1. The document discusses hydrostatic pressure, including how pressure increases linearly with depth in static fluids and how to calculate hydrostatic pressure. 2. An experiment is described to determine the center of pressure on a submerged plane using a balance and measuring water levels and weights. Theoretical calculations of center of pressure are shown for total and partial immersion. 3. Procedures are provided for conducting the experiment, collecting data on water levels and weights, and comparing results to theoretical values to analyze differences.

1. Hydrostatic Pressure
Instructor: Seo, Il Won (35-310)
Web: www.ehlab.snu.ac.kr
Objectives
Objectives
Chapter 1 Hydrostatic Pressure
• To determine the center position of pressure of a plane surface
Chapter 1. Hydrostatic Pressure
immersed in water
T th i t l iti ith th th ti l
• To compare the experimental position with the theoretical
results
Background theory
Background theory
• Fluid pressure ?
What is fluid
statics…?
The pressure at some point within a fluid
• Fluid statics ?
- No relative motion between fluid elements
- No shear stress
Can’t support shearing stresses
Pressure must be acting on
g
normal to surface
Background theory
Background theory
• Hydrostatic pressure ?
y p
The pressure at any given point of a static fluid is called
the hydrostatic pressure
457.204 Elementary Fluid Mechanics and Lab.
Elementary Test

2. Background theory
Background theory
• How to calculate hydrostatic pressure ?
h ( : specific gravity, : depth)
p h
γ
= γ h
Why?
• No shear stress
& Acting normal to surface
Why?
• Force Equilibrium
F F
2
z
F
0
2 2
x
F
p dx p dx
p p dz
x x
⎡ ⎤
∂ ∂
⎛ ⎞ ⎛ ⎞
= − − + =
⎜ ⎟ ⎜ ⎟
⎢ ⎥
∂ ∂
⎝ ⎠ ⎝ ⎠
⎣ ⎦
∑
⎡ ⎤
⎛ ⎞ ⎛ ⎞
1
x
F 2
x
F
0
2 2
z
F
p dz p dz
p p dx dW
z z
⎡ ⎤
∂ ∂
⎛ ⎞ ⎛ ⎞
= − − + − =
⎜ ⎟ ⎜ ⎟
⎢ ⎥
∂ ∂
⎝ ⎠ ⎝ ⎠
⎣ ⎦
∑
1 2
z z
F F dW
= +
•
•
•
Background theory
Background theory
• How to calculate hydrostatic pressure ?
h ( : specific gravity, : depth)
p h
γ
= γ h
Eventually!
p
∂ P i i h i l l
: 0
x
F
p
x
∂
=
∂
∑ Pressure is constant in a horizontal plane
in a static fluid
:
z
F
p
g
z
ρ γ
∂
= =
∂
∑ Increase of pressure with depth in a fluid of
constant density – linearly increase
constant density linearly increase
p z
γ
Δ = Δ
Background theory
Background theory
• Summary
2. Hydrostatic pressure:
Increase linearly with depth
1. Fluid at rest:
Pressure must be acting on
3. Same at all points on a
horizontal plane
Pressure must be acting on
normal to the surface
Example of
Example of hydrostatic
hydrostatic pressure
pressure
• Hydroelectric dam
F diff i h d h bi
From pressure difference with depth, water gets energy to operate turbine.

3. Example of
Example of hydrostatic
hydrostatic pressure
pressure
B t l k & h d t ti
• Basement leaks & hydrostatic pressure
Backfill becomes saturated
Hydrostatic pressure makes a crack
at the basement wall
at the basement wall
Water infiltrates the basement wall
Example of
Example of hydrostatic
hydrostatic pressure
pressure
• There are many kinds of problems….
- Which point should we choose to intake water from dam?
Where should we release water to get sufficient energy?
- Where should we release water to get sufficient energy?
- How can we construct safe dam?
- How can we protect water leak problem?
Basic solutions of these problems come from understanding hydrostatic pressure
s c so u o s o ese p ob e s co e o u de s d g yd os c p essu e
Hydrostatic
Hydrostatic pressure
pressure on
on submerged
submerged plane
plane
• Pressure distribution diagram
Pressure is linearly increase from water surface to bottom
Pressure is linearly increase from water surface to bottom
h
1. What is force, F?
2. Where is the point
f i ?
1
h
1 1
p gh
ρ
=
?
z =
X
of action?
?
F =
Center of the area
Hydrostatic
Hydrostatic pressure
pressure on
on submerged
submerged plane
plane
• Total force acting on surface
- Pressure at
p gh
ρ
=
h
- Force acting on surface
1
h 1 1
p gh
ρ
=
?
z =
F dA X dA
∫ ∫
X
p g
ρ
?
F =
F pdA gX dA
gXA
ρ
ρ
= =
=
∫ ∫
r
- First moments of areas
X dA X A
=
∫
Force X dA X A
=
∫
Force
= Sum of pressure
Center of the pressure

