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Air PocketAir Pocket8th problem8th problem
A vertical air jet from a straw produces A vertical air jet from a straw produces a cavity on a water surface. What a cavity on a water surface. What
parameters determine the volume and parameters determine the volume and the depth of the cavity?the depth of the cavity?
The ProblemThe Problem
The ApparatusThe Apparatus
We used a square-shaped We used a square-shaped aquariumaquarium and and a a compressorcompressor to reproduce the to reproduce the phenomenon.phenomenon.We measured the We measured the depthdepth and the and the width at width at the surfacethe surfaceWe changed the We changed the velocityvelocity of the air jet and of the air jet and the the heightheight of the straw of the strawTwo pipes with different cross-sections Two pipes with different cross-sections were usedwere used
Speed of Air JetSpeed of Air Jet
222211
21 2
1
2
1phgvphgv airairairair
2
2
1vhgp airwaterwater
Neglecting the difference between the heights of the two points:
Bernoulli’s equation is:
air
waterwater hgv
2
The CavityThe Cavity
We approximated the shape of the cavity We approximated the shape of the cavity with a with a paraboloidparaboloid
A paraboloid is a A paraboloid is a parabola rotatedparabola rotated about about the y axisthe y axis
We measured cavity We measured cavity widthwidth, , depthdepth, as the , as the function of function of pipe heightpipe height and and air speedair speed
Cavity Parameters as Functions Cavity Parameters as Functions of Pipe Heightof Pipe Height
Cavity parameters (v=39,09m/s)
0
0,5
1
1,5
2
2,5
3
3,5
0 2 4 6 8 10 12
Height of pipe (cm)
size
(cm
)
Width
Height
Cavity parameters (v=48,38m/s)
0
0,5
1
1,5
2
2,5
3
3,5
4
0 2 4 6 8 10 12
Height of pipe (cm)
size
(cm
) Width
Depth
Cavity parameters (v=56,15m/s)
0
0,5
1
1,5
2
2,5
3
3,5
4
0 2 4 6 8 10 12
Height of pipe (cm)
size
(sm
)
width
depth
Cavity parameters (v=60,34m/s)
0
0,51
1,52
2,53
3,54
4,5
0 2 4 6 8 10 12
Height of pipe (cm)
size
(cm
)width
height
Cavity Parameters as Functions Cavity Parameters as Functions of Air Speedof Air Speed
Cavity parameters (H=10cm)
00,5
11,5
22,5
33,5
44,5
5
0,00m/s
10,00m/s
20,00m/s
30,00m/s
40,00m/s
50,00m/s
60,00m/s
70,00m/s
v
size
(cm
)
width
height
Cavity parameters (h=6cm)
0
0,5
1
1,5
2
2,5
3
3,5
0,00m/s
10,00m/s
20,00m/s
30,00m/s
40,00m/s
50,00m/s
60,00m/s
70,00m/s
v
size
(cm
)
width
height
Cavity parameters (H=2,5cm)
0
0,5
1
1,5
2
2,5
3
3,5
0,00m/s
10,00m/s
20,00m/s
30,00m/s
40,00m/s
50,00m/s
60,00m/s
70,00m/s
v
size
(cm
)
width
height
Cavity parameters (H=1cm)
0
0,5
1
1,5
2
2,5
3
3,5
0,00m/s
10,00m/s
20,00m/s
30,00m/s
40,00m/s
50,00m/s
60,00m/s
70,00m/s
v
size
(cm
)width
height
Cavities with different pipe Cavities with different pipe cross-sectionscross-sections
D=1cm, H=2,5cm
0
0,5
1
1,5
2
2,5
3
3,5
0 10 20 30 40 50 60 70
v
size
(cm
)
width
height
D=1cm, H=1cm
0
0,5
1
1,5
2
2,5
3
3,5
0 10 20 30 40 50 60 70
v
size
(m
)
width
height
D=0,5cm, H=2,5c
0
0,5
1
1,5
2
2,5
3
3,5
0,00m/s
10,00m/s
20,00m/s
30,00m/s
40,00m/s
50,00m/s
60,00m/s
70,00m/s
v
size
(cm
)
width
height
D=0,5cm, H=1cm
0
0,5
1
1,5
2
2,5
3
3,5
0,00m/s
10,00m/s
20,00m/s
30,00m/s
40,00m/s
50,00m/s
60,00m/s
70,00m/s
v
size
(cm
)width
height
Volume of the CavityVolume of the Cavity
A paraboloid is a parabola rotated about A paraboloid is a parabola rotated about the y axis.the y axis.
