Transcript Physics 207: Lecture 2 Notes

Chapter 15, Fluids & Elasticity
This is an actual photo of an iceberg, taken by a rig manager for
Global Marine Drilling in St. Johns, Newfoundland. The water was
calm and the sun was almost directly overhead so that the diver
Physics 207: Lecture 21, Pg 1
Goals:
Lecture 21
• Chapter 15
 Understand pressure in liquids
and gases
 Use Archimedes’ principle to
understand buoyancy
 Understand the equation of
continuity
 Use an ideal-fluid model to study
fluid flow.
 Investigate the elastic
deformation of solids and
liquids
• Assignment
 HW10, Due Wednesday, Apr. 14th
 Thursday: Read all of Chapter 16
Physics 207: Lecture 21, Pg 2
Fluids
 At ordinary temperature, matter exists
in one of three states
 Solid - has a shape and forms a
surface
 Liquid - has no shape but forms a
surface
 Gas - has no shape and forms no
surface
 What do we mean by “fluids”?
 Fluids are “substances that
flow”…. “substances that take the
shape of the container”
 Atoms and molecules are free to
move.
 No long range correlation between
positions.
Physics 207: Lecture 21, Pg 3
Fluids
 An intrinsic parameter of a fluid
 Density
m

V
units :
kg/m3 = 10-3 g/cm3
(water) = 1.000 x 103 kg/m3
= 1.000 g/cm3
(ice)
= 0.917 x 103 kg/m3
= 0.917 g/cm3
(air)
= 1.29 kg/m3
= 1.29 x 10-3 g/cm3
(Hg)
= 13.6 x103 kg/m3
(W or Au) = 19.3 x103 kg/m3
= 13.6 g/cm3
= 19.3 g/cm3
Physics 207: Lecture 21, Pg 4
Fluids
F
p
A
 Another parameter: Pressure
 Any force exerted by a fluid is perpendicular to a surface
of contact, and is proportional to the area of that surface.
 Force (a vector) in a fluid can be expressed in terms
of pressure (a scalar) as:

F  pAnˆ
ˆ
n
A
Physics 207: Lecture 21, Pg 5
What is the SI unit of pressure?
A. Pascal
B. Atmosphere
C. Bernoulli
D. Young
E. p.s.i.
Units :
1 N/m2
1 bar
1 mbar
1 torr
= 1 Pa (Pascal)
= 105 Pa
= 102 Pa
= 133.3 Pa
1 atm = 1.013 x105 Pa
= 1013 mbar
= 760 Torr
= 14.7 lb/ in2 (=PSI)
Physics 207: Lecture 21, Pg 6
Pressure vs. Depth
Incompressible Fluids (liquids)
p
 When the pressure is small,
relative to the bulk modulus of
the fluid, we can treat the
density as constant independent
of pressure:
incompressible fluid
0
y1
F1
y2
p
1
A
p
2
mg F
2
 For an incompressible fluid, the
density is the same everywhere,
but the pressure is NOT!
 p(y) = p0 - y g 
 Gauge pressure (subtract p0)
 pGauge = p(y) - p0
F2 = F1+ m g
= F1+ Vg
F2 /A = F1/A + Vg/A
p2 = p1 - g y
Physics 207: Lecture 21, Pg 7
Pressure vs. Depth

For a uniform fluid in an open container pressure
same at a given depth independent of the container
y
p(y)

