Transcript Folie 1 - Lanzhou University

Team of Austria
Markus Kunesch, Julian Ronacher, Angel Usunov,
Katharina Wittmann, Bernhard Zatloukal
Reporter: Julian Ronacher
No. 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.
Team Austria
powered by:
IYPT 2008 – Trogir, Croatia
Overview
• Experiment
– Experimental setup
– Observations and measurements
• Basic theory
– Conservation of momentum
– Mathematical theory
• Expanded experiments
– Special case
• Combination of theory with the experiments
• References
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First experiments
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First experiments
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Basic theory
• Ice cube begins to spin
– Water rotation
• Ice cube begins to melt
– High water temperature
• Tornado effect
• Conservation of momentum
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Basic 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
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Basic theory
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Basic theory
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Basic theory
• Conservation of momentum
– Mass and radius of the ice cube decrease
– Angular velocity increases
M = torsional moment
m = mass of the ice cube
L = angular momentum
ρ = density of the ice cube
Θ = moment of inertia
h = height of the ice cube
ω = angular velocity
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Mathematic theory
h = constant
Ice cube is completely covered with water

Q = heat energy
R = radius of the ice cube
Qhf = heat of fusion
h = height of the ice cube
t = time
T = temperature
α = heat transmission coefficient
m = mass of the ice cube
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Mathematic theory


ρ = density
h = height
m = mass
α = heat transmission coefficient
V = volume
T = temperature
R = radius
Q = heat of fusion
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Mathematic theory
M = torsional momentum
η = viscosity of water
ω = angular velocity
δ = boundary layer thickness
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Mathematic theory
=>
m = mass
ρ = density
ω = angular velocity
α = heat transmission coefficient
h = height
T = temperature
η = viscosity
Qhf = heat of fusion
δ = boundary layer thickness
t = time
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Mathematic theory
   /(2r )

p  (  ²) / 2  gz  A  gh
ω = angular velocity of the tornado
ρ = density
Γ = circulation in the flowing fluid
g = acceleration
r = radius of the tornado at a specific height
z = height of the ice cube
p = pressure
A = value of p at r = ∞ and z = h
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Mathematic theory
²  2 g ' (h  z )
g '  [ g (  Ice   H 2O )] /  H 2O
ω = angular velocity of the tornado
ρ = density
Γ = circulation in the flowing fluid
g = acceleration
r = radius of the tornado at a specific height
z = height of the ice cube
p = pressure
A = value of p at r = ∞ and z = h
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Expanded experiments
• 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
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Expanded experiments
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Expanded experiments
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Expanded experiments
• 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
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Combination of theory with the experiments
• 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%
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References
• Taschenbuch der Physik; Stöcker; Verlag Harri Deutsch;
5. Auflage
• Mathematik für Physiker; Helmut Fischer; Teubner
Verlag; 5. Auflage
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Extra Slides
• Mathematical background
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Mathematical background
Qhf  Q / mIce
=>
(Qhf * mIce ) / t  2Rh * (TH 2O  TIce )
=>
mIce / t  [2Rh * (TH 2O  TIce )]/ Qhf
=>
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Mathematical background
VIce  R²h

=>
dmIce / dt  (2h / Qhf ) * (TIce  TH 2O ) * mIce /  Iceh
=>
dmIce / dt  (2 * (TIce  TH 2O ) / Qhf ) * h /  Ice * mIce
=>
dmIce / mIce  (2 * (TIce  TH 2O ) / Qhf ) * h /  Ice * dt
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Mathematical background
dmIce / mIce  (2 * (TIce  TH 2O ) / Qhf ) * h /  Ice * dt
=>
2 * mIce  h /  Ice * (2 * (TIce  TH 2O ) / Qhf ) * t  const
=>
mIce  h /  Ice * ( * (TIce  TH 2O ) / Qhf ) * t  mIce (t0 )
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Mathematical background
F   * A * ( /  )
ω
water
F
M  F * R(t )
δ
ice cube
A  2 Rh
=>
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Mathematical background
M  (d(t ) / dt) * (t )  (t ) * (d (t ) / dt)
  mR ² / 2
M   * ( /  ) * 2R²(t )h
  2h / 
=>
 *[ H 2O (t )  Ice (t )]R²(t )  (d / dt) * (m² Ice (t ) / 2Iceh) * Ice  (m² Ice (t ) / 2Iceh) * (d Ice / dt)
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Mathematical background
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