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Cavendish ExperimentCavendish Experiment

Presented by Mark ReeherPresented by Mark Reeher

Lab Partner: Jon RosenfieldLab Partner: Jon RosenfieldFor Physics 521For Physics 521

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Presentation OverviewPresentation Overview

Historical BackgroundHistorical Background

TheoryTheory

Experimental Setup and MethodsExperimental Setup and Methods

ResultsResults

Analysis of ResultsAnalysis of Results– UncertaintiesUncertainties

ConclusionsConclusions

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Brief Timeline of Gravitational Brief Timeline of Gravitational PhysicsPhysics

44thth Century B.C: Aristotle – tendency of Century B.C: Aristotle – tendency of objects to be pulled to Earthobjects to be pulled to Earth

1645: Ismael Bulliadus - inverse square 1645: Ismael Bulliadus - inverse square relationrelation

1665: Sir Isaac Newton - 1665: Sir Isaac Newton - 1798: Henry Cavendish – calculation of 1798: Henry Cavendish – calculation of

Universal Gravitation Constant, GUniversal Gravitation Constant, G Early 1900s: Einstein-Early 1900s: Einstein-

• Inertia-gravitation equivalenceInertia-gravitation equivalence• General relativityGeneral relativity

221

g r

MMG F

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Cavendish ExperimentCavendish Experiment

John Michell – conception of experimentJohn Michell – conception of experiment– Torsion BalanceTorsion Balance

Henry Cavendish – rebuilt balance and Henry Cavendish – rebuilt balance and ran experiment in ran experiment in 1797-17981797-1798

Basic Idea – directly Basic Idea – directly

measure Fmeasure Fgg, find G, find G Found:Found:G = G = 6.754 × 106.754 × 10−11−11 m m33kgkg-1-1ss-2-2

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Committee on Data for Science Committee on Data for Science and Technology’s Valueand Technology’s Value

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Theory – Experimental DesignTheory – Experimental Design

Large masses brought Large masses brought near small massesnear small masses

Gravitational force Gravitational force movement in torsion movement in torsion balancebalance

Study motion to determine Study motion to determine FFgg

With FWith Fgg, measure M, m, r, measure M, m, r– Newton’s gravitational Newton’s gravitational

equationequation– Result = calculated GResult = calculated G

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Slide 7

Derivation - 1Derivation - 1

FαFβ

22

22

11

2

2

22

rrGMmL

r

Mm

r

MmGL

FL

FL

FdFd

Top View

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Slide 8

Derivation - 2Derivation - 2

22

11

2 rrGMmL /2/2 IT

2 I

1

22

2

222

112

11

2

rrLMm

IG

Irr

GMmL

1

22

112

rrLMm

G

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Slide 9

Small Angle ApproximationSmall Angle Approximation

For simplicity, we assume θ is very For simplicity, we assume θ is very smallsmall– Torque dot productTorque dot product– Tan θ = θTan θ = θ

This assumption confirmed by finding This assumption confirmed by finding the largest possible angle of setupthe largest possible angle of setup– θθmaxmax = 0.03884 = 2.226º = 0.03884 = 2.226º– ~0.05% difference between tan θ and θ~0.05% difference between tan θ and θ

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Experimental SetupExperimental Setup

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Experimental SetupExperimental Setup

Large masses

Ametek plotter (converted)

He-Ne laser

Torsion balance enclosure

Vacuum pump (oil)

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Setup DiagramSetup Diagram

Laser

Plotter

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Setup DiagramSetup Diagram

So we need to keep in mind, the plotter reacts to 2θ

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Setup NotesSetup Notes

Torsion enclosure pumped to ~100 mTorrTorsion enclosure pumped to ~100 mTorr

Data recorded automatically in LabviewData recorded automatically in Labview– Photodiode position vs time (4 s intervals)Photodiode position vs time (4 s intervals)

Six total trialsSix total trials– 2 counter-clockwise (positive) torque2 counter-clockwise (positive) torque– 2 clockwise (negative)torque2 clockwise (negative)torque– 2 no mass2 no mass

Fα

FβFα

Fβ

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Slide 19

Given in lab manualGiven in lab manual– m = 0.019 kgm = 0.019 kg– MMrodrod = 0.031 kg (square cross section) = 0.031 kg (square cross section)– L/2 = 15.24 cmL/2 = 15.24 cm

Distance measurements (in inches)Distance measurements (in inches)

DDdd (mirror-diode) = 45 (mirror-diode) = 45 1/321/32”” ω and θ are found from Matlab dataω and θ are found from Matlab data

Results (Our Measurements)Results (Our Measurements)

1 2 3 4d 3.8756 3.9097 3.9517 3.9027D 0.382 0.163 0.278 0.263

d-tube 2.197 2.197 2.197 2.197

1 2

34

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Slide 20

700

750

800

850

900

950

1000

1050

1100

0 10000 20000 30000 40000 50000 60000 70000

Time (s)

Dio

de

Po

siti

on

(ar

b u

nit

s)

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Slide 21

AnalysisAnalysis Data from best fit:Data from best fit:

