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Lecture 21
Sound →Fluids
Standing Waves I
A string is clamped at both ends and
plucked so it vibrates in a standing
mode between two extreme positions
a and b. Let upward motion
correspond to positive velocities.
When the string is in position b, the
instantaneous velocity of points on
the string:
a) is zero everywhere
b) is positive everywhere
c) is negative everywhere
d) depends on the position
along the string
a
b
Standing Waves I
A string is clamped at both ends and
plucked so it vibrates in a standing
mode between two extreme positions
a and b. Let upward motion
correspond to positive velocities.
When the string is in position b, the
instantaneous velocity of points on
the string:
a) is zero everywhere
b) is positive everywhere
c) is negative everywhere
d) depends on the position
along the string
Observe two points:
Just before b
Just after b
Both points change direction
before and after b, so at b all
points must have zero
velocity.
Every point in in SHM, with the
amplitude fixed for each position
Standing Waves II
A string is clamped at both ends and
plucked so it vibrates in a standing
mode between two extreme positions
a and b. Let upward motion
correspond to positive velocities.
When the string is in position c, the
instantaneous velocity of points on
the string:
a) is zero everywhere
b) is positive everywhere
c) is negative everywhere
d) depends on the position
along the string
a
c
b
Standing Waves II
A string is clamped at both ends and
plucked so it vibrates in a standing
mode between two extreme positions
a and b. Let upward motion
correspond to positive velocities.
When the string is in position c, the
instantaneous velocity of points on
the string:
When the string is flat, all points are
moving through the equilibrium
position and are therefore at their
maximum velocity. However, the
direction depends on the location of
the point. Some points are moving
upward rapidly, and some points are
moving downward rapidly.
a) is zero everywhere
b) is positive everywhere
c) is negative everywhere
d) depends on the position
along the string
a
c
b
Beats
Two waves with close
(but not precisely the
same) frequencies will
create a time-dependent
interference
Beats
Beats are an interference pattern in time, rather than in space.
If two sounds are very close in frequency, their sum also
has a periodic time dependence: f beat = f1 - f2
• In tuning a string, a 262-Hz tuning fork is
sounded at the same time as the string is
plucked. Beats are heard with a frequency
of 6 Hz. What is the frequency emitted by
the string?
• In tuning a string, a 262-Hz tuning fork is
sounded at the same time as the string is
plucked. Beats are heard with a frequency
of 6 Hz. What is the frequency emitted by
the string?
fb  f1  f 2  f 2  f1  f b
Which frequency is f1 ?
 f 2  f1  fb  256 Hz or 268 Hz
Beats
The traces below show beats that
occur when two different pairs of
waves interfere. For which case is
the difference in frequency of the
original waves greater?
Pair 1
a) pair 1
b) pair 2
c) same for both pairs
d) impossible to tell by just
looking
Pair 2
Beats
The traces below show beats that
occur when two different pairs of
waves interfere. For which case is
the difference in frequency of the
original waves greater?
a) pair 1
b) pair 2
c) same for both pairs
d) impossible to tell by just
looking
The beat frequency is the difference in frequency between the two
waves: fbeat = f2 – f1.
Pair 1 has the greater beat frequency (more oscillations in same time
period), so pair 1 has the greater frequency difference.
Pair 1
Pair 2
Open and Closed Pipes
You blow into an open pipe
and produce a tone. What
happens to the frequency of
the tone if you close the end
of the pipe and blow into it
again?
a) depends on the speed of sound in
the pipe
b) you hear the same frequency
c) you hear a higher frequency
d) you hear a lower frequency
Open and Closed Pipes
You blow into an open pipe
and produce a tone. What
happens to the frequency of
the tone if you close the end
of the pipe and blow into it
again?
In the open pipe,
a) depends on the speed of sound in
the pipe
b) you hear the same frequency
c) you hear a higher frequency
d) you hear a lower frequency
of a wave “fits”
into the pipe, and in the closed pipe,
only
of a wave fits. Because the
wavelength is larger in the closed
pipe, the frequency will be lower.
Follow-up: What would you have to do
to the pipe to increase the frequency?
