Physics 211: Lecture 28 Todays Agenda

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Physics 211 Lecture 28Todays Agenda

• Description of Fluids at Rest
• Pressure vs Depth
• Archimedes Principle objects in a fluid
• Pascals Principle hydraulic forces

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By what factor does the strength of the Grinch
increase when he finally clues into the meaning
of Christmas? A) 2 B) 3C)10D)12 E) 37
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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 are free to move .. No long
  range correlation between positions.
• What parameters do we use to describe fluids?
• Density m/V
• Pressure F/A

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• Density Pressure are related by the Bulk
  Modulus B
• LIQUID incompressible (density almost constant)
• GAS compressible (density depends a lot on
  pressure)

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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 total force on the fluid
  is zero.
• 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)

but
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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!

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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

I
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Lecture 12, ACT 1
dI
r2

• 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!

r1
p
I
C) r1 r2
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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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Sink or Float?
Objects in water

• The buoyant force is equal to the weight of
  the liquid that is displaced.
• If the buoyant force 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
Pb
styrofoam
C)

• If the object floats right-side up, then it also
  must float upside-down.
• 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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Example Problems
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)?
Solution
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
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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
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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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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
• The change in pressure vs. depth depends only on
  g and the mass density r.

? 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 (F1/A1) is transmitted throughout the
  liquid.

Check that Fd is the same on both sides. Energy
is 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 (Mg/A10) is transmitted to
  the small cylinder.
• This change in pressure determines the change in
  levels. The area of the small cylinder plays no
  role.

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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
• Archimedes Principle objects in a fluid
• Pascals Principle hydraulic forces