Archimedes Principle

1
Archimedes Principle

• Physics 202
• Professor Lee Carkner
• Lecture 2

2
PAL 1 Fluids

• Column of water to produce 1 atm of pressure
• P rgh
• P
• r 1000 kg/m3
• g 9.8 m/s2
• h P/rg
• Double diameter pressure does not change
• On Mars pressure would decrease
• Mars has smaller value of g

3
Archimedes Principle

• What happens if you put an object in a fluid
• Called the buoyant force
• 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

4
Will it Float

• Density
• An object less dense than the fluid will float
• A floating object displaces fluid equal to its
  weight
• A sinking object displaces fluid equal to its
  volume

5
Floating

• How will an object float
• The volume of fluid displaced is proportional to
  the ratio of the densities
• Example ice floating in water
• riVigrwVwg
• VwVi (ri/rw)
• rw 1024 kg/m3 and ri 917 kg/m3

6
Ideal Fluids

• Steady --
• Incompressible -- density is constant
• Nonviscous --
• Irrotational -- constant velocity through a cross
  section
• The ideal fluid approximation is usually not very
  good

7
Moving Fluids

• What happens if the pipe narrows
• Avr constant
• If the density is constant then
• Av constant R volume flow rate
• Constricting a flow increases its velocity
• Because the amount of fluid going in must equal
  the amount of fluid going out
• Or a big slow flow moves as much mass as a small
  fast flow

8
Continuity

• RAvconstant is called the equation of
  continuity
• You can use it to determine the flow rates of a
  system of pipes
• Cant lose or gain any material

9
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

10
Bernoullis Equation

• Consider a pipe that bends up and gets wider at
  the far end with fluid being forced through it
• Wg -Dmg(y2-y1) -rgDV(y2-y1)
• The work of the system due to pressure is
• WpFdpAdDpDV-(p2-p1)DV
• D(1/2mv2)1/2rDV(v22-v12)
• p1(1/2)rv12rgy1p2(1/2)rv22rgy2

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Consequences of Bernoullis

• Fast moving fluids exert less pressure than slow
  moving fluids
• This is known as Bernoullis principle
• Energy that goes into velocity cannot go into
  pressure
• Note that Bernoulli only holds for moving fluids

12
Bernoulli in Action

• Blowing between two pieces of paper
• Convertible top bulging out
• Airplanes taking off into the wind

13
Lift

• If the velocity of the flow is less on the bottom
  than on top there is a net pressure on the bottom
  and thus a net force pushing up
• If you can somehow get air to flow over an object
  to produce lift what happens

14
Deriving Lift

• Use Bernoullis equation
• pt1/2rvt2pb1/2rvb2
• The difference in pressure is
• pb-pt1/2rvt2-1/2rvb2
• (Fb/A)-(Ft/A)1/2r(vt2-vb2)
• L (½)rA(vt2-vb2)

15
Next Time

• Read 15.1-15.3
• Homework Ch 14 P 37 42 47 Ch 15 P 6 7

16
Which of the following would decrease the
pressure you exert on the floor the most

• Doubling your mass
• Doubling the mass of the earth
• Doubling your height
• Doubling the size of your shoes
• Doubling air pressure

17
Which of the following would increase the
pressure of a column of fluid of fixed mass the
most

• Doubling the width of the column
• Halving the density of the fluid
• Halving the mass of the Earth
• Halving the speed of the Earths rotation
• Doubling the height of the column

18
Summary Fluid Basics

• Density rm/V
• PressurepF/A
• On Earth the atmosphere exerts a pressure and
  gravity causes columns of fluid to exert pressure
• Pressure of column of fluid
• pp0rgh
• For fluid of uniform density pressure only
  depends on height

19
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
• FoFi(Ao/Ai) and dodi(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

20
Summary Moving Fluids

• Continuity -- the volume flow rate (RAv) is a
  constant
• fluid moving into a narrower pipe speeds up
• Bernoulli
• p11/2rv12rgy1p21/2rv22rgy2
• Slow moving fluids exert more pressure than fast
  moving fluids