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FLUIDS
What is a fluid? A substance that can flow. A fluid is a substance that cannot maintain its
own shape but takes the shape of its container. Gases and liquids are called fluids because
neither have an orderly arrangement.
In rigid bodies we expressed Newton's laws in terms of mass and force. We describe
fluids in terms of density and pressure
Pressure
force per surface area; symbol is P; SI unit is Pa (Pascal), or N/m2.
P=F/A
where F is force or weight in Newtons
A is cross-sectional area in m2
The pressure at any point in a fluid acts equally in all directions. Also, the force
due to the fluid pressure always acts perpendicularly to any surface the fluid is in
contact with.
Atmospheric pressure= 1.013 x 105 Pa (measured at sea level). This can be
approximated as 101 kPa.
Gauge Pressure Pressure gauges measure the pressure over and above
atmospheric pressure. This is called gauge pressure.
Absolute Pressure To get the absolute pressure at a point, one must add the
atmospheric pressure to the gauge pressure. For example, if gauge pressure is 100
kPa, the absolute pressure at that point is the sum of 100 kPa and 101 kPa, or 201
kPa.
Manometer A U-shaped tube partially filled with liquid used to measure
pressure. The pressure is equal to the difference in height of the two levels of the
liquid according to P = Patm + gh.
Density
the ratio of mass to volume; density is a characteristic property of a any pure
substance. Its SI unit is kg/m3
=m/V
where  is density in kg/m3
m is mass in kilograms
V is volume in m3
(H2O) = 1000 kg/m3
Sometimes densities are given in g/cm3. To convert to the SI unit of kg/m3 simply
multiply by 1000.
Density Lab - Float or Sink: You find out!"
http://www.explorelearning.com/index.cfm?method=cResource.dspDetail&Resou
rceID=17
Specific Gravity The ratio of the density of that substance to the density of water
at 4.0. It has no units.
substance density, kg/m3
substance
density, kg/m3
aluminum
2700
iron and steel
7800
copper
8900
lead
11300
mercury
13600
ethyl alcohol
790
Hydrostatics
the study of fluids at rest
1. Pascal’s principle - any change in pressure at any point in a fluid is
transmitted unchanged throughout the fluid. Or, pressure applied to a
confined fluid increases the pressure throughout the fluid by the same
amount.
F1/A1 = F2/A2
This is the basis for squeezing a tube of toothpaste, hydraulic brakes, and for the
Heimlich maneuver.
2.
Hydrostatic pressure - pressure due to a fluid’s depth
P=gh
where  is density of fluid in kg/m3
h is the height (depth) of fluid
Pressure increases with depth. The pressure at any depth depends only upon that
depth and not upon any horizontal dimension. For example, Hoover Dam holds
back Lake Mead which is 700 ft deep. The bottom of Hoover Dam must
withstand the same pressure if it were only holding back a few thousand gallons
of water 700 ft deep. Also, the pressure at equal depths within a fluid is the same.
3.
Archimede’s principle - an object immersed in a fluid is buoyed up by a force
equal to the weight of the displaced fluid. A fluid provides some support for any object
placed in it. The upward force on an object placed in a fluid is called the buoyant force.
FB =  g V
where  is the density of the fluid in kg/m3
V is the volume of the displace fluid
FB is the buoyant force (The buoyant force occurs because the pressure in a fluid
increases with depth. The upward force on the bottom surface of a submerged
object is greater than the downward pressure on its top surface.)
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If an solid floats partially submerged in a liquid, the volume of liquid
displaced is less than the volume of the object. According to Archimedes
Principle, the weights of the object and its displaced fluid are the same.
The fractional part of an object that is submerged is equal to the ratio of
the density of the solid to the density of the liquid in which is floats (for
example, about 90% of an iceberg is submerged because the density of the
ice is about 90% that of sea water).
An object floats when the buoyant force is equal to its weight.
Objects submerged in a fluid appear to weigh less than they do when
outside the fluid.
Applet demonstrating buoyant force
http://www.sciencejoywagon.com/physicszone/lesson/otherpub/wfendt/buoyforce.htm
Fluid Dynamics
study of fluids in motion
Hydrodynamics
study of water in motion
Aerodynamics
study of air in motion
Lift on an airplane wing
http://www.explorelearning.com/index.cfm?method=cResource.dspDetail&Resou
rceID=24
Equation of Continuity
the volume of fluid passing two points per second is equal
A1 V1 = A2 V2
In a narrow tube, the velocity of the liquid is high; in a wide tube, the velocity of
the liquid is low.
Bernoulli’s principle
as the velocity of a fluid increases, the pressure exerted by that fluid decreases
on top of an airfoil there is low pressure due to high velocity airflow
on the bottom of an airfoil there is high pressure due to low velocity airflow
Fluids in Motion
There are two type of fluid flow, streamline (laminar) and turbulant flow. If the
slow is smooth (layers of fluid slide by each other smoothly), the flow is said to
be steamlined, or laminar. Above a certain speed (which depends upon many
factors), a flow becomes turbulent. Turbulent flow is characterized by the
formation of eddies.
Viscosity Internal friction in a fluid.