4. Hydrostatic
Hydrostatic pressure
pressure on
on submerged
submerged plane
plane
• Position of the total force
O
- Moment at O
2
oo
M X g dA gI Fz
ρ ρ
= = =
∫
1
h
1 1
p gh
ρ
=
X
O
(Second moment of area: )
oo oo
M g I I
z
ρ
∴ = = =
2
oo
X dA I
=
∫
?
z =
X
r
d
z
F g AX AX
ρ
∴ = = =
- From parallel axis theorem
3
F
C
2
oo gg
I I AX
= +
3
12
gg
bd
I =
(Rectangular : )
( )
2
3
/12
/ 2 2
gg gg
oo
I AX I
I bd d
z X r
bd r d
AX AX AX
+ ⎛ ⎞
∴ = = = + = + −
⎜ ⎟
× − ⎝ ⎠
When t=0, Center of pressure is located on 2/3 down
from water surface to bottom of submerged plate
Observation
Observation for
for Total
Total Immersion
Immersion
- In case of rectangular plane
d
2
d
X r
= −
- Resultant Force
F gXA
ρ
=
- Center point of pressure
g
ρ
from water surface
( )
3
/12
/ 2 2
bd d
z r
bd r d
⎛ ⎞
= + −
⎜ ⎟
× − ⎝ ⎠
from pivot
( ) ⎝ ⎠
( )
3
/12
/ 2 2
c
bd d
X r q
bd d
⎛ ⎞
= + − +
⎜ ⎟
⎝ ⎠
o p vo
( Fulcrum)
( )
/ 2 2
c q
bd r d
⎜ ⎟
× − ⎝ ⎠
Theoretical
Theoretical Value
Value for
for Total
Total Immersion
Immersion
Moment arm
Force 1
Force 1
r
Force 2
1
1 2 1 2
2
c c
F L
M M F L F X X
F
= → = → =
1
F mg
=
( ) ( )
2
1 1
2 2
F gr r b g r d r d b
ρ ρ
= × × × − × − × − ×
p gh
ρ
=
r
d
2
2 2
g g
ρ ρ
Observation
Observation for
for Partial
Partial Immersion
Immersion
Moment arm
Force 1
y
- Center point of pressure
Force 2
3
y
Center point of pressure
2
z y
=
y
X a d
= + −
3
z y
3
c
X a d
+
from water surface from pivot (Fulcrum)

5. Theoretical
Theoretical Value
Value for
for Partial
Partial Immersion
Immersion
Moment arm
Force 1
y
Force 2
3
y
1
1 2 1 2
2
c c
F L
M M F L F X X
F
= → = → =
p g y
ρ
1
F mg
=
2
by
p g y
ρ
=
y
2
2
by
F g X A g
ρ ρ
= =
Principle
Principle of
of Experiment
Experiment
• Place a weight until the balance arm is horizontal.
• Record the water level on the quadrant and the weight on the balance pan.
q g p
• Repeat this procedure until the water level reaches the top of the quadrant.
Procedure
Procedure
① Place the quadrant on the two dowel pins and using the clamping screw,
fasten to the balance arm.
② Measure values of a, L, depth d and width b of the quadrant end face.
③ With the Perspex tank on the bench, position the balance arm on the knife edges (pivot).
③ W e e spe o e be c , pos o e b ce o e e edges (p vo ).
④ Hang the balance pan from the end of the balance arm.
⑤ Connect a hose from the drain cock to the sump and another from the bench feed to the
t i l t th t f th P t k
triangular aperture on the top of the Perspex tank.
⑥ Level the tank using the adjustable feet and spirit level.
Procedure
Procedure
⑦ Move the counter balance weight until the balance arm is horizontal.
⑧ Close the drain cock and admit water until the level reaches the bottom edge of the quadrant.
⑨ Place a weight on the balance pan, slowly adding water into the tank
until the balance arm is horizontal.
until the balance arm is horizontal.
⑩ Record the water level on the quadrant and the weight on the balance pan.
⑪ Repeat the above for each increment of weight until the water surface level reaches the top
f h d d f Th h i f i h i i h d
of the quadrant end face. Then remove each increment of weight noting weights and water
levels until the weights have been removed.

6. Results
Results
a
Arm
L
depth
d
Width
b
Mass
m
Measured
length
Theoretical
Value
(m)
L
(m)
d
(m)
b
(m)
m
(kg) cm
X ct
X
Discussion
Discussion
1 Confirm that the center of pressure is below the center of area
1. Confirm that the center of pressure is below the center of area,
and discuss why.
2. Compare the theoretical value with measured length,
discuss why does the difference occurs.
3. Discuss why we do not consider the pressure at the surface of arc.
Why we just calculate the pressure of the rectangular section?
y j p g