Its volume equals its Its volume equals its inverse function’sinverse function’s volume rotated about the x axisvolume rotated about the x axis
We need to calculate the volume of a We need to calculate the volume of a solid solid of revolutionof revolution
2Axy A
xy
Ayx
2
Inverse function of the
parabola
Volume of the CavityVolume of the Cavity
2)( Axxf A
xxf )(1
Volume of the cavityVolume of the cavity
Parameter ‘Parameter ‘AA’ contains the rate of the ’ contains the rate of the width and height of the cavity, in the width and height of the cavity, in the following way:following way:
0
2
2
2
222
00)0(0
)(
hw
Aw
yw
x
Ayx
Axxy
204
w
hA
Volume of the cavityVolume of the cavity
The volume of a solid of revolution:The volume of a solid of revolution:
The volume of the rotated inverse The volume of the rotated inverse parabola:parabola:
This is the volume of the This is the volume of the original original paraboloid, tooparaboloid, too
dxxyV )(2
20
2
00 22h
Ax
Axdx
AA
xV
hhh
Measured VolumesMeasured Volumes
Cavity volume
0
2
4
6
8
10
12
14
0 2 4 6 8 10 12
H (cm)
V (c
m^3
)
39,09 m/s
48,38 m/s
56,15 m/s
60,34 m/s
64,25 m/s
Cavity volume
0
2
4
6
8
10
12
14
0,00 m/s 10,00m/s
20,00m/s
30,00m/s
40,00m/s
50,00m/s
60,00m/s
70,00m/s
v
v (c
m^3
)
10
8,5
6
4
2,5
1
Work Needed for the Formation Work Needed for the Formation of the Cavity of the Cavity
We calculated this work in We calculated this work in twotwo ways ways
First methodFirst method:: Using that Using that ,,
where F is the where F is the buoyancybuoyancy and and
Like dipping the paraboloid from the surface Like dipping the paraboloid from the surface to hto h0 0 depth continually:depth continually:
dhhFW )(
2
2)()( h
AghgVhF waterwater
000
03
0
2
0 62)(
hwater
h
water
h
hA
gdhh
A
gdhhFW
Work Needed for the Formation Work Needed for the Formation of the Cavityof the Cavity
We getWe get
Substituting backSubstituting back : :
306h
A
gW water
202h
AV
03
1gVhW water
Another Method for Determining Another Method for Determining the Workthe Work
When the cavity is created, the When the cavity is created, the mass mass centrecentre of the water which filled the cavity of the water which filled the cavity will will move to the surfacemove to the surface of the water, of the water, which means that its which means that its potential energypotential energy will will rise:rise:
The needed The needed work equals this change of work equals this change of energy:energy:
hVghmgE water
EW
The problem is then, to determine The problem is then, to determine ΔΔhh
Assuming that Assuming that ρρ is constant, is constant, ΔΔh is the h is the difference of the hdifference of the h00 (height of the water (height of the water
surface) and the hsurface) and the hm m mass centre of the mass centre of the
paraboloid:paraboloid:
hhmm can be calculated from the geometry of can be calculated from the geometry of
the paraboloidthe paraboloid
Another Method for Determining Another Method for Determining the Workthe Work
mhhh 0
We calculated the centre of mass of the We calculated the centre of mass of the paraboloid by this formula:paraboloid by this formula:
We got:We got:
Another Method for Determining Another Method for Determining the Workthe Work
0
0
0
2
0
2
)(
)(
h
h
m
dxxy
dxxxy
h
03
2hhm
Substituting back to the equation with the Substituting back to the equation with the work:work:
Another Method for Determining Another Method for Determining the Workthe Work
00
0
3
2
)(
hhVgW
hhVghVgW
EW
water
mwaterwater
03
1gVhW water
Calcuating EfficiencyCalcuating Efficiency
Efficiency is the rate of input work and and Efficiency is the rate of input work and and useful workuseful work
Input work is the mechanical energy Input work is the mechanical energy change of air slowing down at the surfacechange of air slowing down at the surface
Useful work is the work done while Useful work is the work done while creating the cavitycreating the cavity
2
2
1vmW airin 03
1gVhW wateruse
Calculating EfficiencyCalculating Efficiency
Efficiency is the rate of the two:Efficiency is the rate of the two:
Air fill the cavity continously; its mass isAir fill the cavity continously; its mass is
22 3
2
21
31
mv
ghV
mv
ghVk
W
Wk
use
in
Vm air
EfficienciesEfficiencies
Substituting:Substituting:
Efficiency
0
0,01
0,02
0,03
0,04
0,05
0,06
0,07
0,08
0 2 4 6 8 10 12
Height of pipe (cm)
k
39,09 m/s
48,38 m/s
56,15 m/s
60,34 m/s
64,25 m/s
Efficiency
0
0,010,02
0,03
0,04
0,050,06
0,07
0,08
0,00m/s
10,00m/s
20,00m/s
30,00m/s
40,00m/s
50,00m/s
60,00m/s
70,00m/s
v
k
10
8,5
6
4
2,5
1
23
2
v
ghk
air
cavitywater
Volumes and efficiencies of Volumes and efficiencies of different cross-sectionsdifferent cross-sections
D=0,5cm, Volume
0
2
4
6
8
10
12
0,00m/s
10,00m/s
20,00m/s
30,00m/s
40,00m/s
50,00m/s
60,00m/s
70,00m/s
v
V(c
m^
3)
2,5cm
1cm
D=1cm, Voulme
0
2
4
6
8
10
12
0 10 20 30 40 50 60 70
v
V(m
^3) 2,5 cm
1 cm
D=0,5cm, Efficiency
0
0,010,02
0,03
0,04
0,050,06
0,07
0,08
0,00m/s
10,00m/s
20,00m/s
30,00m/s
40,00m/s
50,00m/s
60,00m/s
70,00m/s
v
k
2,5cm
1cm
D=1 cm, Efficiency
0
0,01
0,02
0,03
0,04
0,05
0,06
0,07
0,08
0 10 20 30 40 50 60 70
v
k2,5 cm
1 cm
ConclusionsConclusions
The measurementsThe measurements
Cavity width, depthCavity width, depth
As the function of:As the function of: Speed of the air jetSpeed of the air jet Height of the pipeHeight of the pipe We measured two types of pipem, with We measured two types of pipem, with
different cross-sectionsdifferent cross-sections