Mass Flow Rate the ratio of the mass of a fluid that passes a certain point in a
certain interval of time (or, m/t)
Volume Rate of Flow the ratio of the volume of a fluid that passes a certain point
in a certain interval of time (or, V/t). In SI units, this is m3/sec (or the same
thing as the product of area, A, and velocity, v.)
Bernoulli's Equation Where the velocity of a fluid is high, the pressure is low;
where the velocity is low, the pressure is high. Bernoulli's equation is an
expression of the law of conservation of energy.
P1 + 1/2 v12 + gh1 = P2 + 1/2 v22 + gh2
Torricelli's theorem A liquid leaves a spigot at the bottom os a reservoir with the
same speed that a freely falling object falling through the same height.
v1 = (2g(h2-h1))1/2
AP Multiple Choice Questions Fluid questions were added to the AP B test
beginning in 2002.
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Be able to determine the buoyant force acting on an object when given the
weight in air and the weight in the fluid.
Be able to recognize that the pressure acting on the bottom of a container
is due to the weight of the fluid above it.
Recognize that high velocity fluid has low pressure. Predict the
consequences of low pressure due to high velocity fluid.
AP Free Response Questions Fluid questions were added to the AP B test
beginning in 2002.
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All the ones that have occurred since 2002 have dealt with either a fluidbased laboratory exercise or with fluids at rest.
Be able to calculate gauge pressure.
Be able to calculate absolute pressure.
Be able to calculate pressure.
Kinematics is frequently integrated into fluid problems.
AP Physics B - Fluids Objectives
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Students should understand that fluids exert pressure in all directions.
Students should understand that a fluid at rest exerts pressure perpendicular to any
surface that it contacts.
Students should understand and be able to use the relationship between pressure
and depth in a fluid, P = gh.
Students should understand that the difference in pressure on the upper and lower
surfaces of an object immersed in liquid results in an upward force on the object.
Students should understand and be able to apply Archimedes Principle: the
buoyant force on a submersed object is equal to the weight of the liquid it
displaces.
Students should understand that for laminar flow, the flow rate of a liquid through
its cross section is the same at any point along its path.
Students should understand and be able to apply the equation of continuity,
1A1v1 = 2A2v2.
Students should understand that the pressure of a flowing liquid is low when the
velocity is high, and vice versa.
Students should understand and be able to apply Bernoulli's equation: P + gy +
1/2 v2 = constant
AP Fluids Sample Problems-Hydrostatics
1. Estimate the pressure exerted on a floor by a 50 kg model standing momentarily
on a single spiked heel (area = 0.05 cm2) and compare it to the pressure exerted
by a 1500 kg elephant standing on one foot (area = 800 cm2). Ans: 9.8 x 107 Pa;
1.8375 x 105 Pa
2. A tire gauge reads 220 kPa. What is the absolute pressure within the tire? Ans:
321 kPa
3. What is the pressure due to a column of water 100 m high? Ans: 980,000 Pa
4. The area of the output piston in a hydraulic lift is 20 times that of the input
cylinder. What force would it take to lift a 4000 lb car? Ans: 200 lb
5. A geologist finds that a moon rock whose mass is 8.20 kg has an apparent mass of
6.18 kg when submerged in water. What is the density of the rock? Ans: 4059.41
kg/m3
6. Water and then oil are poured into a U-shaped tube, open at both ends, and do not
mix. They come to equilibrium as shown. What is the density of the oil? The
height of the oil column is 27.2 cm; the distance from the top of the water column
to the top of its tube is 9.41 cm; the top of its tube is even with the top of the oil
column. (Hint: pressures at a and b are equal.) Ans: 654.04 kg/m3
AP Fluids Sample Problems-Hydrodynamics
1. A stream of water emerges from a pipe. The cross-sectional area of the water
stream at the tap is 1.2 cm2 and 43 mm lower than the tap is 0.35 cm2. At what
volume rate of flow does the water flow from the tap? Ans: 3.21 x 10-5 m3/s
2. Water is pumped from a pipeline 2 m above the ground to a water tower 15 m
above the ground. If the pipeline velocity is 8 m/s, its pressure is 3.103 x 105 Pa,
and water enters the tank at a pressure of one atmosphere, with what velocity does
the water enter the tank? Ans: 15.09 m/s
3. A gas is flowing through a pipe whose cross-sectional area is 0.07 m2. The gas
has a density of 1.30 kg/m3. A Venturi meter is used to measure the speed of the
gas. It has a cross-sectional area of 0.05 m2. The pressure difference between the
pipe and the Venturi meter is found to be 120 Pa. Find the speed of the gas in the
pipe and the volume rate of flow of the gas. Ans: 13.87 m/s; 0.97 m3/s
4. An aneurysm is an abnormal enlargement of a blood vessel such as the aorta.
Suppose that, because of an aneurysm, the cross-sectional area of the aorta
increases to a value 1.7 times greater than the original. The speed of the blood
(density of blood is =1060 kg/m3) through a normal portion of the aorta is 0.40
m/s. Assuming the aorta is horizontal (the person is lying down), determine the
amount by which the pressure in the enlarged region exceeds that in the normal
region. Ans: 55.53 Pa