Fluid level is the same everywhere in a connected
container, assuming no surface forces
Physics 207: Lecture 21, Pg 8
Pressure Measurements: Barometer
 Invented by Torricelli
 A long closed tube is filled with mercury
and inverted in a dish of mercury
 The closed end is nearly a vacuum
 Measures atmospheric pressure as
1 atm = 0.760 m (of Hg)
Physics 207: Lecture 21, Pg 9
Exercise
Pressure
 What happens with two fluids??
dI
 Consider a U tube containing liquids of
density 1 and 2 as shown:
2
1
Compare the densities of the liquids:
(A) 1 < 2
(B) 1 = 2
(C) 1 > 2
Physics 207: Lecture 21, Pg 10
Exercise
Pressure
 What happens with two fluids??
 Consider a U tube containing liquids of
density 1 and 2 as shown:
2
 At the red arrow the pressure must be the
dI
1 y
same on either side. 1 x = 2 (d1+ y)
 Compare the densities of the liquids:
(A) 1 < 2
(B) 1 = 2
(C) 1 > 2
Physics 207: Lecture 21, Pg 11
Archimedes’ Principle: A Eureka Moment
 Suppose we weigh an object in air (1) and
in water (2).
W1
W2?
How do these weights compare?
W1 < W2
W1 = W 2
W1 > W2
 Buoyant
force is equal to the
weight of the fluid displaced
Physics 207: Lecture 21, Pg 12
Archimedes’ Principle
 Suppose we weigh an object in air (1) and in water (2).
 How do these weights compare?
W1 < W2
W 1 = W2
 Why?
Since the pressure at the bottom
of the object is greater than that
at the top of the object, the water
exerts a net upward force, the
buoyant force, on the object.
W1 > W 2
W1
W2?
Physics 207: Lecture 21, Pg 14
Sink or Float?
 The buoyant force is equal to the weight of
the liquid that is displaced.
 If the buoyant force is larger than the
weight of the object, it will float; otherwise
it will sink.
y
FB mg
 We can calculate how much of a floating object will be
submerged in the liquid:
 Object is in equilibrium
FB  mg
 liquid  g  Vliquid  object  g  Vobject
Vliquid
Vobject
object

 liquid
Physics 207: Lecture 21, Pg 15
Bar Trick
What happens to the water level when the ice melts?
Expt. 1
A. It rises
Expt. 2
B. It stays the same
piece of rock
on top of ice
C. It drops
Physics 207: Lecture 21, Pg 16
Exercise
V1 = V2 = V3 = V4 = V5
m1 < m2 < m3 < m4 < m5
What is the final position of each block?
Physics 207: Lecture 21, Pg 17
Exercise
V1 = V2 = V3 = V4 = V5
m1 < m2 < m3 < m4 < m5
What is the final position of each block?
Not this
But this
Physics 207: Lecture 21, Pg 18
Pascal’s Principle
 So far we have discovered (using Newton’s Laws):
 Pressure depends on depth: Dp = -  g Dy
 Pascal’s Principle addresses how a change in pressure is
transmitted through a fluid.
Any change in the pressure applied to an enclosed
fluid is transmitted to every portion of the fluid and to
the walls of the containing vessel.
Physics 207: Lecture 21, Pg 24
Pascal’s Principle in action:
Hydraulics, a force amplifier
F1
 Consider the system shown:
F2
 A downward force F1 is applied
to the piston of area A1.
 This force is transmitted through
the liquid to create an upward
force F2.
d2
d1
 Pascal’s Principle says that
increased pressure from F1
(F1/A1) is transmitted
throughout the liquid.
A1
A2
P1 = P2
F1 / A1 = F2 / A2
A2 / A1 = F2 / F1
 F2 > F1 with conservation of energy
Physics 207: Lecture 21, Pg 25
Fluids in Motion
 To describe fluid motion, we need
something that describes flow:
 Velocity v

There are different kinds of fluid flow of varying complexity
 non-steady
/ steady
 compressible / incompressible
 rotational
/ irrotational
 viscous
/ ideal
Physics 207: Lecture 21, Pg 27
Types of Fluid Flow
 Laminar flow
 Each particle of the fluid
follows a smooth path
 The paths of the different
particles never cross each
other
 The path taken by the
particles is called a
streamline
 Turbulent flow
 An irregular flow
characterized by small
whirlpool like regions
 Turbulent flow occurs when
the particles go above some
critical speed
Physics 207: Lecture 21, Pg 28
Types of Fluid Flow
 Laminar flow
 Each particle of the fluid
follows a smooth path
 The paths of the different
particles never cross each
other
 The path taken by the
particles is called a
streamline
 Turbulent flow
 An irregular flow
characterized by small
whirlpool like regions
 Turbulent flow occurs when
the particles go above some
critical speed
Physics 207: Lecture 21, Pg 29
Lecture 21
• Question to ponder:
or float?
Does heavy water (D2O) ice sink
• Assignment
 HW10, due Wednesday, Apr. 14th
 Thursday: Read all of Chapter 16
Physics 207: Lecture 21, Pg 30