Chapter 15: Fluids

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Chapter 15 Fluids

• A fluid is a gas or a liquid.
• A gas expands to fill any container
• A liquid (at fixed pressure and temperature), has
  a fixed volume, but deforms to the shape of its
  container.
• The density r of any substance is its mass M per
  volume V

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Pressure

• Pressure P is the amount of force F per unit area
  A

By the Action-Reaction principle, Pressure is the
inward force per unit area that the container
exerts on the fluid.
Pressure is the outward force per unit area that
the fluid exerts on its container.
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Atmospheric Pressure

• Atmospheric pressure comes from the weight of the
  column of air above us. At sea level,
  atmospheric pressure is (up to 10 lower during
  hurricanes!)
• Pat 1.01 ? 105 N/m2
• 1.01 ? 105 Pa 1 Pascal 1 N/m2
• 14.7 lb/in2
• 1 bar (tire pressure gauges in Europe
  read 1, 2, . bar)
• The mass of the column of atmosphere above each
  square meter of the surface of the earth is
• M A P/g 1m2 1.01 ? 105 N/m2 / 9.81 m
  /s2
• 10.3 ? 103 kg
• This is huge, the mass of 1 m3 of water is only
  103 kg
• The density of air is about 1.0 kg/ m3.
• The mass of a column of air of height h is MrAh.
• The equivalent height of the atmosphere is
• h (M/A)/r 10.3 ? 103 kg /m2/1.0 kg/ m3
  ?10 km
• Actual height is gt100 km because density
  decreases with height

FMg FPA
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Pressure examples

• Estimate the force of the atmosphere on the top
  of your head.
• A (10cm)(15cm)0.015m2
• FPA 1.01 ? 105 N/m2 0.015 m2 1.5 kN
• A (4in)(6in)24 in2
• FPA 15 lb/in224in2 360 lb.
• Is atmospheric pressure on top of a mountain
  greater or less than at sea level?
• Less. At higher altitude, there is less mass
  above.
• If tire inflation pressure is 29.4 lb/in2, what
  is this in bar?
• 29.4 lb/in2 214.7 lb/in2 2.0 bar

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Pressure in a Fluid

• Pressure in a fluid depends only on the depth h
  below the surface.
• P Pat rgh r density of
  fluid
• Weight/Area of fluid
• Weight/Area of atmosphere above fluid
• IF the density of the fluid is constant and it
  has atmospheric pressure (Pat) at its surface.

Mass of fluid above depth h is
(density)(volume) rhA Force of gravity on
fluid above depth h W rghA
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Pressure under water

• To what depth in water must you dive to double
  the pressure exerted on your body?
• P Pat rgh
• rgh Pat , h Pat /rg

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Pressure variation in fluid

• The variation in pressure at two different depths
  is given by
• P2 P1 rgh

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Conceptual Question
When a hole is made in the side of a container
holding water, water flows out and follows a
parabolic trajectory. If the the container is
dropped in free fall, the water flow 1.
diminishes. 2. stops altogether. 3. goes
out in a straight line. 4. curves upward.
Peer Instruction - Mazur
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Fig. 15-22 Water Fountain

• If the fluid emerging from a hole is directed
  straight up, the stream will rise to the level of
  the fluid in the container.
• This is just an application of energy
  conservation.
• This is also equivalent to having two containers,
  joined by a tube.

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Water Fountain

• At the Macarther Mall in Norfolk VA, a jet of
  water is launched at a 45 degree angle, and rises
  1.5 m above the pool at the base of the elevator.
  What is the water pressure driving the jet?
• Energy of an element of mass of water m as it
  leaves the jet is (1/2)mv2. Energy at top of arc
  is mgh (1/2)mvx2 mgh (1/4) mv2
• mgh (1/4) mv2. h v2/(4g)
• If the jet were directed straight up, height
  would be 2h. (vx0) P rg(2h)

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Pascals Principle

• A external pressure P applied to any area of a
  fluid is transmitted unchanged to all points in
  or on the fluid.
• This is just an application of the
  Action-Reaction principle.
• Hydraulic Lift

A Force F1 is applied to area A1, displacing the
fluid by a distance d1. The pressure increase in
the fluid is PF1/A1. The Pressure F1/A1 creates
a force on the car F2 A2 (F1/A1). The volume of
fluid displaced on the left is Vd1 A1. This
equals the volume increase on the right Vd2A2.
Thus the work done by F1 W1 F1d1 , is the
same as the work done by the hydraulic system on
the car W2F2d2 d2(A2 F1/A1)(d2A2 )(F1/A1)(
d1 A1)(F1/A1) F1d1 W1 Energy Conservation
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Archimedes Principle

• Because the pressure in a fluid is greater below
  the object than above, there is an upward buoyant
  force Fb on any object in a fluid.

