lecture no 11 & 12 hydrostatic force and pressure on plates
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Lecture no 11 & 12 HYDROSTATIC FORCE AND PRESSURE ON PLATES. Prepared by Engr.Sarafaraz Khan Turk Lecturer at IBT LUMHS Jamshoro. HYDROSTATIC FORCE AND PRESSURE. - PowerPoint PPT PresentationTRANSCRIPT
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Lecture no 11 & 12
HYDROSTATIC FORCE AND PRESSURE ON PLATES
Prepared by
Engr.Sarafaraz Khan Turk
Lecturer at IBT LUMHS Jamshoro
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HYDROSTATIC FORCE AND PRESSURE
Fluid statics or hydrostatics is the branch of
fluid mechanics that studies fluids at rest. It
embraces the study of the conditions under
which fluids are at rest in stable equilibrium; and
is contrasted with fluid dynamics, the study of
fluids in motion
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HYDROSTATIC FORCE AND PRESSURE
The hydrostatic pressure can be determined from a control
volume analysis of an infinitesimally small cube of fluid. Since
pressure is defined as the force exerted on a test area (p = F/A,
with p: pressure, F: force normal to area A, A: area), and the
only force acting on any such small cube of fluid is the weight of
the fluid column above it, hydrostatic pressure can be calculated
according to the following formula:
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HYDROSTATIC FORCE AND PRESSURE
where:
p is the hydrostatic pressure (Pa),
ρ is the fluid density (kg/m3),
g is gravitational acceleration (m/s2),
A is the test area (m2),
z is the height (parallel to the direction of gravity) of
the test area (m),
z0 is the height of the zero reference point of the
pressure (m).
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Scuba Diving and Hydrostatic Pressure
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Deep-sea divers realize that water
pressure increases as they dive deeper.
This is because the weight of the water above them increases.
HYDROSTATIC FORCE AND PRESSURE
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Pressure on diver at
100 ft?
Danger of emergency
ascent?
,2 3 2
,2 ,2
1998 9.81 100
3.28
1298.5 2.95
101.325
2.95 1 3.95
gage
abs gage atm
kg m mP gz ft
m s ft
atmkPa atm
kPa
P P P atm atm atm
Scuba Diving and Hydrostatic Pressure
1 1 2 2
1 2
2 1
3.954
1
PV PV
V P atm
V P atm
100 ft
1
2
Boyle’s law
If you hold your breath on ascent, your lungvolume would increase by a factor of 4, which would result in embolism and/or death.
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Suppose that a thin plate with area A m2 is
submerged in a fluid of density ρ kg/m3 at a
depth d meters below the surface of the fluid.
HYDROSTATIC FORCE AND PRESSURE
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The fluid directly above the plate has volume
V = Ad
So, its mass is:
m = ρV = ρAd
HYDROSTATIC FORCE AND PRESSURE
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Thus, the force exerted by the fluid on
the plate is
F = mg = ρgAd
where g is the acceleration due to gravity.
HYDROSTATIC FORCE
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The pressure P on the plate is defined
to be the force per unit area:
HYDROSTATIC PRESSURE
FP gd
A
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The SI unit for measuring pressure is newtons
per square meter—which is called a pascal
(abbreviation: 1 N/m2 = 1 Pa).
As this is a small unit, the kilopascal (kPa) is often used.
HYDROSTATIC PRESSURE
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For instance, since the density of water is
ρ = 1000 kg/m3, the pressure at the bottom
of a swimming pool 2 m deep is:
HYDROSTATIC PRESSURE
3 21000kg/m 9.8m/s 2m
19,600Pa
19.6kPa
P gd
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An important principle of fluid pressure is
the experimentally verified fact that, at any
point in a liquid, the pressure is the same in
all directions.
This is why a diver feels the same pressure on nose and both ears.
HYDROSTATIC PRESSURE
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Thus, the pressure in any direction at
a depth d in a fluid with mass density ρ
is given by:
HYDROSTATIC PRESSURE
P gd d
Equation 1
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This helps us determine the hydrostatic
force against a vertical plate or wall or dam
in a fluid.
This is not a straightforward problem.
The pressure is not constant, but increases as the depth increases.
HYDROSTATIC FORCE AND PRESSURE
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A dam has the shape of the trapezoid
shown below.
The height is 20 m. The width is 50 m at the top and 30 m at the bottom.
HYDROSTATIC F AND P Example 1
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Find the force on the dam due to
hydrostatic pressure if the water level
is 4 m from the top of the dam.
HYDROSTATIC F AND P Example 1
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We choose a vertical x-axis with origin
at the surface of the water.
HYDROSTATIC F AND P Example 1
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The depth of the water is 16 m.
So, we divide the interval [0, 16] into subintervals of equal length with endpoints xi.
We choose xi* [xi–1, xi].
HYDROSTATIC F AND P Example 1
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The i th horizontal strip of the dam is
approximated by a rectangle with height Δx
and width wi
HYDROSTATIC F AND P Example 1
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From similar triangles,
HYDROSTATIC F AND P
* *
*
1610or 8
16 20 2 2i i
i
x xaa
x
Example 1
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Hence,
*12
*
2(15 )
2(15 8 )
46
i
i
i
w a
x
x
HYDROSTATIC F AND P Example 1
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If Ai is the area of the strip, then
If Δx is small, then the pressure Pi on the i th
strip is almost constant, and we can use
Equation 1 to write:
HYDROSTATIC F AND P
*(46 )i i iA w x x x
*1000i iP gx
Example 1
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The hydrostatic force Fi acting on the i th
strip is the product of the pressure and
the area:
HYDROSTATIC F AND P
* *1000 (46 )
i i i
i i
F PA
gx x x
Example 1
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Adding these forces and taking the limit as
n → ∞, the total hydrostatic force on the dam
is:
HYDROSTATIC F AND P
* *
1
16
0
16 2
0
1632 7
0
lim 1000 (46 )
1000 (46 )
1000(9.8) (46 )
9800 23 4.43 10 N3
n
i ini
F gx x x
gx x dx
x x dx
xx
Example 1