Download Physics--Chapter 9: Fluid Mechanics

Physics--Chapter 9: Fluid Mechanics
Lecture Notes
I.
Fluids and Buoyant Force
A. Defining a Fluid
1. A fluid is defined as a nonsolid state of matter in which the atoms or
molecules are free to move past each other
2. Fluids are also defined by the property they have of being able to flow and
to alter their shape
3. Liquids are fluids; they have a definite volume but not a definite shape
4. Gases are fluids; they do not have a definite volume or shape
B. Density and Buoyant Force
1. Density
a. Mass density (or just density) is the mass per unit volume of a
substance
b. Mass density = mass/volume; we'll measure it in kg/m 3
c. Equation for mass density ():  = m/V
d. The density of solids and liquids changes very little with pressure and
temperature changes so a standard density can be given
e. The density of gases changes significantly with temperature and
pressure changes so the condition at which density was measured
needs to also be given
2. Buoyant Force
a. Buoyant force is a force that acts upward on an object submerged in a
liquid or floating on the liquid's surface
b. Buoyant force acts in the opposite direction of gravity, so the weight
of an object submerged in a fluid is less than the object's weight
(called its apparent weight)
3. Archimedes' Principle
a. Archimedes' principle states that any object completely or partially
submerged in a fluid experiences an upward buoyant force equal in
magnitude to the weight of the fluid displaced by the object
b. According to Archimedes' principle, the magnitude of the buoyant
force on an object equals the weight of fluid displaced
c. Equationally, Archimedes' principle becomes FB = Fg = mfg where FB
is the buoyant force, Fg is the weight of displaced fluid, mf is the mass
of displaced fluid, and g is 9.81 m/s2
How do you know whether an object will float or sink?
d. For floating objects, the buoyant force is equal to the weight of the
object, so the above equation for floating objects becomes
FB = Fg = mog where Fg is now the weight of the object and mo is the
mass of the object
Physics--Chapter 9: Fluid Mechanics
Lecture Notes
II.
III.
e. Since the apparent weight of a submerged object depends on density,
the relationship between the weight of a submerged object and the
buoyant force on the object can be expressed using their densities:
Fg of the object/FB = o/f where o is the density of the object and f
is the density of the fluid
Fluid Pressure and Temperature
A. Pressure
1. Pressure is simply defined as force per unit area, P = F/A, with units of
N/m2, which is called a pascal (Pa) (1 atm = 105 Pa)
2. Fluids exert pressure; the use of hydraulics is a good example
 An enclosed fluid is put under constant pressure so a smaller force to a
smaller "pad" results in a larger force to a large lift
An equational relationship for how hydraulic systems work
3. Pressure exerted on a submerged object increases with depth
a. Equationally, this can be determined by P = Po + gh
b. In the above equation, P is absolute pressure, Po is atmospheric
pressure,  is the density of the fluid, and h is depth
c. The above equation also reveals that buoyant forces arise because of
the difference in fluid pressure between the top and bottom of an
object (back to Archimedes' principle)
4. Barometers measure atmospheric pressure which results from each
successive layer of air's weight pushing down on the layers below it
a. atmospheric pressure also varies with depth--how does this compare to
water pressure and depth?
b. Atmospheric pressure exerts a force of about 200,000 N or 40,000 lbs
on us!
5. Gas pressure arises from the rate at which particles of gas transfer
momentum to the walls of a container--this transfer rate is equal to the
force exerted on the container, and this force per unit area is gas pressure
B. Temperature and Gas Laws
1. Temperature is defined as a measure of the average kinetic energy of the
particles in a substance
2. We'll use the Kelvin scale
3. Temperature, pressure, and volume of gases are related according to the
ideal gas law
a. In physics, the ideal gas law becomes PV = NkBT where pressure is in
N/m2 or pascals, volume is measured in m3, N is the number of gas
particles, kB is Boltzmann's constant, 1.38 x 10-23 J/K, and temperature
is in Kelvin
b. Also recall the combined gas law, P1V1/T1 = P2V2/T2
Fluids in Motion
Physics--Chapter 9: Fluid Mechanics
Lecture Notes
A. Fluid Flow
1. Described two different ways
a. Laminar flow is when every particle of the fluid that passes a
particular point moves along the same smooth path traveled by the
particles that passed that point earlier--this path is called a streamline
b. Turbulent flow occurs when the flow becomes irregular above a
certain velocity or under conditions hat can cause sudden changes in
velocity--eddy currents are irregular motions of the fluid under
turbulent conditions
2. Ideal Fluids
a. Whether gas or liquid, ideal fluids don’t really exist
b. Ideal fluids are incompressible (density remains constant), nonviscous
(have no internal friction), and have a steady flow (no turbulence or
eddy currents)
c. We use ideal fluids in problem-solving because it is much easier and is
a “close enough” approximation (this assumption breaks down at
temperature and pressure extremes)
B. Conservation Laws in Fluids
1. The mass of fluid flowing into a pipe at one end must equal the mass
flowing out of the pipe in the same time interval
2. Using the equations for density, volume of a cylinder, for steady flow
through a pipe 1A1ν1t = 2A2ν2t, but density (1 and 2) and time
interval (t) would be the same so they are canceled out leaving A1ν1 =
A2ν2
3. This equation, A1ν1 = A2ν2, is referred to as the continuity equation where
A1 and A2 represent the two different cross-sectional areas of the pipe, and
ν1 and ν2 are the fluid speeds at those areas
4. A•ν, which would have the units volume per time, = flow rate
C. Bernoulli
1. Bernoulli’s principle states that the pressure in a fluid decreases as its
velocity increases, OR swiftly moving fluids exert less pressure than do
slowly moving fluids
2. Bernoulli’s equation says that if a fluid is nonviscous, the energy in a
small volume of the fluid does not change as the volume moves along a
given streamline
3. Bernoulli’s equation: P + ½ρν2 + gh = constant; usually used as
P1 + ½ρ1ν12 + 1gh1 = P2 + ½ρ2ν22 + 2gh2