom are respectively:
top:
bottom:
Ft = 7rpwg h t 1'2 }.
Fb= 7rPw9hb1'2
(2.10)
When the cylinder weight is neglected, the net buoyancy force acting on the
cylinder is obtained as:
(2.11 )
where 1 is the cylinder height. This force acts in an upward direction and is
exactly equal to the weight of water contained in the cylinder and does not
depend on the depth of submergence. This conclusion can be extended to any
completely submerged or floating body using the law of buoyancy discovered
by Archimedes in the third century B.C.:
21
Substituting Eq. (2.5) and using the fact that z = -h, we find that the piezometric head, H, is a constant throughout the fluid, i. e.:
H = Pa .
(2.7)
Pwg
If water is kept in containers being filled to the same height, the pressure on
the base is the same and independent on shape of the container (Fig. 2.2c).
The pressure at the container's base is controlled by the depth, h, not by the
weight of water in container as given by Eq. (2.4). The resultant force acting
on the container's base is a product of the corresponding pressure, P, and base
surface A:
F =pA.
(2.8)
Sometimes the pressure at a given point in the water column is calculated
relative to the atmospheric pressure. Therefore, pressure at point P simply is
p = Pwg h .
From Eq. (2.4) it can be deduced that the pressure difference between two
points of different submergence, hI and h2 (hI> h2)' is:
(2.9)
2.2.2 Buoyancy
The forces on submerged objects determine whether they sink, float or remain
where they are. These three stages are usually categorized as negative, positive
or neutral buoyancy. To examine the nature of the buoyancy force, let us
consider a vertical submerged cylinder situated in a large volume of water
(Fig. 2.3). Using Eq. (2.4), and assuming that Pa = 0, the vertical forces on
cylinder top and bottom are respectively:
top:
bottom:
Ft = 7rpwg h t 1'2 }.
Fb= 7rPw9hb1'2
(2.10)
When the cylinder weight is neglected, the net buoyancy force acting on the
cylinder is obtained as:
(2.11 )
where 1 is the cylinder height. This force acts in an upward direction and is
exactly equal to the weight of water contained in the cylinder and does not
depend on the depth of submergence. This conclusion can be extended to any
completely submerged or floating body using the law of buoyancy discovered
by Archimedes in the third century B.C.: