Problem 12. Rolling Magnets. Problem Investigate the motion of a magnet as it rolls down an inclined...

Problem 12Problem 12

Rolling MagnetsRolling Magnets

ProblemProblem

Investigate the motion of a magnet Investigate the motion of a magnet

as it rolls down an inclined planeas it rolls down an inclined plane

OutlineOutline

bull Only rolling motion investigatedOnly rolling motion investigatedbull Two distinct casesTwo distinct cases

bull Nonconducting planeNonconducting planebull Conducting planeConducting plane

bull Quasiinfinite planeQuasiinfinite planebull Finite planeFinite plane

bull Common parametersCommon parametersbull Magnet propertiesMagnet propertiesbull Plane inclinationPlane inclination

The magnetsThe magnets

bull Permanent NdPermanent Nd22FeFe1414B magnetsB magnetsbull Field of magnetization 14 TField of magnetization 14 Tbull Density 7500 kgmDensity 7500 kgm33

bull 3 different sizes3 different sizes

Diameter [cm]Diameter [cm] 254254 100100 095095

Thickness [cm]Thickness [cm] 254254 050050 063063

Case 1 ndash Nonconducting plateCase 1 ndash Nonconducting plate

bull Woodenplastic plateWoodenplastic plate

bull Magnet influenced only by the Earth fieldMagnet influenced only by the Earth field

bull Curved trajectoryCurved trajectory

bull ParametersParameters

bull Plane inclinationPlane inclination

bull Magnet propertiesMagnet properties

bull Much less appealing than second case ndash Much less appealing than second case ndash not studied in detailnot studied in detail

Case 2 ndash Conducting plateCase 2 ndash Conducting plate

Metal slabMetal slab

m

Grow

ing flux

Falling flux

vv ndash magnet

velocityFd ndash drag force

bull In conducting plate ndash eddy currents In conducting plate ndash eddy currents induced due to time-changing field fluxinduced due to time-changing field flux

bull Eddy current field gradient ndash velocity-Eddy current field gradient ndash velocity-dependent drag on magnetdependent drag on magnet

dF

Conducting plate Conducting plate contcont

Two subcasesTwo subcases

bull Magnet moving far from the plate Magnet moving far from the plate edges - ˝infinite˝ plateedges - ˝infinite˝ plate


bull Magnet getting near the edges ndash Magnet getting near the edges ndash boundary effectsboundary effects

1 Infinite conducting plate1 Infinite conducting plate

bull First case much simplerFirst case much simpler

bull Linear motionLinear motion

bull Constant velocity (drag balances Constant velocity (drag balances gravity) ndash simple reference system gravity) ndash simple reference system switchingswitching

bull Main parametersMain parameters

bull Magnet dimensions and Magnet dimensions and magnetizationmagnetization

bull Plate inclinationPlate inclination

bull Plate conductivityPlate conductivity

ExperimentExperiment

bull Measurements Measurements

bull Dependence of terminal velocity on Dependence of terminal velocity on plate inclination for several magnetsplate inclination for several magnets

bull Dependence of terminal velocity on Dependence of terminal velocity on plate conductivityplate conductivity

bull Aluminium plateAluminium plate

bull Velocity measurement ndash solenoid systemVelocity measurement ndash solenoid system

bull Conductivity modification ndash temperature Conductivity modification ndash temperature change change

1 Velocity ndash inclination 1 Velocity ndash inclination contcont

Amplifier amp ADC

PC

Detector solenoids


bull Velocity measurement ndash solenoids detect Velocity measurement ndash solenoids detect passingpassing magnet due to inductionmagnet due to induction

time [s]

0 1 2 3 4 5 6 7

volta

ge [a

u]

-6

-4

-2

0

2

4

6

2 Velocity - conductivity2 Velocity - conductivity

bull Conductivity change Conductivity change

bull Cooling plate in insulating box to 73 K Cooling plate in insulating box to 73 K with liquid Nwith liquid N22

bull As plate warms up magnet is As plate warms up magnet is released and velocity measuredreleased and velocity measured

bull Conductivity measured directly ndash Conductivity measured directly ndash resistance of wire attached to plateresistance of wire attached to plate

