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Chapter 15: Fluids
Volume
Density
Pressure
Buoyancy
Floating
Archimedes Principle
Pascal’s Principle
Fluid Flow
Bernoulli’s Principle
Statics
Dynamics
States of Matter Solid Liquid Gas
What is the most common state of matter in the Universe?
Fluids: Liquids & Gases
•Fluids are substances that are free to flow.
•Atoms and molecules are free to move.
•They take the shape of their containers.
•Cannot withstand or exert shearing forces.
Liquids: Incompressible (density constant)
Gases: Compressible (density depends on pressure)
Parameters to describe Fluids:
Density: = mass/volume
Pressure: P = Force/Area
[P] = N/m2 = 1 Pascal (Pa)
Liquid Units
There are 1000 liters in 1 cubic meter!
1 liter = 10-3 m3 = 103 cm3
1 liter of water has a mass of 1 kg and a weight of 9.8N.
2 0 3
1 1000H
kg kg
liter m
Density • Density of water @4°C:
water = 1g/cm3 = 1000 kg/m3 = 1kg/liter
• Density of air @ 0°C:
Air = 1.29x10-3 g/cm3 = 1.29 kg/m3
Density depends on temperature! Most substances EXPAND upon heating.
m
V
How does that change their densities?
REDUCES DENSITY! m
V
m V
Water: The Exception
• Water @4°C: water =1000 kg/m3
• Ice @ 0°C: ice = 917 kg/m3
Pressure in a fluid is due to the weight
of a fluid. Force
PArea
mg
A
Pressure depends on Depth!
( )V g
A
( )Ah g
A
P gh
Pressure Acts ONLY
Perpendicularly to the Surface
Pressure depends on depth.
Pressure IN a Fluid
•Is due to the weight of the fluid above you
•Depends on Depth and Density Only
•Does NOT depend on how much water is present
•Acts perpendicular to surfaces (no shearing)
•Pressure’s add
•At a particular depth, pressure is exerted equally in ALL directions
including sideways (empirical fact)
Pressure Problem
What is the water pressure 15 m
below the surface of the lake?
Assume it is pure water.
P gh
3 21000 / (9.8 / )15kg m m s m
5 21.47 10 /x N m
147kPa 21 Pascal 1 /N m
Pressure Units
What is the water pressure 150 m
below the surface of the lake?
Assume it is pure water.
P gh
3 21000 / (9.8 / )150kg m m s m
21.47 /N m
Pressure ON a Fluid
Liquids cannot be compressed to a smaller volume.
Liquids are incompressible.
Gases can be compressed to a smaller volume.
Gases are compressible.
The Atmosphere
At sea level,
the atmosphere
has a density of
about 1.29 kg/m3.
The average
density up to
120 km is about
8.59 x10-2 kg/m3.
The Atmosphere
A square meter
extending up through
the atmosphere has a
mass of about
10,000 kg and a weight
of about 100,000 N.
1 N/m2 is a Pascal.
51 1.013 10 14.7atm x Pa psi
Measuring Pressure 51 1.013 10atm x Pa
760h mm
13.6mercury water
mercuryP gh
mercury
Ph
g
2
3 2
101,300 /
13,600 / 9.8 /
N mh
kg m x m s
P gh
Why is the pressure at X equal to atmospheric pressure?
Because if it didn’t, the mercury would
be pushed out of the dish!
31000 /water kg m
Measuring Pressure
Can a barometer be made with Water instead of Mercury?
waterP gh
water
Ph
g
2
3 2
101,300 /
1000 / 9.8 /
N mh
kg m x m s
10.3h m
(Notice: 10.3m is just 13.6 x 760mm!)
13.6mercury water
31000 /water kg m
10.3m
Mercury Barometer Water Barometer
Not to Scale!!!
51 1.013 10atm x Pa 760mm
Barometers
Measuring Air Pressure Fluid in the tube adjusts until the weight of the fluid column
balances the atmospheric force exerted on the reservoir.
Absolute vs. Gauge Pressure
• Guage pressure is
what you measure in
your tires
• Absoulte pressure is
the pressure at B and
is what is used in
PV = nRT
0Guage Pressure: P gh
0Absolute Pressure: P P gh
Measuring Pressure
absolute guage atmosphereP P P
guageP gh
GUAGE PRESSURE is relative to atmospheric pressure.
It is found by measuring h with a manometer.
ABSOLUTE PRESSURE is the total pressure. Pressures ADD!
Water seeks its own level – unless there is pressure!
Tower Power
How high above the ground must the
water level be in the tower to create a
gauge pressure of 3 x 105 Pa
(~ 3 atm) in your house below?
waterP gh
water
Ph
g
5 2
3 2
3 10 /
1000 / 9.8 /
x N mh
kg m x m s
30.6h m
It takes approximately
4.5 psi (.3atm) for a one-
inch diameter pipe to
push a column of water
ten feet in the air, the
standard distance
between the floors of a
house.