– General model:General model:

f(x) = a*exp(-x/b)*cos(c*x+d)+ef(x) = a*exp(-x/b)*cos(c*x+d)+e

– Coefficients (with 95% confidence bounds):Coefficients (with 95% confidence bounds): a = 131 (130.4, 131.6)a = 131 (130.4, 131.6) b = 1.029e+004 (1.006e+004, 1.051e+004)b = 1.029e+004 (1.006e+004, 1.051e+004) c = 0.007577 (0.007575, 0.007579)c = 0.007577 (0.007575, 0.007579) d = 0.004448 (0.0001244, 0.008771)d = 0.004448 (0.0001244, 0.008771) e = 682.1 (681.9, 682.3)e = 682.1 (681.9, 682.3)  

– Goodness of fit:Goodness of fit: SSE: 1000SSE: 1000 R-square: 0.9986R-square: 0.9986 Adjusted R-square: 0.9986Adjusted R-square: 0.9986 RMSE: 1.002RMSE: 1.002

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Slide 22

AnalysisAnalysis I calculation I calculation

Κ calculationΚ calculation

Avg K = 2.60588 x 10Avg K = 2.60588 x 10-7-7 ++ 1.197 x 10 1.197 x 10-11-11 kg m/s kg m/s22

kg3

222

104.4

)4

1

5

2(2)2(

12

1

2

I

mLmrrLMI

III

mmrod

mrod

2 Iω δω K δK 1/δK 2̂ K/δK^2 I

CW1 0.007577 0.000001 2.52608E-07 3.33388E-11 8.99705E+20 2.27273E+14 4.40E-03

CW2 0.0075975 0.0000013 2.53977E-07 4.34577E-11 5.29501E+20 1.34481E+14

CCW1 0.00761 0.0000003 2.54813E-07 1.00452E-11 9.91021E+21 2.52525E+15

CCW2 0.007617 0.0000005 2.55282E-07 1.67574E-11 3.56112E+21 9.09091E+14

NM1 0.0076955 0.0000005 2.60571E-07 1.69301E-11 3.48884E+21 9.09091E+14NM2 0.007696 0.0000005 2.60605E-07 1.69312E-11 3.48839E+21 9.09091E+14

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Slide 23

AnalysisAnalysis rrii calculation (m) calculation (m)

θ calculationθ calculation

Avg eAvg eoo from “NM” values: from “NM” values:eeo o = 3.954” = 3.954” ++ 0.000177” 0.000177”

Define xDefine xii = e = eoo - e - eii

tubeiii dDdr2

1

2

1

r1 0.0868

r2 0.0817

r3 0.0852

r4 0.0841

δr 5.24E-03

03125.45tan

2

1

/)2tan(

01 iee

adjopp

e in inches δe X δxCW1 896.9 4.4845 0.0001 -0.53037 0.000203CW2 888.35 4.44175 0.00065 -0.48763 0.000674CCW1 682.1 3.4105 0.0005 0.543625 0.00053CCW2 682.75 3.41375 0.0004 0.540375 0.000437

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Slide 24

AnalysisAnalysis Now find θ from tanNow find θ from tan-1-1::

Finally… we find G (mFinally… we find G (m33ss-2-2):):

Avg G = (3.89829 x 10Avg G = (3.89829 x 10-10-10++ 1.7129 x 10 1.7129 x 10-11-11)/M)/M

θ dθ/dX dθ/dD δθ

CW1 -0.00589 0.022204 0.000262 4.67E-06

CW2 -0.00541 0.022204 0.00024 8.37E-06CCW1 0.006036 0.022204 -0.00027 7.23E-06CCW2 0.006 0.022204 -0.00027 6.4E-06

1

22

112

rrLMm

G

dG/dK dG/dθ dG/drα dG/drβ G δG sq'd δG

CW1 0.001569954 -6.94676E-08 4.62558E-09 4.8911E-09 4.09111E-10 1.24445E-21 3.53E-11CW2 0.0014434 -6.94676E-08 4.25272E-09 4.49683E-09 3.76133E-10 1.05216E-21 3.24E-11CCW1 0.001494708 6.45259E-08 -4.90547E-09 -4.49737E-09 3.89503E-10 1.21631E-21 3.49E-11CCW2 1.49E-03 6.45259E-08 -4.87614E-09 -4.47048E-09 3.87174E-10 1.20177E-21 3.47E-11

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Slide 25

UncertaintyUncertainty

Total Uncertainty relation for G:Total Uncertainty relation for G:

22

2

2

22

22

rr

Gr

r

GGGG iiii

1

22

112

rrLMm

Gi000000000000

1

22

112

rrLMm

Gi

2

223

114

rrLMmrr

Gi2

223

114

rrLMmrr

Gi

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Slide 26

UncertaintyUncertainty Each of the four variables also had combined Each of the four variables also had combined

uncertainty in their calculationuncertainty in their calculation– All type A aside from distance measurementsAll type A aside from distance measurements

In a few cases, values were averaged:In a few cases, values were averaged:

i i

i i

i i

i

x

x

x

2

2

2

11

1

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Slide 27

ConclusionsConclusions M = 5.701 kg M = 5.701 kg ††

– Gives us:Gives us:

– GGCavendishCavendish = = 6.754 × 106.754 × 10−11−11 m m33kgkg-1-1ss-2-2

– GGCODATA CODATA = 6.67428 = 6.67428 × 10× 10−11−11 m m33kgkg-1-1ss-2-2

Obvious setup interferenceObvious setup interference MMEarthEarth

† conversation with Jose

2131211 1000431083796 skgm-- ..G

kg242

2

1083.5

G

grM

r

mMGmgF

EarthEarth

Earth

Earthg

Accepted value = 5.97 x 1024 kg