Transcript Chapter 11

Chapter 11
Fluids
States of matter
• Solid
• Liquid
• Gas
• Plasma
Fluids
• Fluids – substances that can flow (gases, liquids)
• Fluids conform with the boundaries of any container
in which they are placed
• Fluids lack orderly long-range arrangement of
atoms and molecules they consist of
• Fluids can be compressible and incompressible
Liquids
• Have a definite volume, but no definite shape
• Exist at a higher temperature than solids
• The molecules “wander” through the liquid in a
random fashion
• The intermolecular forces are not strong enough to
keep the molecules in a fixed position
Gases
• Have neither definite volume nor definite shape
• Molecules are in constant random motion
• The molecules exert only weak forces on each other
• Average distance between molecules is large
compared to the size of the molecules
Plasmas
• Matter heated to a very high temperature
• Many of the electrons are freed from the nucleus
• Result is a collection of free, electrically charged
ions
• Plasmas exist inside stars
Density and pressure
• Density
m
  lim
V  0  V
• SI unit of density: kg/m3
• Pressure
P  lim
A0
F
A
• SI unit of pressure: N/m2 = Pa (pascal)
Blaise Pascal
(1623 - 1662)
• Pressure is a scalar – at a given point in a fluid the
measured force is the same in all directions
• For a uniform force on a flat area
F
P
A
Atmospheric pressure
• Atmospheric pressure:
• P0 = 1.00 atm = 1.013 x 105 Pa
• Specific gravity of a substance is the ratio of its
density to the density of water at 4° C (1000 kg/m3)
• Specific gravity is a unitless ratio
Fluids at rest
• For a fluid at rest (static equilibrium) the pressure is
called hydrostatic
• For a horizontal-base cylindrical water sample in a
container
F2  F1  mg
P2 A  P1 A  A( y1  y2 ) g
P2  P1   ( y1  y2 ) g
P  P0  hg
Fluids at rest
• The hydrostatic pressure at a point in a fluid
depends on the depth of that point but not on any
horizontal dimension of the fluid or its container
• Difference between an absolute pressure and an
atmospheric pressure is called the gauge pressure
Pg  P  P0  hg
P  P0  hg
Measuring pressure
• Mercury barometer
P2  P1   ( y1  y2 ) g
y1  0; P1  P0
y2  h; P2  0
P0  hg
• Open-tube manometer
P2  P1   ( y1  y2 ) g
y1  0; P1  P0
y2  h; P2  P
Pg  P  P0  hg
Chapter 11
Problem 29
A 1.00-m-tall container is filled to the brim, partway with mercury and the rest of
the way with water. The container is open to the atmosphere. What must be the
depth of the mercury so that the absolute pressure on the bottom of the
container is twice the atmospheric pressure?
Pascal’s principle
• Pascal’s principle: A change in the pressure applied
to an enclosed incompressible fluid is transmitted
undiminished to every portion of the fluid and to the
walls of its container
• Hydraulic lever
F1 F2
P 

V  A1x1  A2 x2
A1 A2
A2
A1
x1  x2
F1  F2
A1
W  F1x1  F2 x2
A2
• With a hydraulic lever, a given force applied over a
given distance can be transformed to a greater force
applied over a smaller distance
Archimedes’ principle
• Buoyant force:
For imaginary void in a fluid
p at the bottom > p at the top
B  mf g
Archimedes
of Syracuse
(287-212 BCE)
• Archimedes’ principle: when a body is submerged in
a fluid, a buoyant force from the surrounding fluid
acts on the body. The force is directed upward and
has a magnitude equal to the weight of the fluid that
has been displaced by the body
Archimedes’ principle
• Sinking:
mg  B
• Floating:
mg  B
• Apparent weight:
weight apparent  mg  B
• If the object is floating at the surface of a fluid, the
magnitude of the buoyant force (equal to the weight
of the fluid displaced by the body) is equal to the
magnitude of the gravitational force on the body
Chapter 11
Problem 53
One kilogram of glass (ρ = 2.60 × 103 kg/m3) is shaped into a hollow spherical
shell that just barely floats in water. What are the inner and outer radii of the
shell? Do not assume that the shell is thin.
Motion of ideal fluids
Flow of an ideal fluid:
• Steady (laminar) – the velocity of the moving fluid at
any fixed point does not change with time (either in
magnitude or direction)
• Incompressible – density is constant and uniform
• Nonviscous – the fluid experiences no drag force
• Irrotational – in this flow the test body will not rotate
about its center of mass
Equation of continuity
• For a steady flow of an ideal fluid through a tube
with varying cross-section
V  Ax  Avt  A1v1t  A2v2 t
A1v1  A2v2
Av  const
Equation of continuity
Bernoulli’s equation
• For a steady flow of an ideal fluid:
Etot  K  U g  Eint
• Kinetic energy
mv 2 Vv 2
K

2
2
• Gravitational potential energy
U g  mgy  Vgy
• Internal (“pressure”) energy
Eint  VP
Daniel Bernoulli
(1700 - 1782)
Bernoulli’s equation
• Total energy
Etot  K  U g  Eint
Vv

 Vgy  VP
2
2
Etot v

 gy  P  const
V
2
2
v1
2
2
 gy1  P1 
v2
2
2
 gy2  P2
Chapter 11
Problem 72
An airplane has an effective wing surface area of 16 m2 that is generating the
lift force. In level flight the air speed over the top of the wings is 62.0 m/s, while
the air speed beneath the wings is 54.0 m/s. What is the weight of the plane?
Viscous fluid flow
• Viscosity: friction between the layers of a fluid
• Layers in a viscous fluid have different velocities
• The velocity is greatest at the center
• Cohesive forces between the fluid and the walls
slow down the fluid on the outside
Questions?