1. © 2003 By Default! A Free sample background from www.awesomebackgrounds.com Slide 1 Cavendish Experiment Presented by Mark Reeher Lab Partner: Jon Rosenfield For Physics 521

2. © 2003 By Default! A Free sample background from www.awesomebackgrounds.com Slide 2 Presentation Overview Historical Background Historical Background Theory Theory Experimental Setup and Methods Experimental Setup and Methods Results Results Analysis of Results Analysis of Results –Uncertainties Conclusions Conclusions

3. © 2003 By Default! A Free sample background from www.awesomebackgrounds.com Slide 3 Brief Timeline of Gravitational Physics 4 th Century B.C: Aristotle – tendency of objects to be pulled to Earth 4 th Century B.C: Aristotle – tendency of objects to be pulled to Earth 1645: Ismael Bulliadus - inverse square relation 1645: Ismael Bulliadus - inverse square relation 1665: Sir Isaac Newton - 1665: Sir Isaac Newton - 1798: Henry Cavendish – calculation of Universal Gravitation Constant, G 1798: Henry Cavendish – calculation of Universal Gravitation Constant, G Early 1900s: Einstein- Early 1900s: Einstein- Inertia-gravitation equivalenceInertia-gravitation equivalence General relativityGeneral relativity

4. © 2003 By Default! A Free sample background from www.awesomebackgrounds.com Slide 4 Cavendish Experiment Cavendish Experiment John Michell – conception of experiment John Michell – conception of experiment –Torsion Balance Henry Cavendish – rebuilt balance and Henry Cavendish – rebuilt balance and ran experiment in 1797-1798 Basic Idea – directly Basic Idea – directly measure F g, find G Found: Found: G = 6.754 × 10 −11 m 3 kg -1 s -2

5. © 2003 By Default! A Free sample background from www.awesomebackgrounds.com Slide 5 Committee on Data for Science and Technology’s Value

6. © 2003 By Default! A Free sample background from www.awesomebackgrounds.com Slide 6 Theory – Experimental Design Large masses brought near small masses Large masses brought near small masses Gravitational force  movement in torsion balance Gravitational force  movement in torsion balance Study motion to determine F g Study motion to determine F g With F g, measure M, m, r With F g, measure M, m, r –Newton’s gravitational equation –Result = calculated G

7. © 2003 By Default! A Free sample background from www.awesomebackgrounds.com Slide 7 Derivation - 1 FαFα FβFβ Top View

8. © 2003 By Default! A Free sample background from www.awesomebackgrounds.com Slide 8 Derivation - 2

9. © 2003 By Default! A Free sample background from www.awesomebackgrounds.com Slide 9 Small Angle Approximation For simplicity, we assume θ is very small 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 This assumption confirmed by finding the largest possible angle of setup –θ max = 0.03884 = 2.226º –~0.05% difference between tan θ and θ

10. © 2003 By Default! A Free sample background from www.awesomebackgrounds.com Slide 10 Experimental Setup

11. © 2003 By Default! A Free sample background from www.awesomebackgrounds.com Slide 11 Experimental Setup Large masses Ametek plotter (converted) He-Ne laser Torsion balance enclosure Vacuum pump (oil)

12. © 2003 By Default! A Free sample background from www.awesomebackgrounds.com Slide 12

13. © 2003 By Default! A Free sample background from www.awesomebackgrounds.com Slide 13

14. © 2003 By Default! A Free sample background from www.awesomebackgrounds.com Slide 14

15. © 2003 By Default! A Free sample background from www.awesomebackgrounds.com Slide 15

16. © 2003 By Default! A Free sample background from www.awesomebackgrounds.com Slide 16 Setup Diagram Laser Plotter

17. © 2003 By Default! A Free sample background from www.awesomebackgrounds.com Slide 17 Setup Diagram So we need to keep in mind, the plotter reacts to 2θ

18. © 2003 By Default! A Free sample background from www.awesomebackgrounds.com Slide 18 Setup Notes Torsion enclosure pumped to ~100 mTorr Torsion enclosure pumped to ~100 mTorr Data recorded automatically in Labview Data recorded automatically in Labview –Photodiode position vs time (4 s intervals) Six total trials Six total trials – 2 counter-clockwise (positive) torque – 2 clockwise (negative)torque – 2 no mass FαFα FβFβ FαFα FβFβ

19. © 2003 By Default! A Free sample background from www.awesomebackgrounds.com Slide 19 Given in lab manual Given in lab manual –m = 0.019 kg –M rod = 0.031 kg (square cross section) –L/2 = 15.24 cm Distance measurements (in inches) Distance measurements (in inches) D d (mirror-diode) = 45 1/32 ” D d (mirror-diode) = 45 1/32 ” ω and θ are found from Matlab data ω and θ are found from Matlab data Results (Our Measurements) 12 34

20. © 2003 By Default! A Free sample background from www.awesomebackgrounds.com Slide 20

21. © 2003 By Default! A Free sample background from www.awesomebackgrounds.com Slide 21 Analysis Data from best fit: Data from best fit: –General model: f(x) = a*exp(-x/b)*cos(c*x+d)+e f(x) = a*exp(-x/b)*cos(c*x+d)+e –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: SSE: 1000 SSE: 1000 R-square: 0.9986 R-square: 0.9986 Adjusted R-square: 0.9986 Adjusted R-square: 0.9986 RMSE: 1.002 RMSE: 1.002

22. © 2003 By Default! A Free sample background from www.awesomebackgrounds.com Slide 22 Analysis I calculation I calculation Κ calculation Κ calculation Avg K = 2.60588 x 10 -7 + 1.197 x 10 -11 kg m/s 2 Avg K = 2.60588 x 10 -7 + 1.197 x 10 -11 kg m/s 2

23. © 2003 By Default! A Free sample background from www.awesomebackgrounds.com Slide 23 Analysis r i calculation (m) r i calculation (m) θ calculation θ calculation Avg e o from “NM” values: Avg e o from “NM” values: e o = 3.954” + 0.000177” Define x i = e o - e i Define x i = e o - e i

24. © 2003 By Default! A Free sample background from www.awesomebackgrounds.com Slide 24 Analysis Now find θ from tan -1 : Now find θ from tan -1 : Finally… we find G (m 3 s -2 ): Finally… we find G (m 3 s -2 ): Avg G = (3.89829 x 10 -10 + 1.7129 x 10 -11 )/M Avg G = (3.89829 x 10 -10 + 1.7129 x 10 -11 )/M

25. © 2003 By Default! A Free sample background from www.awesomebackgrounds.com Slide 25 Uncertainty Total Uncertainty relation for G: Total Uncertainty relation for G: 000000000000

26. © 2003 By Default! A Free sample background from www.awesomebackgrounds.com Slide 26 Uncertainty Each of the four variables also had combined uncertainty in their calculation 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: In a few cases, values were averaged:

27. © 2003 By Default! A Free sample background from www.awesomebackgrounds.com Slide 27 Conclusions M = 5.701 kg † M = 5.701 kg † –Gives us: –G Cavendish = 6.754 × 10 −11 m 3 kg -1 s -2 –G CODATA = 6.67428 × 10 −11 m 3 kg -1 s -2 Obvious setup interference Obvious setup interference M Earth M Earth † conversation with Jose Accepted value = 5.97 x 10 24 kg