Procedure I

1. Procedure I • Set Up • Level the experiment using the threaded feet, making sure that the mirror is hanging freely in the center of the case, and center the pendulum in the middle of the mesa. • Make sure large masses rotate easily. • Move the large masses through the full range of motion and verify that both masses touch the window of the case. Do this carefully not to cause large disturbances from hitting the glass case with the masses. If not make note of which mass doesn’t touch. • Calibration • Using a strong magnet move the balance through the full range of motion and mark on the chalk board where the small masses touch the glass. • Measure halfway between the two maxima. Note distance between actual equilibrium (which should be measured) and center point. Use right triangles to determine the angle adjustment needed to center the actual equilibrium on the center point and adjust the torsion screw accordingly. • Measure the distance from the mirror to the chalkboard. • Taking Data • Move the large masses to where one is touching the glass. This will be the starting position for the measurement. Before taking data you must wait for the small masses to come to rest. The waiting time can be reduced by slowly bringing a strong magnet near one of the small masses, thereby damping the oscillations. • Once the balance has come to equilibrium, carefully turn on the laser pointer. 1

2. Procedure II • Making measurements • At t=0,after the small masses have stabilized in Position I, make a mark to indicate initial position and switch the Masses to Position II • Make marks every 15sec for 45min for each position • Switch to Position I, and repeat the same process • Marking extrema may improve results • It may be helpful to write down the timeevery few minutes, to keep track of data when calculating • Calculating Equilibrium positions • Take two separate averages of all marks for positions I and II, making sure there are an equal number of maxima in each direction 2

3. Fall 2012 Equilibrium Data • The measurements resulted in G=5.12x10-11 m3/kg/s2—23% less than the accepted value • Given that the experimental error is within 16% , this result is confusing. We do not have an explanation for it at this time. 3

4. Fall 2012 Constant Accel. Data • The measurements from Series 1 resulted in G = 5.23x10-11m3/kg/s2—22% less than the accepted value—and the measurements from Series 2 resulted in G = 6.16x10-11m3/kg/s2—7.7% less than the accepted value. • Given that the experimental error is 10% and 9%, respectively, the first result is confusing and the second is acceptable. We do not have an explanation for this at this time. 4

5. Error Discussion • There were giant problems with error • “Constant acceleration” method had ≈ 1951919335% error • “Equilibrium” method had ≈ 117983750% to 138881057% error • However, there are still errors beyond Pasco’s estimates • Possible sources of error include: • The mirror is not planar, it is concave. • If mirror moves laterally, laser’s incident angle will change. • If laser is not centered properly on the mirror, incident angle will not change linearly with mirror rotation. • Inaccuracies in measuring the equilibrium positions on the graphs • Definitely accounts for some of the error. • The separation of the large and small balls, b, is taken to be constant • it actually changes throughout the experiment. 5

6. Compare With Coulomb Experiment A Note on Previous Presentations… • Previous versions of this PowerPoint contained the following slide: • Forces are much smaller • Typical Coulomb force: 10-4 N (with V= 6kV for both spheres, and distance = 8cm) • Typical Cavendish force: 10-9 N (with distance = 46.5 mm, m1=20g, m2=1.5kg • Both require a correction factor • Coulomb experiment requires a correction factor of 1/(1-a3/R3) • a is the radius of the sphere, R is the distance between spheres • This is because the sphere is not a point charge • Cavendish experiment requires a correction factor of 1/(1-b) • b is the distance between the spheres • This is because there is gravitational force between each small mass and both large masses, but only one is considered in the calculations. • This inserts a correction factor with no explanation of its derivation.