Decibel Level
When Mary talks, she creates an
intensity level of 60 dB at your
location. Alice talks with the same
volume, also giving 60 dB at your
location. If both Mary and Alice talk
simultaneously from the same spot,
what would be the new intensity
level that you hear?
a) more than 70 dB
b) 70 dB
c) 66 dB
d) 63 dB
e) 61 dB
Decibel Level
When Mary talks, she creates an
intensity level of 60 dB at your
location. Alice talks with the same
volume, also giving 60 dB at your
location. If both Mary and Alice talk
simultaneously from the same spot,
what would be the new intensity
level that you hear?
a) more than 70 dB
b) 70 dB
c) 66 dB
d) 63 dB
e) 61 dB
With two voices adding up, the intensity increases by a factor of 2,
meaning that the intensity level is higher by an amount equal to
→ = 10 log(2) = 3 dB. The new intensity level is  = 63 dB.
Workplace Noise
A factory floor operates 120 machines
of approximately equal loudness. A
plant safety inspection shows that the
sound intensity level of 101 dB is too
high, and must be lowered to 91 dB.
How many of the machines, at least,
would need to be turned off to bring
the sound level into compliance?
a) about 10
b) about 12
c) about 60
d) about 108
Fluids
Pressure
Pressure is force
per unit area
Pressure is not the same as force!
The same force applied over a
smaller area results in greater
pressure – think of poking a
balloon with your finger and then
with a needle.
Pressure is a useful concept for discussing fluids,
because fluids distribute their force over an area
Atmospheric Pressure
Atmospheric pressure is due to the weight of the
atmosphere above us.
= 1 pascal (Pa)
Various units to describe pressure:
Pascals
pounds per square inch
bars
Atmospheric Pressure
Atmospheric pressure is due to the
weight of the atmosphere above us.
How much is 1 atm ?
Put a 1 atm block on your hand?
4 in2 area -> ~60 lbs!
Hemi-spheres: ~3 inches radius, ~30 in2 area
~450 lbs!
mass of quarter ~ 0.0057 kg
area of quarter ~ 3x10-4 m2
Pressure from weight of one quarter : 180 N/m2
To get 101kPa, one must be buried under a
stack ~560 quarters, or 14 rolls, deep!
Density, height, and vertical force
How does tension change in
a vertical (massive) rope?
How does normal force
change in stack of blocks?
In a fluid, how does force
change with vertical height?
Density
The density of a material is its
mass per unit volume:
Pressure and Depth
Pressure increases with depth in a fluid due to the
increasing mass of the fluid above it.
Pressure and depth
Pressure in a fluid includes pressure on the fluid
surface (usually atmospheric pressure)
Pressure depends only on depth and external pressure
(and not on shape of fluid column)
Equilibrium only when pressure is the same
Unequal pressure will cause liquid flow:
must have same
pressure at A and B
Oil is less dense, so a taller column of oil is
needed to counter a shorter column of water
The Barometer
A barometer compares the pressure due
to the atmosphere to the pressure due to
a column of fluid, typically mercury.
The mercury column has a vacuum
above it, so the only pressure is due to
the mercury itself.
The barometer equilibrates
where the pressure due to the
column of mercury is equal to
the atmospheric pressure.
Patm = ρgh
Atmospheric pressure
in terms of millimeters
of mercury:
Measuring Pressure
The barometer measures atmospheric pressure
vs. liquid height of known density (vacuum
above the liquid column)
The manometer measures a pressure
relative to atmospheric pressure
Gauge Pressure
ΔP= P - Patm = ρgh
The Straw
You put a straw into a glass of water,
place your finger over the top so that
no air can get in or out, and then left
the straw from the liquid.
You find that the straw retains some
liquid. How does the air pressure P
in the upper part compare to the
atmospheric pressure PA?
a) greater than PA
b) equal to PA
c) less than PA
The Straw
You put a straw into a glass of water,
place your finger over the top so that
no air can get in or out, and then lift
the straw from the liquid.
You find that the straw retains some
liquid. How does the air pressure P
in the upper part compare to the
atmospheric pressure PA?
a) greater than PA
b) equal to PA
c) less than PA
Consider the forces acting at the bottom of the
straw:
PA – P –  g H = 0
This point is in equilibrium, so net force is zero.
Thus, P = PA –  g H
and so we see that
the pressure P inside the straw must be less
than the outside pressure PA.
H
Pascal’s principle
An external pressure applied to an enclosed fluid is
transmitted to every point within the fluid.