Temperature rangeTemperature range 73 ndash 200 K73 ndash 200 K

Conductivity rangeConductivity range 37 ndash 200 MS37 ndash 200 MS

Velocity ndash conductivity Velocity ndash conductivity contcont

bull Apparatus shematicApparatus shematic

Magnet detecting

solenoids

Liquid nitrogen

Aluminium plate

Magnet

Styrofoam box

Magnet insertion slit

Temperature wire


The boxThe box

Box inside with plate and Box inside with plate and solenoidssolenoids

TheoryTheory

bull The geometry in magnet reference The geometry in magnet reference system system

x

z

y

M

dF

M ndash magnetization vector

Fd ndash drag force

xyz ndash magnet reference system

xrsquoyrsquozrsquo ndash plate reference system

t - time

yM ˆMtvxx ˆˆ

yy ˆˆ

zz ˆˆ

Theory Theory contcont

bull Induced field ndash from Maxwell equations Induced field ndash from Maxwell equations in magnet reference systemin magnet reference system

bull For small velocities - field equationFor small velocities - field equationj ndash current density

σ ndash plate conductivity

B0 ndash field of magnet

μ0 ndash permeability of vacuum

v ndash magnet velocity

xv

xv in

in

0

002 BBB

Induced field Source term ndash magnet field

Solution ndash power series in Solution ndash power series in μμ00σσvv

bull For small velocities ndash linear first term For small velocities ndash linear first term dominatesdominates


bull Needed for force ndash Needed for force ndash yy - component - component

bull Numerical integration yieldsNumerical integration yields

x [m]

-0020 -0015 -0010 -0005 0000 0005 0010 0015

z [m

]

-0020

-0015

-0010

-0005

0000

0005

0010

0015

-015 -010 -005 000 005 010 015

Biny [mT]

Magnet radius [cm]Magnet radius [cm] 0505

Magnet thickness [cm]Magnet thickness [cm] 0505

Counductivity [MS]Counductivity [MS] 29852985

Upper plate boundary z = 0Semiinfinite plateMagnet centre of mass z = 05

cmSection y = 0


bull The currents are obtained by The currents are obtained by differentiationdifferentiation

x [m]

-002 -001 000 001 002

z [m

]

-0015

-0010

-0005

0000

0005

0010

0015


bull Drag force ndash for small velocitiesDrag force ndash for small velocities

bull Terminal state ndash balance between Terminal state ndash balance between gravity and drag forcegravity and drag force

sin

g

vT

vFd Λ ndash calculated constantσ ndash plate conductivityv ndash magnet velocity



ΛΛ middotmiddot101099 [kgsS] [kgsS] 528(4)528(4) 119(3)119(3) 295(2)295(2)

ζ ndash magnet massg ndash acceleration of gravityφ ndash plate inclination

Results and comparation Results and comparation contcont

bull For two magnets ndash dependence of For two magnets ndash dependence of terminal velocity on sin terminal velocity on sin φφ linearlinear

plate end height [cm]

-5 0 5 10 15 20 25 30

term

inal

vel

ocity

[ms

]

-005

000

005

010

015

020

025

030

Diameter [cm] 10

Thickness [cm] 05

Counductivity [MS] 2985

Plate thickness [cm] 10


Diameter [cm] 254

Thickness [cm] 254




6 8 10 12 14 16 18

term

inal

vel

ocity

[ms

]

004

006

008

010

012

014


bull For third magnet ndash dependence of For third magnet ndash dependence of terminal velocity on 1conductivity linearterminal velocity on 1conductivity linear

Diameter [cm] 095

Thickness [cm] 063

Plate angle [deg] 285


1conductivity [nm]

5 10 15 20 25 30

term

inal

vel

ocity

[ms

]

004

006

008

010

012

014

016


bull From three measurements the From three measurements the coefficient coefficient ΛΛ is obtained is obtained

bull Agreement is very good ndash justification of Agreement is very good ndash justification of linearizationlinearization

ΛΛ101099

ExperimentExperiment TheoryTheory

121 121 plusmnplusmn 002 002 119(3)119(3)

532 532 plusmnplusmn 02 02 528(4)528(4)

297 297 plusmnplusmn 002 002 295(2)295(2)

2 Boundary effects2 Boundary effects

bull Close to edge ndash nonsymmetric induced Close to edge ndash nonsymmetric induced currentscurrents

