Physics 103: Lecture 18 Fluids

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Physics 103 Lecture 18Fluids

• Todays lecture will cover
• Pascals Principle
• Archimedes Principle
• Fluids in motion Continuity Bernoullis
  equation

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Midterm II

• Midterm Exam II, Thur Nov 5th, 545 - 7 PM
• Arrange alternate times today before/after
  lecture
• Material from Chapters 5-8 inclusive
• One page of notes, both sides, (8.5 x 11)
  allowed
• You can re-use your equation sheet from exam 1 if
  you only wrote on one side.
• 20 multiple choice questions
• Scantron will be used - bring 2 HB pencils
  calculator
• Know your student Id number and section number
  (starts with 3)
• All relevant physics constants will be provided
  as with exam 1
• Exam Rooms
• 165 Bascom 302, 303, 304, 306, 312, 318, 320,
  324 B-10 Ingraham 305, 313, 317, 321, 322, 327, 3
  28, 329, 330
• 3650 Humanities 307, 308, 309, 310, 311, 314,
  315, 319, 323, 326
• Alternative exam room Chamberlin 4320 (lab room)
• (all the same as Exam I)

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Pressure in a fluid or gas

• Impulse to book
• Fx?t ?px ?(Mvx)
• Fx ?(Mvx)/ ?t
• Force is perpendicular to surface
• Force proportional to area of surface
• pressure (p)
• p Force/area N/m2
• 1 N/m2 1 Pascal (Pa)

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Pressure Force per Unit Area

• Which will hurt more?
• If you are pricked by a nail with a force equal
  to your weight
• If your entire weight is supportedby a bed of
  similar nails
• Both will hurt the same

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Pressure
y
Atmospheric Pressure
Even when there is no breeze air molecules are
continuously bombarding everything around -
results in pressure normal atmospheric pressure
1.013 x 105 Pa
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Pressure and Depth

• Examine the darker region, assumed to be a fluid
  ?
• It has a cross-sectional area A
• Extends to a depth h below the surface
• Three external forces act on the region
• -P1A P2A - Mg 0 P2 P1 Mg/A P1 Mgh/V
  P1 ?gh
• At the surface compared to at depth h
• Po is normal atmospheric pressure
• 1.013 x 105 Pa 14.7 lb/in2

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Fluids Summary
Pressure (P) P Force/Area N/m2 1 N/m2 1
Pascal (Pa)
Density Mass/Volume ? M / V units kg/m3
Pressure variation with depth
? P r g h
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Pressure and DepthBarometer a way to measure
atmospheric pressure
p2 p1 ?gh patm ?gh Measure h, determine
patm example--Mercury ? 13,600 kg/m3 patm
1.013 x 105 Pa ? h 0.760 m 760 mm(for 1 atm)
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Pascals Principle

• A change in pressure applied to an enclosed fluid
  is transmitted undiminished to all portions of
  the fluid and to the walls of its container.
• This principle is used in hydraulic system
• P1 P2
• (F1 / A1) (F2 / A2)
• Can be used to derive large gain by making A2
  much larger than A1
• F2 F1 (A2 / A1)
• Work done is the same height by which the
  surface A2 rises is smaller than the change in
  the height of surface with area A1.

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Archimedes Principle

• Buoyant Force (B)
• weight of fluid displaced
• B ?fluid g Vdisplaced
• W ?object g Vobject
• object sinks if ?object gt ?fluid
• object floats if ?object lt ?fluid
• If object floats.
• BW
• Therefore ?fluid g Vdisplaced ?object g
  Vobject
• Therefore Vdisplaced/Vobject ?object / ?fluid

