Physics 211: Lecture 28 Todays Agenda: Fluids

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Physics 211 Lecture 28Todays Agenda Fluids

• Description of Fluids at Rest
• Pressure vs Depth
• Archimedes Principle objects in a fluid
• Will it float
• Pascals Principle hydraulic forces
• Using hydraulics to make your life easier
• Like how a lever works
• Review Lecture Wed usual time, usual place

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Fluids

• What do we mean by fluids?
• Fluids are substances that flow. substances
  that take the shape of the container
• Atoms and molecules must be free to move .. No
  long range correlation between positions (e.g.,
  not a crystal).
• Gas or liquid, not solid
• What parameters do we use to describe fluids?
  (here are a couple)
• Density kg/m3 ? mass/volume
• FYI rWATER 1 g/cm3
• Pressure N/m2 ? force/area
• or Pascals (Pa) name for same thing

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density
pressure

• For a given material, r and P connected by Bulk
  Modulus, B
  
• If you squeeze on it, how much (by what fraction)
  does it compress. (Property of the material)
  (units same as pressure ? Force/Area)
• LIQUID (nearly) incompressible, large B
  (density almost constant)
• GAS compressible, small B (density depends a
  lot on pressure)

Why such a wide range?
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Pressure vs. DepthIncompressible Fluids
(liquids)

• Due to gravity, the pressure depends on depth in
  a fluid

• Consider an imaginary fluid volume (a cube, each
  face having area A)
• The sum of all the forces on this volume must be
  ZERO as it is in equilibrium.
• There are three vertical forces
• The weight (mg)
• The upward force from the pressure on the bottom
  surface (F2)
• The downward force from the pressure on the top
  surface (F1)

(Same fluid)
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Pressure vs. Depth (2)

• For a fluid in an open container
• pressure same at a given depth independent of the
  container

• fluid level is the same everywhere in a connected
  container (assuming no surface forces)
• Why is this so? Why, in equilibrium, does the
  pressure below the surface depend only on depth?

• Imagine a tube that would connect two regions
  at the same depth.

• If the pressures were different, fluid would flow
  in the tube!
• However, if fluid did flow, then the system was
  NOT in equilibrium, since no equilibrium system
  will spontaneously leave equilibrium.

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Lecture 28, ACT 1

• What happens with two different fluids??
  Consider a U tube containing liquids of density
  r1 and r2 as shown
• Compare the densities of the liquids

• If we use the same liquids in a U tube of twice
  the cross-sectional area as the first, compare
  the distances between the levels in the two cases
  (depth of liquid 2 same in both cases).

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Lecture 12, ACT 1

• At the depth of the interface, the pressures
• in each side must be equal.
• Since theres more liquid above this depth on
  the
• left side, that liquid must be less dense!

p
C) r1 gt r2

• The pressure depends ONLY on the depth and the
  density of the fluid.
• e.g. consider case I

dII
r2
r1
p
II
B) dI dII
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Archimedes Principle

• Suppose we weigh an object in air and in water.

• Since the pressure at the bottom of the object is
  greater than that at the top of the object, the
  water exerts a net upward force, the buoyant
  force, on the object.

• The buoyant force is equal to the difference in
  the pressures times the area.

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Archimedes Principle
Free body diagram Add buoyant force to gravity

• Total force is sum of FB and gravity.

displaced volume

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Sink or Float?
Objects in water

• The buoyant force is equal to the weight of
  the liquid that is displaced.
• If the buoyant force of a fully submerged object
  is larger than the weight of the object, it will
  float otherwise it will sink.

• We can calculate how much of a floating object
  will be submerged in the liquid

• Object is in equilibrium

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Sink of Float?
Object is in equilibrium
The Tip of The Iceberg What fraction of an
iceberg is submerged?
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Lecture 28, ACT 2

• A lead weight is fastened to a large styrofoam
  block and the combination floats on water with
  the water level with the top of the styrofoam
  block as shown.
• If you turn the styrofoamPb upside down, what
  happens?

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Lecture 28, ACT 2

• If the object floats right-side up, then it also
  must float upside-down.
• It displaces the same amount of water in both
  cases
• The weight of that water equals the total weight
  of the whole object
• However, when it is upside-down, the Pb displaces
  some water.
• Therefore the styrofoam must displace less water
  than it did when it was right-side up (when the
  Pb displaced no water).