2 Boundary effects 2 Boundary effects contcont

bull Repulsive force occursRepulsive force occurs

bull Magnet follows a quasiperiodical Magnet follows a quasiperiodical trajectorytrajectory

bull Exact modeling very difficultExact modeling very difficult

TheoryTheory

bull Acting on the magnet rolling motionActing on the magnet rolling motion

bull GravityGravity

bull Earth field torqueEarth field torque

bull Friction Friction

GF

frF

From the sideFrom the side

x

EB

E

From aboveFrom above

x

y

m

xy ndash unit vectorsm ndash magnetic

moment


Trajectory equationTrajectory equation

2220 rryxx

DRmB

gr

E

4sin

6

x0 ndash initial x ndash position of magnet

R ndash magnet radius

ρ ndash magnet density

D ndash magnet thickness

bull Trajectory ndash portion of circleTrajectory ndash portion of circle

bull For different initial angles numerical For different initial angles numerical solution neccesarysolution neccesary


bull Linear acceleration while rollingLinear acceleration while rolling

bull Special case magnetic moment initially Special case magnetic moment initially normal to Earth field ndash simple trajectorynormal to Earth field ndash simple trajectory

bull Magnetic field torqueMagnetic field torque

sinsin3

2ga

g ndash acceleration of gravityφ ndash plate inclination

θ ndash angle between magnetic moment and Earth field vector

EBmτ m ndash magnetic moment vector

BE ndash Earth field vector

I ndash moment of inertia of magnet

sinEmBI

Results and comparationResults and comparation


bull Magnet harr an array of infinitely thin Magnet harr an array of infinitely thin dipolesdipoles

bull Force on one dipoleForce on one dipole

Force on magnet in our geometryForce on magnet in our geometryR ndash magnet radius


Biny ndash y - component of induced field

ε - parameter

2

2

0

2

D

DRiny dBMR

F

mmm zyxindd

BmF Bin ndash induced field

xmymzm ndash dipole coordinates

ProblemProblem

Investigate the motion of a magnet Investigate the motion of a magnet

as it rolls down an inclined planeas it rolls down an inclined plane

OutlineOutline




















m

Grow

ing flux

Falling flux

vv ndash magnet




dF






















Amplifier amp ADC

PC

Detector solenoids



time [s]

0 1 2 3 4 5 6 7

volta

ge [a

u]

-6

-4

-2

0

2

4

6










Magnet detecting

solenoids

Liquid nitrogen

Aluminium plate

Magnet

Styrofoam box


Temperature wire


The boxThe box


TheoryTheory


x

z

y

M

dF


Fd ndash drag force



t - time

yM ˆMtvxx ˆˆ

yy ˆˆ

zz ˆˆ








xv

xv in

in

0

002 BBB







x [m]

-0020 -0015 -0010 -0005 0000 0005 0010 0015

z [m

]

-0020

-0015

-0010

-0005

0000

0005

0010

0015

-015 -010 -005 000 005 010 015

Biny [mT]





cmSection y = 0



x [m]

-002 -001 000 001 002

z [m

]

-0015

-0010

-0005

0000

0005

0010

0015




sin

g

vT









-5 0 5 10 15 20 25 30

term

inal

vel

ocity

[ms

]

-005

000

005

010

015

020

025

030

Diameter [cm] 10

Thickness [cm] 05




Diameter [cm] 254

Thickness [cm] 254




6 8 10 12 14 16 18

term

inal

vel

ocity

[ms

]

004

006

008

010

012

014



Diameter [cm] 095

Thickness [cm] 063



1conductivity [nm]

5 10 15 20 25 30

term

inal

vel

ocity

[ms

]

004

006

008

010

012

014

016




ΛΛ101099


121 121 plusmnplusmn 002 002 119(3)119(3)

532 532 plusmnplusmn 02 02 528(4)528(4)

297 297 plusmnplusmn 002 002 295(2)295(2)







TheoryTheory


bull GravityGravity



GF

frF


x

EB

E


x

y

m


moment



2220 rryxx

DRmB

gr

E

4sin

6











sinsin3

2ga






sinEmBI








ε - parameter

2

2

0

2

D

DRiny dBMR

F

mmm zyxindd



OutlineOutline




















m

Grow

ing flux

Falling flux

vv ndash magnet




dF






















Amplifier amp ADC

PC

Detector solenoids



time [s]

0 1 2 3 4 5 6 7

volta

ge [a

u]

-6

-4

-2

0

2

4

6










Magnet detecting

solenoids

Liquid nitrogen

Aluminium plate

Magnet

Styrofoam box


Temperature wire


The boxThe box


TheoryTheory


x

z

y

M

dF


Fd ndash drag force



t - time

yM ˆMtvxx ˆˆ

yy ˆˆ

zz ˆˆ








xv

xv in

in

0

002 BBB







x [m]

