archimedes’ principle physics 202 professor lee carkner lecture 2 “got to write a book, see, to...
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Archimedes’ Principle
Physics 202Professor Lee
CarknerLecture 2
“Got to write a book, see, to prove you’re a philosopher. Then you get your … free official philosopher’s loofah.”
--Terry Pratchett, Small Gods
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PAL #1 Fluids Column of water to produce 1 atm of
pressure
= 1000 kg/m3
h = P/g = 10.3 m Double diameter, pressure does not change
On Mars pressure would decrease
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Archimedes’ Principle The fluid exerts a force on the object
If you measure the buoyant force and the
weight of the displaced fluid, you find: An object in a fluid is supported by a buoyant
force equal to the weight of fluid it displaces Applies to objects both floating and
submerged
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Buoyancy
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Will it Float?
What determines if a object will sink or float?
A floating object displaces fluid equal
to its weight
A sinking object displaces fluid equal
to its volume
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Floating How will an object float? The denser the object, the lower it will float, or:
Example: ice floating in water,
W=Vg
Vi/Vw=w/i
w = 1024 kg/m3 and i = 917 kg/m3
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Iceberg
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Ideal Fluids
Steady --
Incompressible -- Nonviscous -- Irrotational --
Real fluids are much more complicated The ideal fluid approximation is usually not
very good
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Moving Fluids Consider a pipe of cross sectional area A with a fluid
moving through it with velocity v
Mass must be conserved so,
If the density is constant then,Av= constant = R = volume flow rate
Because the amount of fluid going in must equal the
amount of fluid going out
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Continuity
R=Av=constant is called the equation of continuity
You can use it to determine the flow rates of a system of pipes
Can’t lose or gain any material
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Continuity
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The Prancing Fluids
How can we keep track of it all? The laws of physics must be obeyed
Neither energy nor matter can be
created or destroyed
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Bernoulli’s Equation Consider a pipe that bends up and gets wider at
the far end with fluid being forced through it The work of the system due to lifting the fluid is,
The work of the system due to pressure is,
Wp=Fd=pAd=pV=-(p2-p1)V The change in kinetic energy is,
Equating work and KE yields,
p1+(1/2)v12+gy1=p2+(1/2)v2
2+gy2
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Fluid Flow
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Consequences of Bernoulli’s Equation
Fast moving fluids exert less pressure than slow moving fluids
This is known as Bernoulli’s principle Based on conservation of energy
Note that Bernoulli only holds for moving fluids
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Constricted Flow
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Bernoulli in Action
Blowing between two pieces of paper
Convertible top bulging out Shower curtains getting sucked
into the shower
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Shower Physics
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Lift
Consider a thin surface with air flowing above and below it
This force is called lift If you can somehow get air to flow over
an object to produce lift, what happens?
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December 17, 1903
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Deriving Lift Consider a wing of area A, in air of density Use Bernoulli’s equation:
The difference in pressure is:
pb-pt=1/2vt2-1/2vb
2
Pressure is F/A so:
L=Fb-Ft and so:
If the lift is greater than the weight of the plane, you fly
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Summary: Fluid Basics Density ==m/V Pressure=p=F/A On Earth the atmosphere exerts a
pressure and gravity causes columns of fluid to exert pressure
Pressure of column of fluid:p=p0+gh
For fluid of uniform density, pressure only depends on height
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Summary: Pascal and Archimedes
Pascal -- pressure on one part of fluid is transmitted to every other part
Hydraulic lever -- A small force applied for a large distance can be transformed into a large force over a short distance
Fo=Fi(Ao/Ai) and do=di(Ai/Ao) Archimedes -- An object is buoyed up by
a force equal to the weight of the fluid it displaces Must be less dense than fluid to float
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Summary: Moving Fluids
Continuity -- the volume flow rate (R=Av) is a constant fluid moving into a narrower pipe
speeds up Bernoullip1+1/2v1
2+gy1=p2+1/2v22+gy2
Slow moving fluids exert more pressure than fast moving fluids