Zero Pressure: Making A Vacuum Mechanical Vacuum Pump
Minimum pressure produced by mechanical pump:~1Pa
Minimum pressure produced by hi tech: 10-12 Pa
Zero pressure not allowed by Quantum Uncertainty!
Absolute pressure cannot be negative: Pressure pushes not pulls!
Gauge pressure can be negative because it is a relative pressure.
Nature abhors a Vacuum.
-Aristotle
Accelerating or Decelerating?
A little girl holding a helium-filled balloon travels in a car
moving with constant velocity. The windows are up and the car
is relatively airtight. The driver then changes speed. From the
picture did the driver hit the gas or the break? Explain.
Question: Accelerating or
Decelerating?
The car accelerates. The girl’s inertia causes her to jerk back as
the car accelerates forward under her. Similarly, the air in the
car jerks back, creating a low pressure area in the front of the
car so the helium balloon is buoyed forward.
Answer:
Pascal’s Principle: Liquids & Gases A pressure applied to an enclosed fluid is transmitted
undiminished throughout the fluid: P = P everywhere.
1 2
1 2
F F
A A
How much pressure is transmitted in this hydraulic lever?
1 2
1 2
F F
A A
Pascal’s Principle: Liquids & Gases
2
2
10 9.8 /
1
kg m s
m
A = 1m2
98Pa
Buoyant Force
2 2 1 1P A PA
2 1gh A gh A
2 1gA h h
2 1
fluid
displaced
mgA h h
V fluid
displaced
displaced
mgV
V
fluidm g
The difference in pressures at different
depths produces a net upward or
Buoyant Force:
Weight of fluid displaced by object!
2 1BF F F
fluidBF W
The buoyant force on an object equals the
weight of the fluid it displaces.
Archimedes Principle
Note:
For the same volume of displaced
fluid denser fluids exert a greater
buoyant force. This is why ships float
higher in salt water than fresh water.
fluidW Vg
Archimedes Principle
The wood block is floating in the water and
has a mass of .08 kg. What is the volume of
the displaced water?
fluid displacedBF W
fluid fluid blockV g m g
/fluid block fluidV m
blockBF W
fluid displaced blockW W
3.08 /1000 /kg kg m 5 38.0 10fluidV x m
fluid objectm g m g
fluid objectm m
Since object floats, BF
(weight of fluid displaced)
equals the weight of the
object. So the masses of the
object and fluid displaced
are the same.
fluidBF W
The buoyant force on an object equals the
weight of the fluid it displaces.
Archimedes Principle
Wobject < BF Object will Float
Wobject > BF Object will Sink
Wobject = BF Object will Be Neutral
Archimedes Principle
Object will Float if:
object object fluid displacedV g V g
object displaced fluid displacedV g V g
object fluid
But the Volume of the object is the same as the
Volume of the fluid displaced:
An object will float if its density is less than the fluid it is in!
object fluid dispalcedW BF W
Archimedes Principle
object fluid
object fluid
object fluid
Sink
Float
Neutral
Archimedes Principle
Buoyant force also depends on SHAPE which
determines the VOLUME of displaced fluid.
This is how Iron Ships can float!!
They have a shape that displaces enough water to produce a
Buoyant Force to balance the weight!
Buoyant Force Question
A boulder is thrown into a deep lake. As
it sinks deeper and deeper into the water,
does the buoyant force on it increase?
NO! The buoyant force does not change as the boulder
sinks because the boulder displaces the same amount of
fluid no matter how deep it is in the water!
Question
A piece of iron placed on a block of wood makes it float
lower in the water. If the iron were instead suspended
beneath the wood, would it float as low, lower or higher?
Question A piece of iron placed on a block of wood
makes it float lower in the water. If the iron
were instead suspended beneath the wood,
would it float as low, lower or higher?
Answer
Higher. When the iron is on top, its whole weight pushes the
wood into the water. But when the iron is submerged,
buoyancy on it reduces its effective weight and less of the
block is pulled beneath the water line.
Archimedes Principle
Fraction of Object Floating
displaced
object
V
V
/
/
fluid fluid
object object
m
m
object
fluid
submerged
object
V
VSince object floats, BF
(weight of fluid displaced)
equals the weight of the
object. Thus the masses of
the object and fluid
displaced are the same.
fluid objectm g m g
fluid objectm m
Archimedes Principle
Fraction of Object Floating
submerged
object
V
V
If the above object is floating in water, what is it’s density?
object
fluid
What is the fraction submerged? 1/2
What is the density in terms of water? 1/2
Archimedes Principle
Fraction of Object Floating
submerged
object
V
V
What is the density of the object?
object
fluid
What fraction is submerged? ~1/3
Density is 1/3 that of water.
Why does the water stop when
the top is closed?
Pressure is greater in the fluid at the spout due to weight of water
so water flows. Hand covers top and water keeps flowing until
the pressure is reduced to 1 atm by increasing volume of air
above the fluid just like with a closed barometer!