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Archimedes Principle
Weight of a glass filled to the brim with water
is Wb. A cube of ice is placed in it, causing
some water to spill. After the splilled water is
cleaned up, the weight of the glass with ice cube
is Wa. How do the weights compare 1. Wb gt Wa.
2. Wb lt Wa. 3. Wb Wa.
Archimedes Principle The buoyant force on an
object equals the weight of the fluid it
displaces. Weight of water displaced Buoyant
force Weight of ice
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Preflight 7
Suppose you float a large ice-cube in a glass of
water, and that after you place the ice in the
glass the level of the water is at the very brim.
When the ice melts, the level of the water in the
glass will 1. Go up causing the water to spill.
2. Go down. 3. Stay the same.
Archimedes Principle The buoyant force on an
object equals the weight of the fluid it
displaces. Weight of water displaced Buoyant
force Weight of ice When ice melts it will turn
into water of same volume
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Fluid Flow
Fluid flow without friction

• Volume flow rate DV/Dt ADd/Dt Av (m3/s)
• Continuity A1 v1 A2 v2
• i.e., flow rate the same everywhere
• e.g., flow of river or a garden hose
• Water through a narrow hose moves faster

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Faucet

• A stream of water gets narrower as it falls from
  a faucet (try it see).
• Explanation the equation of continuity

The velocity of the liquid increases as the water
falls due to gravity. If the volume flow rate is
conserved, them the cross-sectional area must
decrease in order to compensate
The density of the water is the same no matter
where it is in space and time, so as it falls
down and accelerates because of gravity,the water
is in a sense stretched, so it thins out at the
end.
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Bernoullis Equation

• Pressure drops in a rapidly moving fluid
• whether or not the fluid is confined to a tube
• For incompressible, frictionless fluid

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Applications of Bernoullis Principle

• Wings and sails
• Higher velocity on one side of sail or wing
  versus the other results in a pressure difference
  that can even allow the boat to sail into the
  wind
• Water pressure/velocity at your house
• Velocity measurement

