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Liquids & Buoyant Force
Notes (p275 HRW)
Liquids
Study of liquids
• Hydrostatics – liquids that are stationary
• Archimedes, Pascal
• Hydrodynamics – liquids that are moving
• Bernouilli
Fundamental Characteristic
Density ()
How is density defined? • mass/volume
• = m/V
What are density units in SI • kg/m3 (standard) or gm/cc (common)
Example • water = 1000 kg/m3 (or 1 gm/cc)
• gold = 19.3 x water
• lead = 11.3 x
• iron = 7.8 x
• ice = 0.92 x
• balsa = 0.12 x
• air = 1.2 kg/m3
Lower density floats
on higher density
Pressure (p)
Definition:
• force perpendicular to plane/area over which force is acting
• P = Force/Area = F/A
• SI Units = pascals (Newtons/meter2)
• Air Pressure
• Standard air pressure @ sea level and 20C, expressed as:
• 1.01 x 105 Pa
• 1 atmosphere
• 760 mm (or 76 cm) of mercury
• 30 inches of mercury
• Measured by a barometer or manometer
• Average tire pressure ~ 200 kPa or 30 psi (lbs/in2)
Hydrostatic (Gauge) Pressure
Fluid pressure increases with depth because the water at a depth must support the weight of water above it. • Ex.
• Diving to the bottom of the deep end of a pool – what do you feel?
• Air pressure
P = ρgh, where • ρ = density of fluid
• g = 9.8 m/sec2
• h = depth of fluid (ex. height of water column)
• This is gauge pressure
Container shape?
NO EFFECT!
Absolute/Total Pressure
Add atmospheric pressure (Pair) which acts on the surface of the fluid and the total pressure becomes • Ptotal = Pair + ρgh
Generically • The pressure at the bottom of a
column (Pb) equals the pressure at the top (Pt) PLUS the pressure due to the column, or
• Pb = Pt + ρgh
Practice - submarines
A sub dives to a depth of 200 m. How
much water pressure must the hull be
able to withstand, or what is the (gauge)
water pressure at 200 m?
• Solve P = ρgh
• P = 1000 x 9.8 x 200
• P = 1.96 x 106 Pa
Every sq meter of surface must
withstand ~ 2 million newtons
Note: in this example we have NOT included the air pressure pushing on the top of the water!
Submarines
Maximum depth – classified, but
generally believed to be ~1500 ft for US
and ~2500 ft for Russian
Sub disasters –
• Thresher (‘63), sank off Cape Cod in 8400 ft
after joint to outer hull failed, flooding sub
• Kursk (2000), Barents Sea, 350 ft after
torpedo accident
Practice
How deep (h) must a diver go before he experiences another atmosphere (1.01 x 105 Pa) of water pressure?
Solve for h in water (ρ = 1000 kg/m3) • P = ρgh
• 1.01 x 105 = 1000 x 9.8 x h
• h = 10.31 m (~34 feet)
Roughly, every 30 feet of dive adds 1 atmosphere of pressure to a diver
Interesting facts on diving!
Deepest free dive
• 124 m (~400 ft) feet
Deepest assisted dive
• 214 m (~700 ft)
Breath holding record
• >19 minutes
Mini-lab
Density determination of 3 samples
• Measure
• Write-up
• Submit
Pascal’s Principle
Pressure applied to an enclosed fluid is
transmitted equally and undiminished to
every part of the fluid, as well as the walls
of the container.
• P1 = P2, (Pressure a point 1 = pressure at point 2)
2
2
1
1
A
F
A
F
Application of Pascal’s Principle
Hydraulic lift – service stations
Examples of Pascal’s Principle
Examples
Example
The small piston of a hydraulic lift has an
area of 0.2 m2. If a car weighing 1.2e4 N
sits on the large piston, area 0.9 m2, how
large a force must be applied to the small
piston to support the car?
F1/A1 = F2/A2
• F1/0.2 = 1.2e4/0.9
• F1 = 0.2*1.2e4/0.9 = 2.7e3 N
What is buoyancy?