Archimedes Principle The upward buoyant force
on an object is equal to the weight of the
displaced fluid. Fb rgV Nota bene r is the
density of the (displaced) fluid, not the density
of the object (in green). This result does not
depend upon the shape of the immersed object.
F2/A F1 /A rgh F2 F1 rghA
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Flotation
When an object floats, the magnitude of the
upward buoyant force equals its weight.
Therefore an object floats when it displaces an
amount of fluid equal to its weight. In order to
float, an object must have a density less than or
equal to that of the fluid in which it is
immersed.
WMg rblockVg, V volume of block F
fraction of block submerged Volume displaced
fV Weight of displaced water rwaterfVgFb Fb-W
Ma0 (equilibrium) Fb W ? rwaterf Vg
rblockVg f rblock/rwater
How do steel ships float?
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Conceptual Question
A boat is floating in a lake. The boat has a
large rock in it. If the rock is thrown
overboard, does the level of the water in the
lake h increase, decrease or remain the same?
Inside the boat, The rock displaces a volume of
water equal in mass to the rock. At the bottom of
the lake, the rock displaces only a volume of
water equal in mass to the rock. The density of
the rock is about 4 times larger than the density
of water. The height h of the water on the shore
(not on the side of the boat) goes DOWN when you
through the rock overboard.
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Walker Problem 29, pg. 494
A 0.12-kg balloon is filled with helium (density
0.179 kg/m3). If the balloon is a sphere with
a radius of 5.2 m, what is the maximum weight it
can lift? Density of air 1.29 kg/m3.
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Barometer

• Fill a tube with fluid, and then invert the tube.
  If the fluid tries to flow out, it creates a
  vacuum at the (sealed) top. Therefore P0 at
  top, but P Patmosphere at bottom.
• Pressure varies with depth from top as P rfluid
  g y
• As long as P(h) rfluid g y lt Pat, the fluid
  will NOT flow out!
• In equilibrium, the height h measures the ambient
  Pressure
• P rfluid g h

y

• Mercury barometer rHg 13,600 kg/ m3
• h P/ rfluid g (1.01 ? 105 N/m2 )/(1.36 ?
  104 kg/m3)(9.81 m/s2) 757mm
• 1 Torr 1 mm of Hg, Standard Atmospheric
  pressure 760 mm of Hg
• Water barometer rWater 1000 kg/ m3, h10.3 m !!

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The Continuity Equation
If you have continuous flow of a fluid, then the
rate of mass flow is the same at every
point. r1A1v1 r2A2v2 (general case all
liquids and gasses)
If the density does not change, which is true for
most liquids A1v1 A2v2 (liquids)
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Bernoullis Principle

• Conservation of energy in a flowing fluid leads
  to Bernoullis Equation (work done by pressure
  change in mechanical energy of a small volume of
  fluid)
• P1 ½rv12 rgy1 P2 ½rv22 rgy2
• Here we assume that the density does not change.

Example lift on an airplane wing
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Water Pipe

• Water is flowing continuously in the pipe shown
  below.
• Where is the velocity of the water greatest?
• (b) Where is the pressure in the water greatest?

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Walker Problem 48, pg. 495
A horizontal pipe contains water at a pressure of
110 kPa flowing with a speed of 1.4 m/s. When
the pipe narrows to one-half its original
diameter, what is (a) the speed and (b) the
pressure of the water?
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Quiz
a)
b)
c)

• Three containers a), b), c) each have the same
  bottom surface area, A, and are each filled with
  water to the same height h. Neglect the mass of
  the containers.
• Which container has the greatest mass of water.
  (or they all equal)?
• Which container has the greatest pressure of
  water pushing on the bottom surface (or they all
  equal)?
• For container b), if the pressure of the water on
  the bottom is Pb, is the force of the container
  pushing down on the shelf greater, equal, or less
  than Pb A? Hint Is the volume of water in b)
  greater, equal or less than A h?