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Sailing Against the Wind

• v2gtv1
• Therefore
• P1gtP2
• Pressure difference causes a force

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Problem 1
(a) Calculate the approximate force on a square
meter of sail, given the horizontal velocity of
the wind is 6 m/s parallel to its front surface
and 3.5 m/s along its back surface. Take the
density of air to be 1.29 kg/m3. (b) Discuss
whether this force is great enough to be
effective for propelling a sail boat.
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Problem 2
(a) What is the pressure drop due to Bernoulli
effect as water goes into a 3 cm diameter nozzle
from a 9 cm diameter fire hose while carrying a
flow of 40 L/s? (b) To what maximum height above
the nozzle can this water rise neglecting air
resistance.
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Extra
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Heavy Ice
Two identical glasses are filled to the same
level with water. One glass has a cube of
regular ice floating in it and the other has a
cube of special ice-9, heavier than water, which
sinks to the bottom. Which of the two glasses
weighs more? 1. The glass with the regular ice
2. The glass with the ice-9 3. Both glasses
weigh the same
The ice-9 sinks. The buoyant force equal to the
weight of the displaced water is not sufficient
to counter the weight of the ice-9.
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Archimedes Principle
The buoyant force on an immersed body has the
same magnitude as 1. The weight of the
body. 2. The weight of the fluid displaced by
the body 3. The difference between the weights
of the body and the displaced
fluid. 4. The average pressure of the fluid
times the surface area of the body.
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Buoyant force and depth
Imagine holding two identical bricks under water.
Brick A is just beneath the surface of the water,
while brick B is at a greater depth. The force
needed to hold brick B in place is 1.
larger 2. the same as 3. Smaller than the
force required to hold brick A in place.
The buoyant force on each brick is equal to the
weight of the water it displaces and does not
depend on depth.
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A 200-ton ship enters the lock of a canal. The
fit between the sides of the lock and the ship is
so tight that the weight of the water left in the
lock after it closes is much less than 200 tons.
Can the ship still float if the quantity of water
left in the lock is much less than the ships
weight? 1. Yes, as long as the water gets up to
the ships waterline. 2. No, the ship touches
bottom because it weighs more than the water in
the lock.
What matters is not the weight of the water left
in the lock, but the weight of the water forced
out of the lock by the ship. As long as the
density of the ship is less than that of water,
and the water gets to the waterline, it floats.
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Preflight 8
Which weighs more 1. A large bath tub filled to
the brim with water. 2. A large bath tub filled
to the brim with water with a toy battle
ship floating on it. 3. Both weigh the same.
Archimedes Principle The buoyant force on an
object equals the weight of the fluid it
displaces. Weight of water displaced Buoyant
force Weight of the ship in water
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Preflight 9
An oil tanker is floating in a port. The oil
tanker is loaded with oil. The density of oil is
less than that of water. The waterline (i.e., a
line marked on the outside of a ship) of a loaded
tanker compared to that of the empty tanker is
1. lower. 2. the same. 3. higher.
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Preflight 10
Marbles dropped into a partially filled bathtub
sink to the bottom. The downward force on the
bottom of the tub with the marbles, compared to
the situation without the marbles is 1. lower.
2. the same. 3. higher.
The downwards force with the marbles is the sum
of the weights of the marbles and the water
column above it. Marbles sank. Therefore, their
weight is higher compared to the volume occupied
by water which originally filled their place.
Therefore, the total force is higher.
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Melting Ice and Volume
Two identical glasses are filled to the same
level with water. One of the two glasses has ice
cubes floating in it. When the ice cubes melt, in
which glass is the level of the water higher?
1. The glass without ice cubes 2. The glass
with ice cubes originally 3. The level is the
same in both.
The weight of water from molten ice cubes is
equal to the water originally displaced by the
cubes.
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Floating Balls
Two identical glasses are filled to the same
level with water. One of the two glasses has
plastic balls floating in it. If the density of
the plastic balls is less than that of ice, which
of the two glasses weighs more? 1. The glass
without plastic balls 2. The glass with plastic
balls 3. Both glasses weigh the same
The plastic balls displace exactly their own
weight in water, so the two glasses weigh the
same amount.
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Sunken Balls
Two identical glasses are filled to the same
level with water. Solid steel balls are at the
bottom in one of the glasses. Which of the two
glasses weighs more? 1. The glass without steel
balls 2. The glass with steel balls 3. Both
glasses weigh the same
The steel balls sink. The buoyant force equal to
the weight of the displaced water is not
sufficient to counter the weight of the steel
balls. Therefore, the glass with steel balls
weighs more.
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Preflight 4
Two hoses, one of 20-mm diameter, the other of
15-mm diameter are connected one behind the other
to a faucet. At the open end of the hose, the
flow of water measures 10 liters per minute.
Through which pipe does the water flow faster?
1. The 20-mm hose 2. The 15-mm hose 3. Water
flows at the same speed in both cases 4. The
answer depends on which of the two hoses comes
first in the flow
When a tube narrows, the same volume occupies a
greater length. For the same volume to pass
through points 1 and 2 in a given time, the
velocity must be greater at point 2. The process
is reversible.
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Preflight 5

• A large bucket full of water has two equal
  diameter drains. The water level in the bucket is
  kept constant by constantly refilling it. One is
  a hole in the side of the bucket at the bottom,
  and the other is a pipe coming out of the bucket
  near the top, which is bent downward such that
  the bottom of this pipe even with the other hole,
  like in the picture below
• Though which drain is the water spraying out with
  the highest speed?
• 1. The hole
• 2. The pipe
• 3. Same

Since the pressures at the two drains are the
same and the liquid leaves the pipe at the same
height, their speeds are the same.
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Preflight 6
An artery with cross sectional area of 1 cm2
branches into 20 smaller arteries each with 0.5
cm2 cross sectional area. If the velocity of
blood in thicker artery is v, what is the
velocity of the blood in the thinner
arteries? 1. 0.1 v 2. 0.2 v 3. 0.5 v 4.
v 5. 2 v
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Problem
(a) Calculate the approximate force on a square
meter of sail, given the horizontal velocity of
the wind is 6 m/s parallel to its front surface
and 3.5 m/s along its back surface. Take the
density of air to be 1.29 kg/m3. (b) Discuss
whether this force is great enough to be
effective for propelling a sail boat.
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Torricellis Theorem
P1, v1, h1
h
P2P1 , v2 , h2
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Velocity Measurement Pitot tube
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Notes on Moduli

• Solids have Youngs, Shear, and Bulk moduli
• Liquids have only bulk moduli