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MR. SURRETTE
VAN NUYS HIGH SCHOOL
CHAPTER 9: FLUID MECHANICS
CLASSNOTES
FLUIDS
A fluid is a substance whose molecules move freely past one another and has the tendency to assume the
shape of its container. In general, liquids and gases are fluids.
DENSITY
The density  of a substance is mass per unit volume. It has units of kilograms per cubic meter (or
grams per cubic centimeter) in the metric system:
=m/V
PRESSURE
Pressure is force exerted per unit area:
P=F/A
Pressure [Pascals] = Force [Newtons] / Area [m2]
BASIC LAW OF FLUID PRESSURE
The pressure at the bottom of a fluid can be expressed as:
P = gh
( = density of fluid, h = depth of fluid
g = 9.8 m/s2)
FLUID PRESSURE
The pressure at any point in a fluid depends only on its density and its depth. It acts equally in all
directions.
ABSOLUTE PRESSURE
The absolute pressure, P, at a depth, h, below the surface of a liquid which is open to the atmosphere is
greater than atmospheric pressure, Pa, by an amount that depends on the depth below the surface.
ABSOLUTE PRESSURE EQUATION
P = Pa + gh
( = density of the liquid)
Example 1. What is the absolute pressure at the bottom of a 5 m deep swimming pool? (Note the
pressure contribution from the atmosphere is 1.01 x 105 N/m2, the density of water is 103 kg/m3, and g =
9.8 m/s2)
1A.
(1) P = Pa + gh
(2) P = (1.01 x 105 N/m2)+(1.0 x 103 kg/m3)
(9.8 m/s2)(5 m)
(3) P = 1.5 x 105 N/m2
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PHYSICS
MR. SURRETTE
VAN NUYS HIGH SCHOOL
PASCAL’S PRINCIPLE
Pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and the
walls of the containing vessel:
F1 / A1 = F2 / A2
PASCAL’S PRINCIPLE
Pressure is equal (P1 = P2)
Example 2. A hydraulic lift raises a 2000 kg automobile when a 500 N force is applied to the smaller
piston. If the smaller piston has an area of 10 cm2, what is the cross-sectional area of the larger piston?
2A.
(1) F1 / A1 = F2 / A2
(2) A2 = (F2A1)/F1
(3) F2 = w = mg
(4) A2 = (mgA1) / F1
(5) A2 = (2000 kg)(9.8 m/s2)(1.0 x 10-3 m2) / 500 N
(6) A2 = 3.92 x 10-2 m2
ARCHIMEDE’S PRINCIPLE
Archimede’s Principle is the law of buoyancy. When an object is submerged in a fluid (completely or
partially), there exists an upward force on the object that is equal to the weight of the fluid that is
displaced by the object.
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PHYSICS
MR. SURRETTE
VAN NUYS HIGH SCHOOL
ARCHIMEDE’S PRINCIPLE
BUOYANT FORCE
One common expression for the upward buoyant force is:
FB = Vg
(FB = buoyant force,  = density of liquid
V = volume of submerged object,
g = 9.8 m/s2)
BUOYANT FORCE
Another expression for buoyant force is:
FB = w – w’
(w = actual weight of object
w’ = apparent weight of submerged object)
DENSITY OF OBJECTS
The density of a submerged object can be determined:
object = liquid / (w – w’)
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PHYSICS
MR. SURRETTE
VAN NUYS HIGH SCHOOL
Example 3. An object of mass 0.5 kg is suspended from a scale and submerged in a liquid. What is the
buoyant force that the fluid exerts if the reading on the scale is 3.0 N?
3A.
(1) w = mg
(2) w = (0.5 kg)(9.8 m/s2)
(3) w = 4.9 N
(4) FB = w – w’
(5) FB = 4.9 N – 3.0 N
(6) FB = 1.9 N upwards
FLOW RATE
The product Av is called the flow rate. The flow rate at any point along a pipe carrying an
incompressible fluid is constant:
A1v1 = A2v2
Example 4. If the flow rate of a liquid going through a 2.00 cm radius pipe is measured at
0.8 x 10-3 m3/s, what is the average fluid velocity in the pipe?
4A.
(1) Flow rate = Av
(2) v = Flow rate / A
(3) v = Flow rate / r2
(4) v = 0.8 x 10-3 m3/s / (2.0 x 10-2 m)2
(5) v = 0.64 m/s
TORRICELLI’S THEOREM
The speed of efflux is the same as the speed a body would acquire in falling freely through a height h:
v = (2gh)1/2
TORRICELLI’S THEOREM
Torricelli’s Theorem can be analyzed another way: Point 1 is at the surface of water in a tank and Point
2 is the position of the hole.
TORRICELLI’S THEOREM
y1 is the vertical distance from the ground to Point 1 and y2 is the vertical distance from the ground to
Point 2.