Hydraulic lift
F1 A1  P  F2 A2
Assume fluid is “incompressible”
Pascal’s principle
Hydraulic lift
F1 A1  P  F2 A2
Are we getting “something for nothing”?
Assume fluid is “incompressible”
so Work in = Work out!
Buoyancy
A fluid exerts a net upward force on any object it
surrounds, called the buoyant force.
This force is due to the
increased pressure at the
bottom of the object
compared to the top.
Consider a cube
with sides = L
Archimedes’ Principle
Archimedes’ Principle: An object completely immersed
in a fluid experiences an upward buoyant force equal in
magnitude to the weight of fluid displaced by the object.
Buoyant Force When a Volume V is
Submerged in a Fluid of Density ρfluid
Fb = ρfluid gV
Applications of Archimedes’ Principle
An object floats when it displaces an
amount of fluid equal to its weight.
wood
block
brass
block
equivalent
mass of water
equivalent
mass of water
Applications of Archimedes’ Principle
An object made of material that is denser than
water can float only if it has indentations or pockets
of air that make its average density less than that of
water.
An object floats when it displaces an
amount of fluid equal to its weight.
Applications of Archimedes’ Principle
The fraction of an object that is submerged when it
is floating depends on the densities of the object
and of the fluid.
Measuring the Density
The King must know: is his crown true gold?
Get the mass
from W = T1
= mg
Get the volume from
( T1 - T2 ) = V(ρwater g)
On Golden Pond
A boat carrying a large chunk of gold
is floating on a lake. The chunk is
then thrown overboard and sinks.
What happens to the water level in
the lake (with respect to the shore)?
a) rises
b) drops
c) remains the same
d) depends on the size of
the gold
On Golden Pond
A boat carrying a large chunk of gold
is floating on a lake. The chunk is
then thrown overboard and sinks.
What happens to the water level in
the lake (with respect to the shore)?
Initially the chunk of gold “floats” by
sitting in the boat. The buoyant force is
equal to the weight of the gold, and this
will require a lot of displaced water to
equal the weight of the gold. When
thrown overboard, the gold sinks and
only displaces its volume in water. This
is not so much water—certainly less
than before—and so the water level in
the lake will drop.
a) rises
b) drops
c) remains the same
d) depends on the size of
the gold
Wood in Water
Two beakers are filled to the brim with water. A wooden block
is placed in the beaker 2 so it floats. (Some of the water will
overflow the beaker and off the scale). Both beakers are then
weighed. Which scale reads a larger weight?
a
b
c same for both
Wood in Water
Two beakers are filled to the brim with water. A wooden block
is placed in the beaker 2 so it floats. (Some of the water will
overflow the beaker and off the scale). Both beakers are then
weighed. Which scale reads a larger weight?
The block in 2 displaces an amount of
a
b
water equal to its weight, because it is
floating. That means that the weight of
the overflowed water is equal to the
weight of the block, and so the beaker
in 2 has the same weight as that in 1.
c same for both
Wood in Water II
Earth
A block of wood floats in a container of
water as shown on the right. On the Moon,
how would the same block of wood float in
the container of water?
Moon
a
b
c
Wood in Water II
Earth
A block of wood floats in a container of
water as shown on the right. On the Moon,
how would the same block of wood float in
the container of water?
Moon
A floating object displaces a
weight of water equal to the
object’s weight. On the Moon,
the wooden block has less
weight, but the water itself also
has less weight.
a
b
c
A wooden block is held at the bottom of a bucket filled with water.
The system is then dropped into free fall, at the same time the
force pushing the block down is also removed. What will happen
to the block?
a) the block will float to the surface.
b) the block will stay where it is.
c) the block will oscillate between the
surface and the bottom of the bucket
A wooden block of cross-sectional area A, height H, and
density ρ1 floats in a fluid of density ρf .
If the block is displaced downward and then released, it
will oscillate with simple harmonic motion. Find the
period of its motion.
h
A wooden block of cross-sectional area A, height H, and
density ρ1 floats in a fluid of density ρf .
If the block is displaced downward and then released, it
will oscillate with simple harmonic motion. Find the
period of its motion.
Vertical force: Fy = (hA)g ρf - (HA)g ρ1
at equilibrium: h0 = Hρ1/ρf
h = h0 - y
Total restoring force: Fy = -(Agρf)y
Analogous to mass on a spring, with κ = Agρf
h