-0020 -0015 -0010 -0005 0000 0005 0010 0015

z [m

]

-0020

-0015

-0010

-0005

0000

0005

0010

0015

-015 -010 -005 000 005 010 015

Biny [mT]





cmSection y = 0



x [m]

-002 -001 000 001 002

z [m

]

-0015

-0010

-0005

0000

0005

0010

0015




sin

g

vT









-5 0 5 10 15 20 25 30

term

inal

vel

ocity

[ms

]

-005

000

005

010

015

020

025

030

Diameter [cm] 10

Thickness [cm] 05




Diameter [cm] 254

Thickness [cm] 254




6 8 10 12 14 16 18

term

inal

vel

ocity

[ms

]

004

006

008

010

012

014



Diameter [cm] 095

Thickness [cm] 063



1conductivity [nm]

5 10 15 20 25 30

term

inal

vel

ocity

[ms

]

004

006

008

010

012

014

016




ΛΛ101099


121 121 plusmnplusmn 002 002 119(3)119(3)

532 532 plusmnplusmn 02 02 528(4)528(4)

297 297 plusmnplusmn 002 002 295(2)295(2)







TheoryTheory


bull GravityGravity



GF

frF


x

EB

E


x

y

m


moment



2220 rryxx

DRmB

gr

E

4sin

6











sinsin3

2ga






sinEmBI








ε - parameter

2

2

0

2

D

DRiny dBMR

F

mmm zyxindd


















m

Grow

ing flux

Falling flux

vv ndash magnet




dF






















Amplifier amp ADC

PC

Detector solenoids



time [s]

0 1 2 3 4 5 6 7

volta

ge [a

u]

-6

-4

-2

0

2

4

6










Magnet detecting

solenoids

Liquid nitrogen

Aluminium plate

Magnet

Styrofoam box


Temperature wire


The boxThe box


TheoryTheory


x

z

y

M

dF


Fd ndash drag force



t - time

yM ˆMtvxx ˆˆ

yy ˆˆ

zz ˆˆ








xv

xv in

in

0

002 BBB







x [m]

-0020 -0015 -0010 -0005 0000 0005 0010 0015

z [m

]

-0020

-0015

-0010

-0005

0000

0005

0010

0015

-015 -010 -005 000 005 010 015

Biny [mT]





cmSection y = 0



x [m]

-002 -001 000 001 002

z [m

]

-0015

-0010

-0005

0000

0005

0010

0015




sin

g

vT









-5 0 5 10 15 20 25 30

term

inal

vel

ocity

[ms

]

-005

000

005

010

015

020

025

030

Diameter [cm] 10

Thickness [cm] 05




Diameter [cm] 254

Thickness [cm] 254




6 8 10 12 14 16 18

term

inal

vel

ocity

[ms

]

004

006

008

010

012

014



Diameter [cm] 095

Thickness [cm] 063



1conductivity [nm]

5 10 15 20 25 30

term

inal

vel

ocity

[ms

]

004

006

008

010

012

014

016




ΛΛ101099


121 121 plusmnplusmn 002 002 119(3)119(3)

532 532 plusmnplusmn 02 02 528(4)528(4)

297 297 plusmnplusmn 002 002 295(2)295(2)







TheoryTheory


bull GravityGravity



GF

frF


x

EB

E


x

y

m


moment



2220 rryxx

DRmB

gr

E

4sin

6











sinsin3

2ga






sinEmBI








ε - parameter

2

2

0

2

D

DRiny dBMR

F

mmm zyxindd
























Amplifier amp ADC

PC

Detector solenoids



time [s]

0 1 2 3 4 5 6 7

volta

ge [a

u]

-6

-4

-2

0

2

4

6










Magnet detecting

solenoids

Liquid nitrogen

Aluminium plate

Magnet

Styrofoam box


Temperature wire


The boxThe box


TheoryTheory


x

z

y

M

dF


Fd ndash drag force



t - time

yM ˆMtvxx ˆˆ

yy ˆˆ

zz ˆˆ








xv

xv in

in

0

002 BBB







x [m]

-0020 -0015 -0010 -0005 0000 0005 0010 0015

z [m

]

-0020

-0015

-0010

-0005

0000

0005

0010

0015

-015 -010 -005 000 005 010 015

Biny [mT]





cmSection y = 0



x [m]