How High?
Flow Rate The volume of fluid per second that flows through an area A.
VF Av
t
Continuity Equation Incompressible Fluid ( doesn’t change)
1 1 2 2Av A v
1 2v v
Continuity Equation
1 2v v
1v
2v
Fluid Flow A swelling in a blood vessel doubles its cross sectional area. By
how much will the blood flow change as it moves through the
enlarged area? The speed of blood (density = 1060 kg/m3)
through a normal part of the vessel is v = .5 m/s.
11 1 2 2 2 1
2
10.5 / 0.25 /
2
AA v A v v v m s m s
A
Continuity Equation
Calculate the velocity to which water can be squirted with the
hose with a constant flow rate of 0.500 liters per second if it :
a) emerges from the hose (r = 0.900cm) without the thumb
b) emerges with the thumb reducing the radius to r = 0.250 cm.
c)What is the maximum height the water can reach if it is shot
straight up with the thumb?
Water with air bubbles flows through a pipe
that gets narrower. In the narrow region the
water gains speed and the bubbles are
a) larger
b) smaller
c) the same size
Bernoulli’s Bubbles
Lower pressure!!
WHY?
Bernoulli’s Principle
Where the speed of a fluid increases,
internal pressure in the fluid decreases.
Bernoulli’s Ball
Why does this happen?
Bernoulli’s Principle
Bernoulli’s Equation
2
1 1 1
1constant
2Fd mv mgh
211 1
1constant
2
Fd m mv gh
V V V
2
1 1 1
1constant
2P v gh
Conservation of energy for a smooth steady fluid flow.
constantW KE PE
Efflux Speed A large tank is filled with water up to a depth of 15 m above
ground. A spout located 10.0 m above the bottom of the tank is
then opened. With what speed will water emerge from the spout?
The tank is so large that the liquid level at the top (open to air)
changes very slowly. The pressures are both at 1 atm.
2 2
1 1 1 2 2 2
1 1
2 2P v gh P v gh
1
2
2
1 2 2
1
2gh v gh
2 1 22 ( )v g h h
22 9.8 / (5 )m s m
2 9.9 /v m s
Why does the water stop when
the top is closed?
Pressure is greater in the fluid at the spout due to weight of
water so water flows. Hand covers top and water keeps
flowing until the pressure is reduced to 1 atm by increasing
volume of air above the fluid.
Bernoulli’s Principle: LIFT
Sailing Problem Calculate the approximate force on a square meter of sail, given
the horizontal velocity of the wind is 6.00 m/s parallel to its
front surface and 3.50 m/s along its back surface. 31.29 /Air kg m
( )Front BackF P P A
Question Here is a glass of ice water with an ice cube floating in it.
What fraction of the ice cube is submerged in the water?
3 3917 / , 1000 /ice waterkg m kg m
Question Here is a glass of ice water with an ice cube floating in it.
What fraction of the ice cube is submerged in the water?
submerged
object
V
V
object
fluid
3
3
917 /.917
1000 /
object
fluid
kg m
kg m
90%
3 3917 / , 1000 /ice waterkg m kg m
Question The glass is full to the brim with ice water and ice cubes.
When the ice melts, will the water level decrease? Overflow?
Remain the same? WHY?
Question Here is a glass of ice water with an ice cube floating
in it. When the ice cube melts, will the water level
rise, drop or remain the same? Assume constant
temperature.
Since the cube floats, the BF equals its weight. BF = Wice
Weight of the water displaced equals the weight of the ice cube.
mwaterg = miceg
Mass of the water displaced equals the mass of the ice cube.
When the ice melts, it is water and since it has the same mass and
density as the displaced water, it has the same volume, filling up the
space where of the displaced water. The level remains unchanged.
W
BF
Answer
Volume above and below
How does the volume of the billions of
hexagonal open spaces in the structures of ice
crystals in a piece of ice compare to the portion
of ice that floats above the water line?
water =1000 kg/m3
ice = 917 kg/m3
Answer: The volume is the same! When the ice melts, the open
spaces are filled in by the amount of ice that extends above the water
level. This is also why the water level doesn’t rise when ice melts.
If you release a ball inside a freely-falling elevator, it
stays in front of you instead of "falling to the floor"
because you, the ball, and the elevator are all
accelerating downward at the same acceleration, g. If
you similarly release a helium-filled balloon, the
balloon will
a) also stay in front of you
b) press against the ceiling
c) press against the floor
If you release a ball inside a freely-falling elevator, it
stays in front of you instead of "falling to the floor"
because you, the ball, and the elevator are all
accelerating downward at the same acceleration, g. If
you similarly release a helium-filled balloon, the
balloon will
a) also stay in front of you
b) press against the ceiling
c) press against the floor
Question
Answer: a) The balloon stays in front of you because
it, along with you and the air and the elevator, is in free fall.
When the air is in free fall it has no weight, therefore it cannot
provide a buoyant force. Therefore, the balloon falls with
everything else.