Lecture 21

Sound →Fluids

Standing Waves IStanding Waves IA 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

b

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.

Standing Waves IStanding Waves IA 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

Every point in in SHM, with the amplitude fixed for each position

a

b

c

Standing Waves IIStanding Waves IIA 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

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 locationdirection depends on the location of the point. Some points are moving upward rapidly, and some points are moving downward rapidly.

a

b

c

Standing Waves IIStanding Waves IIA 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

Two waves with close (but not precisely the same) frequencies will create a time-dependent interference

Beats

Beats

If two sounds are very close in frequency, their sum also has a periodic time dependence: f beat = f1 - f2

Beats are an interference pattern in time, rather than in space.

• 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?

1 2 2 1

1

2 1

?

256 268

b b

b

f f f f f f

Which frequency is f

f f f Hz or Hz

Pair 1 Pair 2

a) pair 1

b) pair 2

c) same for both pairs

d) impossible to tell by just looking

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?

BeatsBeats

Pair 1 Pair 2

The beat frequency is the difference in frequencydifference in frequency between the

two waves: ffbeatbeat = = ff22 – – ff11..

Pair 1 has the greater beat frequencygreater beat frequency (more oscillations in same time period), so pair 1 has the greater frequency differencegreater frequency difference.

a) pair 1

b) pair 2

c) same for both pairs

d) impossible to tell by just looking

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?

BeatsBeats

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

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?

Open and Closed PipesOpen and Closed Pipes

In the open pipeopen pipe, of a waveof a wave “fits”

into the pipe, and in the closed closed

pipepipe, only of a wave of a wave fits.

Because the wavelength is larger in wavelength is larger in

the closed pipethe closed pipe, the frequency will frequency will

be lowerbe lower.

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

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?

Open and Closed PipesOpen and Closed Pipes

Follow-upFollow-up: What would you have to : What would you have to do to the pipe to increase the do to the pipe to increase the

frequency?frequency?

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 LevelDecibel 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?

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.

Decibel LevelDecibel Level

a) more than 70 dB

b) 70 dB

c) 66 dB

d) 63 dB

e) 61 dB

a) about 10

b) about 12

c) about 60

d) about 108

Workplace NoiseWorkplace 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?

Fluids

Pressure

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 not the same as force!

Pressure is force per unit area

Pressure is a useful concept for discussing fluids, because fluids distribute their force over an area

Atmospheric PressureAtmospheric pressure is due to the weight of the

atmosphere above us.

= 1 pascal (Pa)

Pascals

pounds per square inch

bars

Various units to describe pressure:

Atmospheric PressureAtmospheric pressure is due to the weight of the atmosphere above us.

Hemi-spheres: ~3 inches radius, ~30 in2 area

~450 lbs!

How much is 1 atm ?

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!

4 in2 area -> ~60 lbs!Put a 1 atm block on your hand?

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 DepthPressure 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 BarometerA 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.

Patm = ρghAtmospheric pressure in terms of millimeters of mercury:

The barometer equilibrates where the pressure due to the column of mercury is equal to the atmospheric pressure.

Measuring PressureThe 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

a) greater than PA

b) equal to PA

c) less than PA

The StrawThe StrawYou 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?

Consider the forces acting at the bottom of

the straw: PPAA – – PP – – ρρ g H g H = = 00

This point is in equilibrium, so net force is This point is in equilibrium, so net force is

zero.zero.

Thus, PP = = PPAA – – ρρ g H g H and so we see that the pressure pressure PP inside the straw must be lessless than the outside pressure outside pressure PPAA.

H

a) greater than PA

b) equal to PA

c) less than PA

The StrawThe StrawYou 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?

Pascal’s principleAn external pressure applied to an enclosed fluid is

transmitted to every point within the fluid.

Hydraulic lift

Assume fluid is “incompressible”

1 1 2 2PF A F A

Pascal’s principle

Hydraulic lift

Are we getting “something for nothing”?

Assume fluid is “incompressible”

so Work in = Work out!

1 1 2 2PF A F A

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

Buoyant Force When a Volume V is Submerged in a Fluid of Density

ρfluid

Fb = ρfluid gV

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.

Applications of Archimedes’ Principle

An object floats when it displaces an amount of fluid equal to its weight.

equivalent mass of water

wood block

equivalent mass of water

brass block

Applications of Archimedes’ PrincipleAn 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’ PrincipleThe 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

Get the volume from( T1 - T2 ) = V(ρwater g)

Get the mass from W = T1 = mg

The King must know: is his crown true gold?

On Golden PondOn Golden Pond

a) rises

b) drops

c) remains the same

d) depends on the size

of the gold

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

Initially the chunk of gold “floats” by sitting in the boat. The buoyant force is equal to the weightweight of the gold, and this will require a lot of displaced require a lot of displaced waterwater to equal the weight of the gold. When thrown overboard, the gold sinks and only displaces its only displaces its volumevolume in in waterwater. This is not so much water—certainly less than before—and so the water level in the lake will drop.

On Golden PondOn 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)?

Wood in WaterWood 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

csame for both

The block in 2 displaces an amount displaces an amount

of water equal to its weightof water equal to its weight, because

it is floating. That means that the

weight of the overflowed water is weight of the overflowed water is

equal to the weight of the blockequal to the weight of the block, and

so the beaker in 2 has the same beaker in 2 has the same

weight as that in 1weight as that in 1.

Wood in WaterWood 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

csame for both

Wood in Water IIWood in Water II

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?

Earth

Moon

a b c

A floating object displaces a

weight of water equal to the weight of water equal to the

object’s weightobject’s weight. On the

Moon, the wooden block has

less weightless weight, but the water

itself also has less weightalso has less weight.

Wood in Water IIWood in Water II

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

Earth

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 h

at equilibrium: h0 = Hρ1/ρf

Total restoring force: Fy = -(Agρf)y

h = h0 - y

Analogous to mass on a spring, with κ = Agρf