Understanding Fluid Mechanics: Archimedes, Pascal, and Bernoulli

 
Fluids: Archimedes’ Principle, Pascal’s Law, Bernoulli’s Principle
 
Fluids and Buoyant Force
 
Archimedes’
Principle
 
“Any object completely or partially
submerged in a fluid experiences
an upward force equal in
magnitude to the weight of the
fluid displaced by the object.”
- 
Archimedes’ Principle
 
For Floating Objects
 
Buoyant Force:
 
For Floating Objects:
F
B
 = F
g (displaced)
 = m
f 
g
F
B
 = F
g (object)
 = m
o 
g
 
where m
f
 = mass of fluid displaced
 
Image source: 2008 Yupi666, Wikimedia Commons
http://commons.wikimedia.org/wiki/File:Buoyancy.svg
Archimedes’ Principle:
The buoyant force is equal to the
weight of the displaced water.
 
Image source: Bradley W. Carroll. Used with permission.
http://physics.weber.edu/carroll/archimedes/principle.htm
 
Image source: Bradley W. Carroll. Used with permission.
http://physics.weber.edu/carroll/archimedes/principle.htm
ball:
 displaced water weighs less than the ball
hull:
 displaced water weight equals hull weight
 
Buoyant Force
 
Buoyant force 
is also equal to the difference between the
weight of an object in air and weight of an object in fluid.
F
B
 = W
air
 - W
fluid
 
In other words, the apparent loss in weight of a body
immersed in a fluid is equal to the weight of the displaced
fluid.
 
Image source: Bill Winfield. Used with permission.
 
Other Relationships
 
Net force (F
net
) 
is the object’s apparent weight:
F
net
 = F
B
 – F
g (object)
F
net
 = 
f
v
f 
- 
ρ
o
v
o
) g
 
In solving buoyancy problems, the following derived expression is used:
 
where: m = 
ρ
v
 
Pascal’s
Law
 
Pressure
 
Pressure
 is a measure of how much force is applied over a given
area.
 
units
:
1 Pa (Pascal) = 1 N/m
2
1 atm = 105 Pa
 
“Pressure applied to a fluid
in a closed container is
transmitted equally to every
point of the fluid and to
the walls of the container.”
- 
Pascal’s Law
 
o
Pressure applied
anywhere to a fluid
causes a force to be
transmitted equally in all
directions.
o
Change in pressure
disperses equally
throughout the fluid.
o
Force acts at right angles
to any surface in contact
with the fluid.
 
Image source: Bill Winfield. Used with permission.
 
A
1
 = 1 m
2
F
1
 = 10 N
P
1
 = ___?
 
A
2
 
=
 
1
0
 
m
2
P
2
 
=
 
_
_
_
_
?
F
2
 
=
 
_
_
_
_
?
 
Bernoulli’s
Principle
 
Types of Fluid Flow
 
o
Laminar:
 When fluid particles move along the same smooth path.
The path is called a streamline.
 
Source: Wikimedia Commons 
http://commons.wikimedia.org/wiki/File:Toky.png
 
Types of Fluid Flow
 
o
Turbulent:
 When fluid particles flow irregularly causing changes
in velocity. They form eddy currents.
 
Source: Wikimedia Commons 
http://commons.wikimedia.org/wiki/File:Toky.png
 
Continuity equation:
A
1
v
1
 = A
2
v
2
P + ½ 
ρ
v
2
 + 
ρ
gh = constant
 
Bernoulli’s equation:
 
“The pressure in a fluid decreases
as the fluid’s velocity increases.”
- 
Bernoulli’s Principle
 
Bernoulli’s equation at different points
in a horizontal pipe:
 
Image source: 2013 Emily Sappington, University of Houston
 
Point 1
 
Point 3
 
Point 2
P
1
 + ½ 
ρ
v
1
2
 = P
2
 + ½ 
ρ
v
2
2
 
Bernoulli’s Equation
 
Source: NASA 
http://www.grc.nasa.gov/WWW/k-12/airplane/bern.html
 
Bernoulli’s equation at two different
points of varying height
P
1
 + ½ 
ρ
v
1
2
 + 
ρ
gh
1
 = P
2
 + ½ 
ρ
v
2
2
 + 
ρ
gh
2
 
Source: 
http://commons.wikimedia.org/wiki/File:BernoullisLawDerivationDiagram.svg
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Explore the principles of fluid mechanics including Archimedes' Principle, Pascal's Law, and Bernoulli's Principle. Learn about mass density, buoyant force, and the behavior of floating objects in liquids. Discover how the buoyant force is equal to the weight of displaced fluids, and how it affects objects submerged in liquids. Dive into the fascinating world of fluids and their interaction with various forces.


