Transcript Physics 131: Lecture 14 Notes

Physics 151: Lecture 30
Today’s Agenda

Today’s Topic:
Fluids in Motion
Bernoulli’s Equation and applications
Physics 151: Lecture 30, Pg 1
Example / Fluid Statics

A beaker of mass mbeaker containing oil of
mass moil (density = roil) rests on a scale. A
block of iron of mass miron is suspended
from a spring scale and completely
submerged in the oil as in Figure on the
right.

Determine the equilibrium readings of
both scales.
Physics 151: Lecture 30, Pg 2
See text: 14.5
Fluids in Motion

Up to now we have described fluids in terms
of their static properties:
density r
pressure p

To describe fluid motion, we need something
that can describe flow:
velocity v

There are different kinds of fluid flow of varying complexity
 non-steady
/ steady
 compressible
/ incompressible
 rotational
/ irrotational
 viscous
/ ideal
Physics 151: Lecture 30, Pg 3
See text: 14.5
Ideal Fluids



Fluid dynamics is very complicated in general (turbulence,
vortices, etc.)
Consider the simplest case first: the Ideal Fluid
no “viscosity” - no flow resistance (no internal friction)
incompressible - density constant in space and time
Simplest situation: consider
ideal fluid moving with steady
flow - velocity at each point in
the flow is constant in time
streamline
A
A
2
1
v1

In this case, fluid moves on
streamlines
v2
Physics 151: Lecture 30, Pg 4
See text: 14.6
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
streamline
A
A
2
1
v1
v2
Flow obeys continuity equation
volume flow rate Q = A·v
tube.
A1v1 = A2v2

is constant along flow
• follows from mass conservation if flow is incompressible.
Physics 151: Lecture 30, Pg 5
See text: 14.7
Conservation of Energy for
Ideal Fluid

Recall the standard work-energy relation W  K
Apply the principle to a section of flowing fluid with
volume dV and mass dm = r dV (here W is work done
on fluid)
v
W  Wgravity  Wpressure
Wgravity  dm g( y 2  y1 )
  r dV g( y 2  y1 )
y
W  K 
1 dm v 2
2
2
2
 1 dm v1
 Bernoulli Equation
2
2
v
Wpressure  p1 A1 dx1  p2 A2 dx2
 ( p1  p2 ) dV

y
2
dV
1
p
2
1
p
1
2
2
2
1
2
r dV( v 2  v1 )
2
p1  12 r v1 r gy1  p2  12 r v 2 r gy 2
Physics 151: Lecture 30, Pg 6
Lecture 30 Act 1
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
1) 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
c) 1/4 v1
Physics 151: Lecture 30, Pg 7
Lecture 30 Act 2
Bernoulli’s Principle

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
2) What is the pressure in the 1/2” pipe relative to the
1” pipe?
a) smaller
b) same
c) larger
Physics 151: Lecture 30, Pg 8
DEMO SLIDE

The smaller the diameter the lower is the pressure
Physics 151: Lecture 30, Pg 9
Example

A Pitot tube (see Fig. below) can be used to determine the
velocity of air flow by measuring the difference between
the total pressure and the static pressure. If the fluid in the
tube is mercury, density rHg = 13 600 kg/m3, and h = 5.00
cm, find the speed of air flow. (Assume that the air is
stagnant at point A and take rair = 1.25 kg/m3.)
Physics 151: Lecture 30, Pg 10
Venturi Meter
v=?
Can we know what
is v from what we
can measure ?
h
rHg
rair
Physics 151: Lecture 30, Pg 11
Example


A tank containing a liquid of
density r has a hole in its side at
a distance h below the surface of
the liquid. The hole is open to
the atmosphere and its diameter
is much smaller than the
diameter of the tank.
What is the speed with of the
liquid as it leaves the tank.
h
r
v=?
Physics 151: Lecture 30, Pg 12
Example

Figure on the right shows a stream
of water in steady flow from a
kitchen faucet. At the faucet the
diameter of the stream is 0.960 cm.
The stream fills a 125-cm3 container
in 16.3 s. Find the diameter of the
stream 13.0 cm below the opening of
the faucet.
Physics 151: Lecture 30, Pg 13
Example

Water is forced out of a fire extinguisher by air pressure, as
shown in Figure below. How much gauge air pressure in
the tank (above atmospheric) is required for the water jet to
have a speed of 30.0 m/s when the water level in the tank is
0.500 m below the nozzle?
Physics 151: Lecture 30, Pg 14
Recap of today’s lecture

14.4-7
Streamlines
Bernoulli’s Equation and applications
Physics 151: Lecture 30, Pg 15