Transcript Physics 207: Lecture 2 Notes

Lecture 21
Goals:
• 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
 HW9, Due Wednesday, Apr. 8th
 Thursday: Read all of Chapter 16
Physics 207: Lecture 21, Pg 1
Pressure vs. Depth
Incompressible Fluids (liquids)
p
 In many fluids the bulk modulus is
0
such that we can treat the density as
a constant independent of pressure:
y1
p1
An incompressible fluid
F1
y2
A
p
2
 For an incompressible fluid, the
density is the same everywhere, but
the pressure is NOT!
 p(y) = p0 - y g r = p0 + d g r
 Gauge pressure (subtract p0,
pressure 1 atm, i.e. car tires)
mg F2
F2 = F1+ m g
= F1+ rVg
F2 /A = F1/A + rVg/A
p2 = p1 - rg y
Physics 207: Lecture 21, Pg 2
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 3
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 4
Exercise
Pressure
 What happens with two fluids??
 Consider a U tube containing liquids of
density r1 and r2 as shown:
r2
 At the red arrow the pressure must be the
dI
r1 y
same on either side. r1 x = r2 (d1+ y)
 Compare the densities of the liquids:
(A) r1 < r2
(B) r1 = r2
(C) r1 > r2
Physics 207: Lecture 21, Pg 5
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 6
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 8
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
r liquid  g  Vliquid = robject  g  Vobject
Vliquid
Vobject
robject
=
r liquid
Physics 207: Lecture 21, Pg 9
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 10
Exercise
V1 = V2 = V3 = V4 = V5
m1 < m2 < m3 < m4 < m5
What is the final position of each block?
Physics 207: Lecture 21, Pg 11
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 12
Pascal’s Principle
 So far we have discovered (using Newton’s Laws):
 Pressure depends on depth: Dp = r 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 18
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
A1
A2
 Pascal’s Principle says that
increased pressure from F1
(F1/A1) is transmitted
throughout the liquid.
 F2 > F1 with conservation of energy
Physics 207: Lecture 21, Pg 19
Exercise
 Consider the systems shown on right.
 In each case, a block of mass M is
placed on the piston of the large
cylinder, resulting in a difference di
in the liquid levels.
 If A2 = 2 A1, how do dA and dB compare?A
V10 = V1 = V2
dA A1 = dB A2
dA A1 = dB 2A1
dA
A10
1
dB
A2
M
M
A10
dA = dB 2
Physics 207: Lecture 21, Pg 20
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 22
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 23
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 24
Ideal Fluids
 Streamlines do not meet or cross
 Velocity vector is tangent to streamline
 Volume of fluid follows a tube of flow
bounded by streamlines
A2
A1
v1
v2
 Streamline density is proportional to
velocity
 Flow obeys continuity equation
Volume flow rate (m3/s)
Q = A·v
is constant along flow tube.
A1v1 = A2v2
Follows from mass conservation if flow is incompressible.
Mass flow rate is just r Q (kg/s)
Physics 207: Lecture 21, Pg 25
Exercise
Continuity
 A housing contractor saves some
money by reducing the size of a
pipe from 1” diameter to 1/2”
diameter at some point in your
house.
v1
v1/2
 Assuming the water moving in the pipe is an ideal fluid,
relative to its speed in the 1” diameter pipe, how fast is the
water going in the 1/2” pipe?
(A) 2 v1
(B) 4 v1
(C) 1/2 v1
(D) 1/4 v1
Physics 207: Lecture 21, Pg 26
Exercise
Continuity
v1
(A) 2 v1

(B) 4 v1
(C) 1/2 v1
v1/2
(D) 1/4 v1
For equal volumes in equal times then ½ the diameter implies ¼
the area so the water has to flow four times as fast.
 But if the water is moving 4 times as fast then it has
16 times as much kinetic energy.
 Something must be doing work on the water
(the pressure drops at the neck) and we recast the work as
P DV = (F/A) (A Dx) = F Dx
Physics 207: Lecture 21, Pg 27
Conservation of Energy for
Ideal Fluid
W
= (P1– P2 ) DV and
P1
W
= ½ Dm v22 – ½ Dm v12
P2
= ½ (r DV) v22 – ½ (r DV) v12
(P1– P2 ) = ½ r v22 – ½ r v12
P1+ ½ r v12 = P2+ ½ r v22 = const.
and with height variations:
Bernoulli Equation  P1+ ½ r v12 + r g y1 = constant
Physics 207: Lecture 21, Pg 29
Lecture 21
• Question to ponder:
or float?
Does heavy water (D2O) ice sink
• Assignment
 HW9, due Wednesday, Apr. 8th
 Thursday: Read all of Chapter 16
Physics 207: Lecture 21, Pg 30