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Team of AustriaMarkus Kunesch, Julian Ronacher, Angel Usunov,
Katharina Wittmann, Bernhard Zatloukal
IYPT 2008 – Trogir, Croatia
No. 13 Spinning IceNo. 13 Spinning Ice
Pour very hot water into a cup and stir it so the water rotates slowly. Place a small ice cube at the centre of the rotating water. The ice cube will spin faster than the water around it. Investigate the parameters that influence the ice rotation.
Reporter: Julian Ronacher
Team Austriapowered by:
OverviewOverview
• Experiment– Experimental setup– Observations and measurements
• Basic theory– Conservation of momentum– Mathematical theory
• Expanded experiments– Special case
• Combination of theory with the experiments• References
Team of Austria – Problem no. 13 – Spinning Ice 22
Basic theoryBasic theory
• Ice cube begins to spin– Water rotation
• Ice cube begins to melt– High water temperature
• Tornado effect• Conservation of momentum
55Team of Austria – Problem no. 13 – Spinning Ice
Basic theoryBasic theory
• Tornado effect– Cold water is flowing down to the ground
• Spinning round
– Water from the side of the ice cube has to fill the gap• Ice cube gets accelerated
66Team of Austria – Problem no. 13 – Spinning Ice
Basic theoryBasic theory
• Conservation of momentum– Mass and radius of the ice cube decrease– Angular velocity increases
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M = torsional moment
L = angular momentum
Θ = moment of inertia
ω = angular velocity
m = mass of the ice cube
ρ = density of the ice cube
h = height of the ice cube
M = torsional moment
L = angular momentum
Θ = moment of inertia
ω = angular velocity
m = mass of the ice cube
ρ = density of the ice cube
h = height of the ice cube
Team of Austria – Problem no. 13 – Spinning Ice
Mathematic theoryMathematic theory
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h = constant
Ice cube is completely covered with water
Q = heat energy
Qhf = heat of fusion
t = time
α = heat transmission coefficient
R = radius of the ice cube
h = height of the ice cube
T = temperature
m = mass of the ice cube
Team of Austria – Problem no. 13 – Spinning Ice
Mathematic theoryMathematic theory
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ρ = density
m = mass
V = volume
R = radius
h = height
α = heat transmission coefficient
T = temperature
Q = heat of fusion
Team of Austria – Problem no. 13 – Spinning Ice
Mathematic theoryMathematic theory
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M = torsional momentum
η = viscosity of water
ω = angular velocity
δ = boundary layer thickness
Team of Austria – Problem no. 13 – Spinning Ice
Mathematic theoryMathematic theory
1313Team of Austria – Problem no. 13 – Spinning Ice
=>
m = mass
ω = angular velocity
h = height
η = viscosity
δ = boundary layer thickness
ρ = density
α = heat transmission coefficient
T = temperature
Qhf = heat of fusion
t = time
Mathematic theoryMathematic theory
)2/( r
1414Team of Austria – Problem no. 13 – Spinning Ice
ω = angular velocity of the tornado
Γ = circulation in the flowing fluid
r = radius of the tornado at a specific height
p = pressure
hgAzgp 2/²)(
ρ = density
g = acceleration
z = height of the ice cube
A = value of p at r = ∞ and z = h
Mathematic theoryMathematic theory
)('2² zhg
1515Team of Austria – Problem no. 13 – Spinning Ice
ω = angular velocity of the tornado
Γ = circulation in the flowing fluid
r = radius of the tornado at a specific height
p = pressure
OHOHIcegg 22 /)](['
ρ = density
g = acceleration
z = height of the ice cube
A = value of p at r = ∞ and z = h
Expanded experimentsExpanded experiments
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• Special case– Angular velocity of the ice
cube and the water are the same
– No relative movement between ice cube and water
– Although the ice cube becomes faster than the water
Team of Austria – Problem no. 13 – Spinning Ice
Expanded experimentsExpanded experiments
1919
• Water accelerates the ice cube– viscosity
• Ice cube still independent from the water– No tornado effect– Ice cube can become faster
• By loss of mass and radius• Tornado effect again
Team of Austria – Problem no. 13 – Spinning Ice
Combination of theory with the experimentsCombination of theory with the experiments
2020
• Theory– Tornado effect
• Angular velocity of the ice cube: 2,05 1/sec
– Conservation of momentum• Angular velocity of the ice cube: 0,73 1/sec
– All together: 2,78 1/sec• Experiments
– Angular velocity of the ice cube: 2,9 1/sec– Measurement error: 5%
Team of Austria – Problem no. 13 – Spinning Ice
±
ReferencesReferences
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• Taschenbuch der Physik; Stöcker; Verlag Harri Deutsch; 5. Auflage
• Mathematik für Physiker; Helmut Fischer; Teubner Verlag; 5. Auflage
Team of Austria – Problem no. 13 – Spinning Ice
Extra SlidesExtra Slides
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• Mathematical background
Team of Austria – Problem no. 13 – Spinning Ice
Mathematical background
)(*2/)*( 2 IceOHIcehf TThRtmQ
hfIceOHIce QTThRtm /)](*2[/ 2
2323Team of Austria – Problem no. 13 – Spinning Ice
Icehf mQQ /
=>
=>
=>
Mathematical background
hRVIce ²
2424Team of Austria – Problem no. 13 – Spinning Ice
=>
=>
=>
hmTTQhdtdm IceIceOHIcehfIce /*)(*)/2(/ 2
dthQTTmdm IcehfOHIceIceIce */*)/)(*2(/ 2
IceIcehfOHIceIce mhQTTdtdm */*)/)(*2(/ 2
Mathematical background
2525Team of Austria – Problem no. 13 – Spinning Ice
=>
=>
dthQTTmdm IcehfOHIceIceIce */*)/)(*2(/ 2
consttQTThm hfOHIceIceIce *)/)(*2(*/*2 2
)(*)/)(*(*/ 02 tmtQTThm IcehfOHIceIceIce
Mathematical background
2626Team of Austria – Problem no. 13 – Spinning Ice
=>
)(* tRFM
water
ice cube
δ
ω
F)/(** AF
hRA 2
Mathematical background
2727Team of Austria – Problem no. 13 – Spinning Ice
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)/)((*)()(*)/)(( dttdttdttdM
htRM )²(2*)/(* /2 h
)/(*)2/)(²(*)2/)(²(*)/()²()]()([* 2 dtdhtmhtmdtdtRtt IceIceIceIceIceIceIceOH
2/²mR