7. A Note on Previous Presentations… • Fcorrection = G m1 m2 / h2 • h =[(2d)2 + (b sin θ)2]1/2 • Torquecorrection = d x Fcorrection = d Fcorrection sin θ • Torquegrav-corrected = 2 F d + 2 d Fcorrection sin θ = κθ =2G m1 m2 d / b2 + 2G m1 m2 d / [(2d)2 + (b sin θ)2] • G = κθ / (2 m1 m2 d / b2 + 2 m1 m2 d / [(2d)2 + (b sin θ)2]= κ deltaS *(4L)-1 * (2 m1 m2 d / b2 + 2 m1 m2 d / [(2d)2 + (b sin θ)2])-1= 4 pi2 I deltaS *(4L*T2)-1 * (2 m1 m2 d / b2 + 2 m1 m2 d / [(2d)2 + (b sin θ)2])-1= 8 pi2 m2 (d2 + 2/5r2)deltaS *(4L*T2*d)-1 * (2 m1 m2 / b2 + 2 m1 m2 / [(2d)2 + (b sin θ)2])-1= pi2 (d2 + 2/5r2)deltaS / (L T2 d m1) / (1/ b2 + 1 / [(2d)2 + (b sin [deltaS / (4L)])2])

8. A Note on Previous Presentations… • Using Mathematica to equate the result on the previous slide with the result derived earlier, the correction factor is (4 d^2 + b^2 Sin[deltaS/(4 L)]^2)/(b^2 + 4 d^2 + b^2 Sin[deltaS/(4 L)]^2)So multiplying the original result by this factor gives the corrected result for G. Using the values stated earlier, this gives a correction factor equal to approximately 0.83. • Since this is not negligible, we must include it in our final result.

9. Procedure I • Set Up • Level the experiment using the threaded feet, making sure that the mirror is hanging freely in the center of the case , and center the pendulum in the middle of the mesa. • Make sure that when the large masses are moved that the small masses only experience small oscillations. • Move the large masses through the full range of motion and verify that both masses touch the window of the case. Do this carefully not to cause large disturbances from hitting the glass case with the masses. If not make note of which mass doesn’t touch. • Calibration • Using a strong magnet move the balance through the full range of motion and mark on the graph paper where the small masses touch the glass. • To center the natural equilibrium position, we would move one small tick mark. Then we would watch which direction the masses moved towards and would move it another small tick if the movement was away from the center. Initially, one could move 2 small tick marks if one was not near the center already. The angular variation in equilibrium points from switching the mass-positions was approx. 0.01 rad = 0.8º. The total angular variation from maximums was 2.5º. • Measure the distance from the mirror to the midpoint between the marks where the small masses touch the glass. • Taking Data • Move the large masses to where one is touching the glass. This will be the starting position for the measurement. Before taking data you must wait for the small masses to come to rest. The waiting time can be reduced by slowly bringing a strong magnet near one of the small masses, thereby damping the oscillations. • Once the balance has come to equilibrium, carefully turn on the laser pointer. 9

10. Procedure II • Making measurements • At t=0,after the small masses have stabilized in Position I, make a mark to indicate initial position and switch the Masses to Position II • Make marks every 15sec for 2min, then every 30sec for 30min, moving down a row for each precession • Switch to Position I, and repeat the same process • For better results, make marks every 15sec for 45min for each position • Marking extrema may improve results () • It may be helpful to write down the timeevery few minutes, to keep track of data when calculating 10

11. Procedure III • Calculating Equilibrium positions • Two methods: amplitude and frequency • For amplitude, take two separate averages of all marks for positions I and II, making sure there are an equal number of maxima in each direction • For frequency, average the marks closest to ¼ and ¾ the time of each period 11

12. Spring 2010 Equilibrium Data • The measurements resulted in G=7.03 x 10-11- 4% greater than the accepted value • Given that the experimental error is within 10%, these are excellent results • Note that the equilibrium position on the first data set is above the apparent value • This may be due to experimental error • Probably due to the method of calculating the equilibrium position 12

13. Spring 2010 Constant Accel. Data • The measurements resulted in G= 8.68 x 10-11 - 30% greater than the accepted value • Given that the experimental error is around 30%, these results are very good 13

14. Error Discussion I • Neither method had serious problems with error • “Constant acceleration” method had ≈ 15% to 30% error (ours was 30%) • “Equilibrium” method had ≈ 10% to 20% error (ours was 4%) • However, there are still errors beyond Pasco’s estimates • Possible sources of error include: • The mirror is not planar, it is concave. • If mirror moves laterally, laser’s incident angle will change. • If laser is not centered properly on the mirror, incident angle will not change linearly with mirror rotation. • Inaccuracies in measuring the equilibrium positions on the graphs • Definitely accounts for some of the error. • The separation of the large and small balls, b, is taken to be constant • it actually changes throughout the experiment. 14