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  1. Fluids: Archimedes Principle, Pascals Law, Bernoullis Principle

  2. Fluids and Buoyant Force Fluids: Matter that flows (liquid and gas). Mass Density: Mass per unit volume of a substance. It is often represented by the Greek letter (rho). = m o o v Buoyant Force: The upward force on objects that are partially or completely submerged in fluids. o

  3. Archimedes Principle

  4. Any object completely or partially submerged in a fluid experiences an upward force equal in magnitude to the weight of the fluid displaced by the object. - Archimedes Principle

  5. For Floating Objects Buoyant Force: FB= Fg (displaced)= mf g where mf= mass of fluid displaced For Floating Objects: FB= Fg (object)= mog

  6. Image source: 2008 Yupi666, Wikimedia Commons http://commons.wikimedia.org/wiki/File:Buoyancy.svg

  7. Archimedes Principle: The buoyant force is equal to the weight of the displaced water. Image source: Bradley W. Carroll. Used with permission. http://physics.weber.edu/carroll/archimedes/principle.htm

  8. ball: displaced water weighs less than the ball hull: displaced water weight equals hull weight Image source: Bradley W. Carroll. Used with permission. http://physics.weber.edu/carroll/archimedes/principle.htm

  9. Buoyant Force Buoyant force is also equal to the difference between the weight of an object in air and weight of an object in fluid. FB= Wair- Wfluid In other words, the apparent loss in weight of a body immersed in a fluid is equal to the weight of the displaced fluid.

  10. Image source: Bill Winfield. Used with permission.

  11. Other Relationships Net force (Fnet) is the object s apparent weight: Fnet= FB Fg (object) Fnet= ( fvf- ovo) g where: m = v In solving buoyancy problems, the following derived expression is used: = 0 f Fg (object) FB

  12. Pascals Law

  13. Pressure Pressure is a measure of how much force is applied over a given area. P = F A units: 1 Pa (Pascal) = 1 N/m2 1 atm = 105 Pa

  14. Pressure applied to a fluid in a closed container is transmitted equally to every point of the fluid and to the walls of the container. - Pascal s Law

  15. o Pressure applied anywhere to a fluid causes a force to be transmitted equally in all directions. o Change in pressure disperses equally throughout the fluid. o Force acts at right angles to any surface in contact with the fluid. A1= 1 m2 F1= 10 N P1= ___? A2= 10 m2 P2= ____? F2= ____? Image source: Bill Winfield. Used with permission.

  16. Bernoullis Principle

  17. Types of Fluid Flow o Laminar: When fluid particles move along the same smooth path. The path is called a streamline. Source: Wikimedia Commons http://commons.wikimedia.org/wiki/File:Toky.png

  18. Types of Fluid Flow o Turbulent: When fluid particles flow irregularly causing changes in velocity. They form eddy currents. Source: Wikimedia Commons http://commons.wikimedia.org/wiki/File:Toky.png

  19. The pressure in a fluid decreases as the fluid s velocity increases. - Bernoulli s Principle Continuity equation: A1v1= A2v2 Bernoulli s equation: P + v2+ gh = constant

  20. Bernoullis equation at different points in a horizontal pipe: P1+ v12= P2+ v22 Point 3 Point 1 Point 2 Image source: 2013 Emily Sappington, University of Houston

  21. Bernoullis Equation Source: NASA http://www.grc.nasa.gov/WWW/k-12/airplane/bern.html

  22. Bernoullis equation at two different points of varying height P1+ v12+ gh1= P2+ v22+ gh2 Source: http://commons.wikimedia.org/wiki/File:BernoullisLawDerivationDiagram.svg

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