-002 -001 000 001 002

z [m

]

-0015

-0010

-0005

0000

0005

0010

0015




sin

g

vT









-5 0 5 10 15 20 25 30

term

inal

vel

ocity

[ms

]

-005

000

005

010

015

020

025

030

Diameter [cm] 10

Thickness [cm] 05




Diameter [cm] 254

Thickness [cm] 254




6 8 10 12 14 16 18

term

inal

vel

ocity

[ms

]

004

006

008

010

012

014



Diameter [cm] 095

Thickness [cm] 063



1conductivity [nm]

5 10 15 20 25 30

term

inal

vel

ocity

[ms

]

004

006

008

010

012

014

016




ΛΛ101099


121 121 plusmnplusmn 002 002 119(3)119(3)

532 532 plusmnplusmn 02 02 528(4)528(4)

297 297 plusmnplusmn 002 002 295(2)295(2)







TheoryTheory


bull GravityGravity



GF

frF


x

EB

E


x

y

m


moment



2220 rryxx

DRmB

gr

E

4sin

6











sinsin3

2ga






sinEmBI








ε - parameter

2

2

0

2

D

DRiny dBMR

F

mmm zyxindd





time [s]

0 1 2 3 4 5 6 7

volta

ge [a

u]

-6

-4

-2

0

2

4

6










Magnet detecting

solenoids

Liquid nitrogen

Aluminium plate

Magnet

Styrofoam box


Temperature wire


The boxThe box


TheoryTheory


x

z

y

M

dF


Fd ndash drag force



t - time

yM ˆMtvxx ˆˆ

yy ˆˆ

zz ˆˆ








xv

xv in

in

0

002 BBB







x [m]

-0020 -0015 -0010 -0005 0000 0005 0010 0015

z [m

]

-0020

-0015

-0010

-0005

0000

0005

0010

0015

-015 -010 -005 000 005 010 015

Biny [mT]





cmSection y = 0



x [m]

-002 -001 000 001 002

z [m

]

-0015

-0010

-0005

0000

0005

0010

0015




sin

g

vT









-5 0 5 10 15 20 25 30

term

inal

vel

ocity

[ms

]

-005

000

005

010

015

020

025

030

Diameter [cm] 10

Thickness [cm] 05




Diameter [cm] 254

Thickness [cm] 254




6 8 10 12 14 16 18

term

inal

vel

ocity

[ms

]

004

006

008

010

012

014



Diameter [cm] 095

Thickness [cm] 063



1conductivity [nm]

5 10 15 20 25 30

term

inal

vel

ocity

[ms

]

004

006

008

010

012

014

016




ΛΛ101099


121 121 plusmnplusmn 002 002 119(3)119(3)

532 532 plusmnplusmn 02 02 528(4)528(4)

297 297 plusmnplusmn 002 002 295(2)295(2)







TheoryTheory


bull GravityGravity



GF

frF


x

EB

E


x

y

m


moment



2220 rryxx

DRmB

gr

E

4sin

6











sinsin3

2ga






sinEmBI








ε - parameter

2

2

0

2

D

DRiny dBMR

F

mmm zyxindd












Magnet detecting

solenoids

Liquid nitrogen

Aluminium plate

Magnet

Styrofoam box


Temperature wire


The boxThe box


TheoryTheory


x

z

y

M

dF


Fd ndash drag force



t - time

yM ˆMtvxx ˆˆ

yy ˆˆ

zz ˆˆ








xv

xv in

in

0

002 BBB







x [m]

-0020 -0015 -0010 -0005 0000 0005 0010 0015

z [m

]

-0020

-0015

-0010

-0005

0000

0005

0010

0015

-015 -010 -005 000 005 010 015

Biny [mT]





cmSection y = 0



x [m]

-002 -001 000 001 002

z [m

]

-0015

-0010

-0005

0000

0005

0010

0015




sin

g

vT









-5 0 5 10 15 20 25 30

term

inal

vel

ocity

[ms

]

-005

000

005

010

015

020

025

030

Diameter [cm] 10

Thickness [cm] 05




Diameter [cm] 254

Thickness [cm] 254




6 8 10 12 14 16 18

term

inal

vel

ocity

[ms

]

004

006

008

010

012

014



Diameter [cm] 095

Thickness [cm] 063



1conductivity [nm]

5 10 15 20 25 30

term

inal

vel

ocity

[ms

]