In physics, buoyancy is
the upward force acting
on an object in a fluid,
and can be:
• Positive
• Neutral
• Negative
Alligators & crocodiles?
What causes buoyancy?
Buoyancy is the result of the difference
in pressure exerted on the top and
bottom of an object.
Question
If you place a rock into a
bucket of water, say ½ filled,
what will you observe about
the water line in the bucket?
It will rise to reflect the
volume of the rock – if the
rock was 0.1 m3, the water
would rise 0.1 m3, b/c the
rock is submerged.
This is an example of Archimedes’ Principle…
Archimedes Principle - example
In air the stone
weighs 44 N
In water the stone
weighs 36 N
Buoyant force is the
difference = 8N ↑
Buoyancy Example
Weight in
air Apparent
weight
Fb = Weight in air – weight in water
Buoyancy - Archimedes
Archimedes’ Principle • When a body is partly or wholly submerged in a fluid,
it will experience a buoyant force (Fb)equal to the weight of the fluid displaced.
• Buoyant (Upthrust) force (Fb) • Fb = Weightfluid displaced = (massfluid)*(g)
• Fb = (fluid)*(Vfluid displaced)*(g) (Note: =m/V or m= *V)
• Fb = (V)*g directed upward!
• Case 1 – object is submerged • Fb = (fluid)*(Vfluid displaced)*(g)
• Case 2 – object is partially submerged (floating) • Fb = (object)*(Vobject)*(g)
Buoyancy Summary Table
IF THEN
ρobject > ρfluid Wobj↓ > Fb ↑ sinking
ρobject = ρfluid Wobj↓ = Fb ↑ neither
sink nor float
ρobject < ρfluid Wobj↓ < Fb ↑ float
ρ = density, W = weight Fb = buoyant force
Example of Archimedes
Principle & Buoyant Force
If a house brick of density (ρ) 2000 kg/m3
and volume (V) of 0.00123 m3 is placed in
a bucket of water, what is the buoyant force
(Fb) acting on a brick?
Solve: Fb = ρVg
Fb = 1000 x 0.00123 x 9.8
Fb = 12.05 N
Buoyant Force Practice
An ice cube is floating in a glass of water
(ρ=1000 kg/m3). The ice, whose density is
917 kg/m3, has dimensions of 0.03 x 0.02 x
0.02 m. What is the buoyant force on the ice?
Solve: Note: the ice is floating thus the
buoyant force↑ = weight of the ice↓
• Fb = (ρV)*g (use ρ, V for ice, not water)
• Fb = 917*(0.03*0.02*0.02)*9.8
• Fb = 0.11 N↑
Buoyant Force Practice
A ferry boat is 4 m wide and 6 m long. When a truck pulls onto it, the boat sinks 4 cm in the water. • What is the weight of the truck?
Use Archimedes Principle: • Weight of truck = weight of water displaced (find
this displacement)
• Weight = mass x g • Mass of water (m) = density x volume
• m= 1000 x (4 x 6 x 0.04) = 960 kg
• Weight = m x g = 960 x 9.8 = 9,408N
Buoyant Force Practice
Example
• A piece of metal weighs 7.84N in air and
6.86N when completely immersed in water.
• What is the buoyant force?
• Fb = 7.84 – 6.86 = 0.98N
• What is the density (ρ) of the object?
• Fb = ρVg (ρ, V refer to the fluid when submerged)
• V = Fb /ρg = 0.98/(1000*9.8) = 0.0001 m3
• Mass = Weight/9.8 = 0.8 kg
• ρ = m/V = 0.8/0.0001 = 8000 kg/m3
Buoyancy - icebergs
If an object is floating on the
surface, then the volume that
is below the surface can be
determined as follows:
• Vf/Vo = o/f, where
• Vf= volume of object that is below
surface
• Vo= volume of the object
• o = density of the object
• f = density of the fluid
% Iceberg in salt water that is below surface = .92/1.025 = 89%