Cavendish experiment
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Cavendish Experiment. Presented by Mark Reeher. Lab Partner: Jon Rosenfield For Physics 521. Presentation Overview. Historical Background Theory Experimental Setup and Methods Results Analysis of Results Uncertainties Conclusions. Brief Timeline of Gravitational Physics.

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Cavendish Experiment

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Cavendish Experiment

Presented by Mark Reeher

Lab Partner: Jon Rosenfield

For Physics 521


Presentation Overview

  • Historical Background

  • Theory

  • Experimental Setup and Methods

  • Results

  • Analysis of Results

    • Uncertainties

  • Conclusions


Brief Timeline of Gravitational Physics

  • 4th Century B.C: Aristotle – tendency of objects to be pulled to Earth

  • 1645: Ismael Bulliadus - inverse square relation

  • 1665: Sir Isaac Newton -

  • 1798: Henry Cavendish – calculation of Universal Gravitation Constant, G

  • Early 1900s: Einstein-

    • Inertia-gravitation equivalence

    • General relativity


Cavendish Experiment

  • John Michell – conception of experiment

    • Torsion Balance

  • Henry Cavendish – rebuilt balance and

    ran experiment in

    1797-1798

  • Basic Idea – directly

    measure Fg, find G

  • Found:

    G = 6.754 × 10−11 m3kg-1s-2


Committee on Data for Science and Technology’s Value


Theory – Experimental Design

  • Large masses brought near small masses

  • Gravitational force  movement in torsion balance

  • Study motion to determine Fg

  • With Fg, measure M, m, r

    • Newton’s gravitational equation

    • Result = calculated G


Top View

Derivation - 1


Derivation - 2


Small Angle Approximation

  • For simplicity, we assume θ is very small

    • Torque dot product

    • Tan θ = θ

  • This assumption confirmed by finding the largest possible angle of setup

    • θmax = 0.03884 = 2.226º

    • ~0.05% difference between tan θ and θ


Experimental Setup


Experimental Setup

Torsion balance enclosure

Large masses

Vacuum pump (oil)

He-Ne laser

Ametek plotter (converted)


Setup Diagram

Laser

Plotter


Setup Diagram

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


Setup Notes

  • Torsion enclosure pumped to ~100 mTorr

  • Data recorded automatically in Labview

    • Photodiode position vs time (4 s intervals)

  • Six total trials

    • 2 counter-clockwise (positive) torque

    • 2 clockwise (negative)torque

    • 2 no mass


Results (Our Measurements)

  • Given in lab manual

    • m = 0.019 kg

    • Mrod = 0.031 kg (square cross section)

    • L/2 = 15.24 cm

  • Distance measurements (in inches)

  • Dd (mirror-diode) = 45 1/32”

  • ω and θ are found from Matlab data

1

2

4

3


Analysis

  • Data from best fit:

    • General model:

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

    • Coefficients (with 95% confidence bounds):

             a =         131  (130.4, 131.6)

             b =  1.029e+004  (1.006e+004, 1.051e+004)

             c =    0.007577  (0.007575, 0.007579)

             d =    0.004448  (0.0001244, 0.008771)

             e =       682.1  (681.9, 682.3)

    • Goodness of fit:

        SSE: 1000

        R-square: 0.9986

        Adjusted R-square: 0.9986

        RMSE: 1.002


Analysis

  • I calculation

  • Κ calculation

  • Avg K = 2.60588 x 10-7+ 1.197 x 10-11 kg m/s2


Analysis

  • ri calculation (m)

  • θ calculation

  • Avg eo from “NM” values:

    eo = 3.954” + 0.000177”

  • Define xi = eo - ei


Analysis

  • Now find θ from tan-1:

  • Finally… we find G (m3s-2):

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


Uncertainty

  • Total Uncertainty relation for G:

000000000000


Uncertainty

  • Each of the four variables also had combined uncertainty in their calculation

    • All type A aside from distance measurements

  • In a few cases, values were averaged:


Conclusions

  • M = 5.701 kg †

    • Gives us:

    • GCavendish = 6.754 × 10−11 m3kg-1s-2

    • GCODATA = 6.67428 × 10−11 m3kg-1s-2

  • Obvious setup interference

  • MEarth

Accepted value = 5.97 x 1024 kg

† conversation with Jose


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