004

006

008

010

012

014

016




ΛΛ101099


121 121 plusmnplusmn 002 002 119(3)119(3)

532 532 plusmnplusmn 02 02 528(4)528(4)

297 297 plusmnplusmn 002 002 295(2)295(2)







TheoryTheory


bull GravityGravity



GF

frF


x

EB

E


x

y

m


moment



2220 rryxx

DRmB

gr

E

4sin

6











sinsin3

2ga






sinEmBI








ε - parameter

2

2

0

2

D

DRiny dBMR

F

mmm zyxindd




The boxThe box


TheoryTheory


x

z

y

M

dF


Fd ndash drag force



t - time

yM ˆMtvxx ˆˆ

yy ˆˆ

zz ˆˆ








xv

xv in

in

0

002 BBB







x [m]

-0020 -0015 -0010 -0005 0000 0005 0010 0015

z [m

]

-0020

-0015

-0010

-0005

0000

0005

0010

0015

-015 -010 -005 000 005 010 015

Biny [mT]





cmSection y = 0



x [m]

-002 -001 000 001 002

z [m

]

-0015

-0010

-0005

0000

0005

0010

0015




sin

g

vT









-5 0 5 10 15 20 25 30

term

inal

vel

ocity

[ms

]

-005

000

005

010

015

020

025

030

Diameter [cm] 10

Thickness [cm] 05




Diameter [cm] 254

Thickness [cm] 254




6 8 10 12 14 16 18

term

inal

vel

ocity

[ms

]

004

006

008

010

012

014



Diameter [cm] 095

Thickness [cm] 063



1conductivity [nm]

5 10 15 20 25 30

term

inal

vel

ocity

[ms

]

004

006

008

010

012

014

016




ΛΛ101099


121 121 plusmnplusmn 002 002 119(3)119(3)

532 532 plusmnplusmn 02 02 528(4)528(4)

297 297 plusmnplusmn 002 002 295(2)295(2)







TheoryTheory


bull GravityGravity



GF

frF


x

EB

E


x

y

m


moment



2220 rryxx

DRmB

gr

E

4sin

6











sinsin3

2ga






sinEmBI








ε - parameter

2

2

0

2

D

DRiny dBMR

F

mmm zyxindd



TheoryTheory


x

z

y

M

dF


Fd ndash drag force



t - time

yM ˆMtvxx ˆˆ

yy ˆˆ

zz ˆˆ








xv

xv in

in

0

002 BBB







x [m]

-0020 -0015 -0010 -0005 0000 0005 0010 0015

z [m

]

-0020

-0015

-0010

-0005

0000

0005

0010

0015

-015 -010 -005 000 005 010 015

Biny [mT]





cmSection y = 0



x [m]

-002 -001 000 001 002

z [m

]

-0015

-0010

-0005

0000

0005

0010

0015




sin

g

vT









-5 0 5 10 15 20 25 30

term

inal

vel

ocity

[ms

]

-005

000

005

010

015

020

025

030

Diameter [cm] 10

Thickness [cm] 05




Diameter [cm] 254

Thickness [cm] 254




6 8 10 12 14 16 18

term

inal

vel

ocity

[ms

]

004

006

008

010

012

014



Diameter [cm] 095

Thickness [cm] 063



1conductivity [nm]

5 10 15 20 25 30

term

inal

vel

ocity

[ms

]

004

006

008

010

012

014

016




ΛΛ101099


121 121 plusmnplusmn 002 002 119(3)119(3)

532 532 plusmnplusmn 02 02 528(4)528(4)

297 297 plusmnplusmn 002 002 295(2)295(2)







TheoryTheory


bull GravityGravity



GF

frF


x

EB

E


x

y

m


moment



2220 rryxx

DRmB

gr

E

4sin

6











sinsin3

2ga






sinEmBI








ε - parameter

2

2

0

2

D

DRiny dBMR

F

mmm zyxindd










xv

xv in

in

0

002 BBB







x [m]

-0020 -0015 -0010 -0005 0000 0005 0010 0015

z [m

]

-0020

-0015

-0010

-0005

0000

0005

0010

0015

-015 -010 -005 000 005 010 015

Biny [mT]





cmSection y = 0



x [m]

-002 -001 000 001 002

z [m

]

-0015

-0010

-0005

0000

0005

0010

0015




sin

g

vT









-5 0 5 10 15 20 25 30

term

inal

vel

ocity

[ms

]