TORRICELLI’S THEOREM EQUATION
v = (2g(y1-y2))1/2
(Notice that (y1 – y2) = h = fluid depth)
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PHYSICS
MR. SURRETTE
VAN NUYS HIGH SCHOOL
TORRICELLI’S THEOREM
Example 5. A liquid filled tank has a hole on its vertical surface just above the bottom edge. If the
surface of the liquid is 0.4 m above the hole, at what speed will the stream of liquid emerge from the
hole?
Question 5A.
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PHYSICS
MR. SURRETTE
VAN NUYS HIGH SCHOOL
5A.
(1) v = (2gh)1/2
(2) v = [(2)(9.8 m/s2)(0.4 m)]1/2
(3) v = 2.8 m/s
BERNOULLI’S EQUATION
Bernoulli’s equation relates to the Work Energy Theorem and to the Conservation of Energy. It is
derived from Newtonian mechanics.
BERNOULLI’S EQUATION
p1 + ½ v12 + gy1 = p2 + ½ v22 + gy2
[p1 = initial pressure, p2 = final pressure
 = density of fluid, g = 9.8 m/s2
v1 = initial velocity, v2 = final velocity
y1 = initial height, y2 = final height]
Example 6. Water (density = 1 x 103 kg/m3) is flowing through a pipe whose radius is 0.04 m with a
speed of 15 m/s. This same pipe goes up to the second floor of the building, 3 m higher, and the
pressure remains unchanged. What is the cross-sectional area of the pipe on the second floor?
6A.
(1) First Equation: p1 + ½ v12 + gy1 = p2 + ½ v22 + gy2
(2) ½ v12 + gy1 = ½ v22 + gy2
(3) v12 = v22 + 2gy2
(4) v2 = [(v1)2 – 2gy2]1/2
(5) v2 = [(15 m/s)2 – 2(9.8 m/s2)(3 m)]1/2
(6) v2 = 12.9 m/s
(7) Second Equation: A1v1 = A2v2
(8) A2 = A1v1 / v2
(9) A1 = r2
(10) A2 = (r2)v1 / v2
(11) A2 = (0.04 m)2(15 m/s) / (12.9 m/s)
(12) A2 = 5.84 x 10-3 m2
BERNOULLI’S PRINCIPLE
Bernoulli’s principle states that swiftly moving fluids exert less pressure than slowly moving fluids.
AIR FOILS
An air foil is a body (like an airplane wing or propeller blade) designed to provide a desired reaction
force when in motion.
AIR MOLECULE MOTION
A popular expression in science is: “Nature abhors a vacuum.” This is very true for air. Like all other
gases, air quickly fills in all available space.
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PHYSICS
MR. SURRETTE
VAN NUYS HIGH SCHOOL
AIR MOLECULE MOTION
At the molecular level, air molecules randomly bounce off each other and prevent gaps.
AIR MOLECULE DISTANCE
Airplane wings are air foils designed to create greater distance for air molecules to travel above the wing
than below the wing.
AIR MOLECULE TIME
Because air molecules quickly fill in available space, it takes the same amount of time for a parcel of air
(think of a small invisible cloud) to split, move around both sides of an airplane wing, and then
reassemble on the other side of the wing.
VELOCITY EQUATION
As we saw in Chapter 1 (Kinematics), velocity is equal to distance divided by time:
v=d/t
WINGS AND AIR VELOCITY
Air molecules above an airplane wing travel a greater distance in the same amount of time as air
molecules below an airplane wing.
WINGS AND VELOCITY EQUATION
Since v = d/t:
(1) d ABOVE WING > d BELOW WING
(2) V ABOVE WING = d ABOVE WING / t CONSTANT
(3) V BELOW WING = d BELOW WING / t CONSTANT
WINGS AND VELOCITY EQUATION
Therefore:
V ABOVE WING > V BELOW WING
(v2 > v1 in diagram below)
BERNOULLI’S PRINCIPAL AND FLIGHT
According to Bernoulli’s principle, faster air exerts less pressure than slower air.
BERNOULLI’S PRINCIPAL AND FLIGHT
Since V ABOVE WING > V BELOW WING, air pressure below the wing must be greater than air pressure above
the wing:
P ABOVE WING < P BELOW WING
(P2 < P1 in diagram below)
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PHYSICS
MR. SURRETTE
VAN NUYS HIGH SCHOOL
APPLICATION OF BERNOULLI’S PRINCIPLE
CROSS-SECTION OF WING (AIR FOIL)

APPLICATION OF BERNOULLI’S PRINCIPLE
P = P1 – P2. When the force exerted by P exceeds the weight of the airplane, the airplane is
provided lift and flies.
BERNOULLI’S TUBE
The velocity of a fluid at the constricted end of a pipe is greater than the velocity at the wider ends if
steady flow is maintained.
BERNOULLI’S TUBE
BERNOULLI’S TUBE
At Point 1, the velocity is less, but the pressure is greater. At Point 2, the velocity is greater, but the
pressure is less.
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PHYSICS