Transcript Slide 1

Today (Chapter 10, Fluids)
Review Concepts from Tuesday
Continuity Equation
Bernoulli’s Equation
Applications/Examples
Tomorrow (Chapters 6-10)
Review for Exam 2
Pressure
When an object is submerged in a fluid (air, water, etc.) it will
feel a pressure from the fluid. This pressure is caused by the
collision of molecules of the fluid with the surface of the
object.
From our own experience, we know that the pressure deeper
in a fluid is more than the pressure at the surface.
What is the difference in the pressure on the top of the object
and on the bottom of the object?
P2  P1  gh
Quiz Question!
A submarine is constructed so that it can
safely withstand a pressure of 1.6 x 107 Pa.
How deep can this submarine descend in the
ocean if the average density of sea water is
1025 kg/m3?
Fluids and Effect of Gravity
• U-Tube Example 10.3 on Page 316.
• Consider a U-Tube with both ends open to the
atmosphere. Let there be two liquids, water
with density 1000 kg/m3 and an oil with
density 700 kg/m3. Let the oil be filled 0.10 m
above the oil-water boundary.
• Determine the difference in the level of the two
liquids, d.
Pascal’s Principle
(why hydraulics work)
Any change in the pressure applied to a completely
enclosed fluid is transmitted undiminished to all parts
of the fluid and the enclosing walls.
In other words, pressure applied at one end of a
hydraulic system is added to every point in the system.
F1
A1
F2
A2
Archimedes’ Principle
Buoyancy (why things float)
Any fluid applies a buoyant force to an object that is partially
or completely immersed in it.
The magnitude of the buoyant force equals
the weight of the fluid that the object displaces.
In other words, if the object can displace enough fluid,
it will generate enough buoyant force to counteract
it’s weight and it will float!
FB  Wfl uid di spl ace d
FB  gV
A solid block of wood that normally
floats in water is pushed down and held
under water by a physics 218 student.
Does the water level in the container rise
or fall? (rise)
Before
Is the buoyant force on the wood greater
than, equal to, or less than the weight of
the object? (greater)
After
A flat-bottomed barge, loaded with coal, has a mass of 3×105 kg.
The barge is 20.0 m long and 10.0 m wide. It floats in the fresh
water. What is the depth of the barge below the waterline?
FB  mg
mdisplaced water g  mbarge g
 waterVdisplaced water  mbarge
 water wlh  mbarge
h
mbarge
 water wl

300000kg
 1.5m
3
1000kg / m 20m10m
A block of birch wood floats in oil with 90% of its volume
submerged. What is the density of the oil? The density of the birch
is 0.67g/cm3.
FB  mbirch g
mdisplaced oil g  mbirch g
oilVdisplaced oil g  birchVbirch g
oilVdisplaced oil  birchVbirch
oil 0.9Vbirch  birchVbirch
oil 0.9  birch
oil  0.744 g / cm3
Continuity Equation
What comes in, must go out.
x1  v1t
x2  v2t
m2  2 A2 x2  2 A2v2t
m1  1 A1 x1  1 Av
1 1t
1 Av
1 1t  2 A2v2 t
1 Av
1 1  2 A2v2
A1v1 
Constant
A2v2 density
Continuity Equation
What comes in, must go out.
Mass flow rate
m
  Av
t
Units: kg/s
Volume flow rate
V
 Av
t
Units: m3/s
A1v1  A2v2
Continuity Equation
(why a fire hose works)
The mass flow rate has the same value at every position along
a tube that has one entry point and one exit point.
1 A1v1  2 A2 v 2
In other words, if you reduce the area that the fluid can flow
through, it has to flow faster!
A1v1  A2v2
What comes in, must go out.
(If you shove 2 gallons of water in one end of a pipe in one
second, 2 gallons of water must come out the other end of
the pipe in one second.)
• Continuity Equation Example
A1v1  A2v2
Water is flowing through a pipe
with a cross-sectional area of 4.0 cm2 and
connects to a faucet with an opening of
area 0.50 cm2.
If the water is flowing at a speed of
5.0 m/s in the pipe, what is the speed as
it leaves the faucet?
Bernoulli’s Equation
(why a heavy airplane can fly)
In a (steady, irrotational) flow of a (nonviscous,
incompressible) fluid of density , the pressure, fluid speed,
and elevation (y) at any two points are related by:
1 2
1 2
P1  v1  gy1  P2  v2  gy2
2
2
In other words, if we are talking about two points
with the same elevation:
a quickly flowing fluid has a lower pressure
than a slowly flowing fluid.
Application: Bernoulli’s Equation
• What is the upward “lift” force on an airplane
wing?
•
•
•
•
•
Assume the area of the wing is 60 m2
The speed of air over the wing is 250 m/s
The speed of air under the wing is 200 m/s
Let the wing be 1 meter thick
Assume the density of the air to be 1.29 kg/m3
1 2
1 2
P1  v1  gy1  P2  v2  gy2
2
2
Difference in Pressures
• The pressure in the atmosphere is not a
constant, but fluctuates as the weather changes.
• Consider a window of a house where the
pressure inside is 101,000 Pa and the pressure
outside has decreased to 96000 Pa. Let the area
of the window be 1.5 m2.
• What is the force exerted on the window due to
this difference in pressures?
Fluids and Gravity
A new water tower is built on a very tall hill. At a
nearby house, the pipes are leaking because the
pressure in the water is too high.
Assume water pipes leak when the pressure
exceeds five times atmospheric pressure.
Determine the relative height of the water tower in
comparison to the house.
Archimedes’s Principle and Buoyancy
• Consider a wooden block of density
700 kg/m3 and a volume of 2.0 m3.
• Determine the force required to hold
it completely under water.
Archimedes’s Principle and Buoyancy
• Concept: Apparent Weight
• Consider a piece of aluminum attached to a
string and suspended in a pool of oil
(density 750 kg/m3).
• Let the apparent weight of the aluminum be
540 N.
• Determine the volume of the aluminum.