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At what depth is the water pressure two
atmospheres? (It is one atmosphere at the
surface.) What is the pressure at the bottom of
the deepest oceanic trench (about 104 meters)?
Example Problems
Solution
d is the depth. The pressure increases one
atmosphere for every 10 meters. This assumes
that water is incompressible.
P2 P1 rgd 2.02?105 Pa 1.01?105 Pa
103 kg/m39.8m/s2d d 10.3
m P2 1.01?105 Pa 103 kg/m39.8m/s2104 m
9.81?107 Pa 971 Atm
For d 104 m
If water were compressible, would the pressure at
the bottom of the ocean be greater or smaller
than the result of this calculation?
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Example Problems (2)
Have you ever tried to submerge a beach ball (r
50 cm) in a swimming pool? Its difficult. How
big a downward force must you exert to get it
completely underwater?
Solution
F rg4pr3/3 5131 N 523 kgg
Im ignoring the weight of the beach ball. The
force is the weight of a 523 kg object.
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More Fun With Bouyancy

• Two cups are filled to the same level with water.
  One of the two cups has plastic balls floating
  in it. Which cup weighs more?

• Archimedes principle tells us that the cups weigh
  the same.
• Each plastic ball displaces an amount of water
  that is exactly equal to its own weight.

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Still More Fun!

• A plastic ball floats in a cup of water with half
  of its volume submerged. Oil (roil lt rball
  ltrwater) is slowly added to the container until
  it just covers the ball.
• Relative to the water level, the ball moves up.
  Why?

• For oil to cover the ball, the ball must have
  displaced some oil.
• Therefore, the buoyant force on the ball
  increases.
• Therefore, the ball moves up (relative to the
  water).
• Note that we assume the bouyant force of the air
  on the ball is negligible (it is!) the bouyant
  force of the oil is not.

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Pascals Principle

• So far we have discovered (using Newtons Laws)
• Pressure depends on depth Dp rgDy
• Since pressure depends on depth, an object in a
  liquid experiences an upward buoyant force FB
  Wliquid displaced
• Pascals Principle addresses how a change in
  pressure is transmitted through a fluid.

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Pascals Principle
Hydraulic jack

• Pascals Principle is most often applied to
  incompressible fluids (liquids)
• Increasing p at any depth (including the surface)
  gives the same increase in p at any other depth

? Hydraulic lifts
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Pascals Principle (2)

• Consider the system shown
• A downward force F1 is applied to the piston of
  area A1.
• This force is transmitted through the liquid to
  create an upward force F2.
• Pascals Principle says that increased pressure
  from F1 DP(F1/A1) is transmitted throughout
  the liquid.

Check that Fd is the same on both
sides. Displaced volumes are the same, so
Works like a lever
energy conserved
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Lecture 28, ACT 3

• Consider the systems shown to the right.
• In each case, a block of mass M is placed on the
  piston of the large cylinder, resulting in a
  difference di between the liquid levels.
• If A2 2A1, compare dA and dB.

A) dA (1/2)dB
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Lecture 28, ACT 3Solution

• The change in pressure DP (Mg/A10) is
  transmitted to the small cylinder in both cases.
• The pressure at the level of the top of the fluid
  in the big cylinder must equal the pressure at
  the same absolute height (right across) in the
  smaller cylinders (depends on height)
• This change in pressure determines the change in
  levels.
• DP (rA1dAg/A1) (rA2dBg/A2)
• The area of the small cylinders cancels in these
  formulae
• DP (rdAg) (rdBg) (Mg/A10)

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Using Fluids to Measure Pressure

• Use Barometer to measure Absolute Pressure

• Top of tube evacuated (p0)
• Bottom of tube submerged into pool of mercury
  open to atmosphere (pp0)
• Pressure dependence on depth

• Use Manometer to measure Gauge Pressure

• Measure pressure of volume (p1) relative to the
  atmospheric pressure (º gauge pressure )
• The height difference (Dh) measures the gauge
  pressure

1 atm 760 mm (29.9 in) Hg 10.3 m
(33.8 ft) H20
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Recap of Todays Lecture

• Description of Fluids at Rest
• Pressure vs Depth (Text 13-2)
• Archimedes Principle objects in a fluid (Text
  13-3)
• Pascals Principle hydraulic forces (Text
  13-2)
• Look at textbook problems Ch. 13 7, 27, 31, 39,
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