-005

000

005

010

015

020

025

030

Diameter [cm] 10

Thickness [cm] 05




Diameter [cm] 254

Thickness [cm] 254




6 8 10 12 14 16 18

term

inal

vel

ocity

[ms

]

004

006

008

010

012

014



Diameter [cm] 095

Thickness [cm] 063



1conductivity [nm]

5 10 15 20 25 30

term

inal

vel

ocity

[ms

]

004

006

008

010

012

014

016




ΛΛ101099


121 121 plusmnplusmn 002 002 119(3)119(3)

532 532 plusmnplusmn 02 02 528(4)528(4)

297 297 plusmnplusmn 002 002 295(2)295(2)







TheoryTheory


bull GravityGravity



GF

frF


x

EB

E


x

y

m


moment



2220 rryxx

DRmB

gr

E

4sin

6











sinsin3

2ga






sinEmBI








ε - parameter

2

2

0

2

D

DRiny dBMR

F

mmm zyxindd






x [m]

-0020 -0015 -0010 -0005 0000 0005 0010 0015

z [m

]

-0020

-0015

-0010

-0005

0000

0005

0010

0015

-015 -010 -005 000 005 010 015

Biny [mT]





cmSection y = 0



x [m]

-002 -001 000 001 002

z [m

]

-0015

-0010

-0005

0000

0005

0010

0015




sin

g

vT









-5 0 5 10 15 20 25 30

term

inal

vel

ocity

[ms

]

-005

000

005

010

015

020

025

030

Diameter [cm] 10

Thickness [cm] 05




Diameter [cm] 254

Thickness [cm] 254




6 8 10 12 14 16 18

term

inal

vel

ocity

[ms

]

004

006

008

010

012

014



Diameter [cm] 095

Thickness [cm] 063



1conductivity [nm]

5 10 15 20 25 30

term

inal

vel

ocity

[ms

]

004

006

008

010

012

014

016




ΛΛ101099


121 121 plusmnplusmn 002 002 119(3)119(3)

532 532 plusmnplusmn 02 02 528(4)528(4)

297 297 plusmnplusmn 002 002 295(2)295(2)







TheoryTheory


bull GravityGravity



GF

frF


x

EB

E


x

y

m


moment



2220 rryxx

DRmB

gr

E

4sin

6











sinsin3

2ga






sinEmBI








ε - parameter

2

2

0

2

D

DRiny dBMR

F

mmm zyxindd





x [m]

-002 -001 000 001 002

z [m

]

-0015

-0010

-0005

0000

0005

0010

0015




sin

g

vT









-5 0 5 10 15 20 25 30

term

inal

vel

ocity

[ms

]

-005

000

005

010

015

020

025

030

Diameter [cm] 10

Thickness [cm] 05




Diameter [cm] 254

Thickness [cm] 254




6 8 10 12 14 16 18

term

inal

vel

ocity

[ms

]

004

006

008

010

012

014



Diameter [cm] 095

Thickness [cm] 063



1conductivity [nm]

5 10 15 20 25 30

term

inal

vel

ocity

[ms

]

004

006

008

010

012

014

016




ΛΛ101099


121 121 plusmnplusmn 002 002 119(3)119(3)

532 532 plusmnplusmn 02 02 528(4)528(4)

297 297 plusmnplusmn 002 002 295(2)295(2)







TheoryTheory


bull GravityGravity



GF

frF


x

EB

E


x

y

m


moment



2220 rryxx

DRmB

gr

E

4sin

6











sinsin3

2ga






sinEmBI








ε - parameter

2

2

0

2

D

DRiny dBMR

F

mmm zyxindd






sin

g

vT









-5 0 5 10 15 20 25 30

term

inal

vel

ocity

[ms

]

-005

000

005

010

015

020

025

030

Diameter [cm] 10

Thickness [cm] 05




Diameter [cm] 254

Thickness [cm] 254




6 8 10 12 14 16 18

term

inal

vel

ocity

[ms

]

004

006

008

010

012

014



Diameter [cm] 095

Thickness [cm] 063



1conductivity [nm]

5 10 15 20 25 30

term

inal

vel

ocity

[ms

]

004

006

008

010

012

014

016




ΛΛ101099


121 121 plusmnplusmn 002 002 119(3)119(3)

532 532 plusmnplusmn 02 02 528(4)528(4)

297 297 plusmnplusmn 002 002 295(2)295(2)







