No. 13 Spinning Ice

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

First experimentsFirst experiments

33Team of Austria – Problem no. 13 – Spinning Ice

First experimentsFirst experiments

44Team of Austria – Problem no. 13 – Spinning Ice

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

77Team of Austria – Problem no. 13 – Spinning Ice

Basic theoryBasic theory

88Team of Austria – Problem no. 13 – Spinning Ice

Basic theoryBasic theory

• Conservation of momentum– Mass and radius of the ice cube decrease– Angular velocity increases

99

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

1010

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

1111

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

1212

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

1616

• 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

1717Team of Austria – Problem no. 13 – Spinning Ice

Expanded experimentsExpanded experiments

1818Team 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

2121

• 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

2222

• 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

=>

)/)((*)()(*)/)(( dttdttdttdM

htRM )²(2*)/(* /2 h

)/(*)2/)(²(*)2/)(²(*)/()²()]()([* 2 dtdhtmhtmdtdtRtt IceIceIceIceIceIceIceOH

2/²mR

Mathematical background

2828Team of Austria – Problem no. 13 – Spinning Ice