TheoryTheory


bull GravityGravity



GF

frF


x

EB

E


x

y

m


moment



2220 rryxx

DRmB

gr

E

4sin

6











sinsin3

2ga






sinEmBI








ε - parameter

2

2

0

2

D

DRiny dBMR

F

mmm zyxindd






-5 0 5 10 15 20 25 30

term

inal

vel

ocity

[ms

]

-005

000

005

010

015

020

025

030

Diameter [cm] 10

Thickness [cm] 05




Diameter [cm] 254

Thickness [cm] 254




6 8 10 12 14 16 18

term

inal

vel

ocity

[ms

]

004

006

008

010

012

014



Diameter [cm] 095

Thickness [cm] 063



1conductivity [nm]

5 10 15 20 25 30

term

inal

vel

ocity

[ms

]

004

006

008

010

012

014

016




ΛΛ101099


121 121 plusmnplusmn 002 002 119(3)119(3)

532 532 plusmnplusmn 02 02 528(4)528(4)

297 297 plusmnplusmn 002 002 295(2)295(2)







TheoryTheory


bull GravityGravity



GF

frF


x

EB

E


x

y

m


moment



2220 rryxx

DRmB

gr

E

4sin

6











sinsin3

2ga






sinEmBI








ε - parameter

2

2

0

2

D

DRiny dBMR

F

mmm zyxindd




Diameter [cm] 254

Thickness [cm] 254




6 8 10 12 14 16 18

term

inal

vel

ocity

[ms

]

004

006

008

010

012

014



Diameter [cm] 095

Thickness [cm] 063



1conductivity [nm]

5 10 15 20 25 30

term

inal

vel

ocity

[ms

]

004

006

008

010

012

014

016




ΛΛ101099


121 121 plusmnplusmn 002 002 119(3)119(3)

532 532 plusmnplusmn 02 02 528(4)528(4)

297 297 plusmnplusmn 002 002 295(2)295(2)







TheoryTheory


bull GravityGravity



GF

frF


x

EB

E


x

y

m


moment



2220 rryxx

DRmB

gr

E

4sin

6











sinsin3

2ga






sinEmBI








ε - parameter

2

2

0

2

D

DRiny dBMR

F

mmm zyxindd





Diameter [cm] 095

Thickness [cm] 063



1conductivity [nm]

5 10 15 20 25 30

term

inal

vel

ocity

[ms

]

004

006

008

010

012

014

016




ΛΛ101099


121 121 plusmnplusmn 002 002 119(3)119(3)

532 532 plusmnplusmn 02 02 528(4)528(4)

297 297 plusmnplusmn 002 002 295(2)295(2)







TheoryTheory


bull GravityGravity



GF

frF


x

EB

E


x

y

m


moment



2220 rryxx

DRmB

gr

E

4sin

6











sinsin3

2ga






sinEmBI








ε - parameter

2

2

0

2

D

DRiny dBMR

F

mmm zyxindd






ΛΛ101099


121 121 plusmnplusmn 002 002 119(3)119(3)

532 532 plusmnplusmn 02 02 528(4)528(4)

297 297 plusmnplusmn 002 002 295(2)295(2)







TheoryTheory


bull GravityGravity



GF

frF


x

EB

E


x

y

m


moment



2220 rryxx

DRmB

gr

E

4sin

6











sinsin3

2ga






sinEmBI








ε - parameter

2

2

0

2

D

DRiny dBMR

F

mmm zyxindd









TheoryTheory


bull GravityGravity



GF

frF


x

EB

E


x

y

m


moment



2220 rryxx

DRmB

gr

E

4sin

6











sinsin3

2ga






sinEmBI








ε - parameter

2

2

0

2

D

DRiny dBMR

F

mmm zyxindd





2220 rryxx

DRmB

gr

E

4sin

6











sinsin3

2ga






sinEmBI








ε - parameter

2

2

0

2

D

DRiny dBMR

F

mmm zyxindd







sinsin3

2ga






sinEmBI








ε - parameter

2

2

0

2

D

DRiny dBMR

F

mmm zyxindd










ε - parameter

2

2

0

2

D

DRiny dBMR

F

mmm zyxindd



Problem 12. Rolling Magnets. Problem Investigate the motion of a magnet as it rolls down an inclined...

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Transcript of Problem 12. Rolling Magnets. Problem Investigate the motion of a magnet as it rolls down an inclined...