15. Error Discussion II • Uncertainty in the “b” value given by apparatus manual • Value given for separation between masses in manual is a constant • b changed throughout experiment as arm rotated • The equilibrium points were not at the center between windows • At position 1, the equilibrium was .571 m (~3.9o) from the center position • At position 2, the equilibrium was .469 m (~3.2o) from the center position • Total change in b (window to center) in our experiment was 0.19 cm, or 0.4% of accepted value of b • Not a significant source of error 15

16. Procedure I • Set Up • Level the experiment using the threaded feet, making sure that the mirror is hanging freely in the center of the case , and center the pendulum in the middle of the mesa. • Make sure that when the large masses are moved that the small masses only experience small oscillations. • Move the large masses through the full range of motion and verify that both masses touch the window of the case. Do this carefully not to cause large disturbances from hitting the glass case with the masses. If not make note of which mass doesn’t touch. • Calibration • Using a strong magnet move the balance through the full range of motion and mark on the graph paper where the small masses touch the glass. • To center the natural equilibrium position, we would move one small tick mark. Then we would watch which direction the masses moved towards and would move it another small tick if the movement was away from the center. Initially, one could move 2 small tick marks if one was not near the center already. The angular variation in equilibrium points from switching the mass-positions was approx. 0.01 rad = 0.8º. The total angular variation from maximums was 2.5º. • Measure the distance from the mirror to the midpoint between the marks where the small masses touch the glass. • Taking Data • Move the large masses to where one is touching the glass. This will be the starting position for the measurement. Before taking data you must wait for the small masses to come to rest. The waiting time can be reduced by slowly bringing a strong magnet near one of the small masses, thereby damping the oscillations. • Once the balance has come to equilibrium, carefully turn on the laser pointer. 16

17. Procedure II • Making measurements • At t=0,after the small masses have stabilized in Position I, make a mark to indicate initial position and switch the Masses to Position II • Make marks every 15sec for 2min, then every 30sec for 30min, moving down a row for each precession • Switch to Position I, and repeat the same process • For better results, make marks every 15sec for 45min for each position • Marking extrema may improve results () • It may be helpful to write down the timeevery few minutes, to keep track of data when calculating 17

18. Procedure III • Calculating Equilibrium positions • Two methods: amplitude and frequency • For amplitude, take two separate averages of all marks for positions I and II, making sure there are an equal number of maxima in each direction • For frequency, average the marks closest to ¼ and ¾ the time of each period 18

19. Spring 2010 Equilibrium Data • The measurements resulted in G=7.03 x 10-11- 4% greater than the accepted value • Given that the experimental error is within 10%, these are excellent results • Note that the equilibrium position on the first data set is above the apparent value • This may be due to experimental error • Probably due to the method of calculating the equilibrium position 19

20. Spring 2010 Constant Accel. Data • The measurements resulted in G= 8.68 x 10-11 - 30% greater than the accepted value • Given that the experimental error is around 30%, these results are very good 20

21. Error Discussion I • Neither method had serious problems with error • “Constant acceleration” method had ≈ 15% to 30% error (ours was 30%) • “Equilibrium” method had ≈ 10% to 20% error (ours was 4%) • However, there are still errors beyond Pasco’s estimates • Possible sources of error include: • The mirror is not planar, it is concave. • If mirror moves laterally, laser’s incident angle will change. • If laser is not centered properly on the mirror, incident angle will not change linearly with mirror rotation. • Inaccuracies in measuring the equilibrium positions on the graphs • Definitely accounts for some of the error. • The separation of the large and small balls, b, is taken to be constant • it actually changes throughout the experiment. 21

22. Error Discussion II • Uncertainty in the “b” value given by apparatus manual • Value given for separation between masses in manual is a constant • b changed throughout experiment as arm rotated • The equilibrium points were not at the center between windows • At position 1, the equilibrium was .571 m (~3.9o) from the center position • At position 2, the equilibrium was .469 m (~3.2o) from the center position • Total change in b (window to center) in our experiment was 0.19 cm, or 0.4% of accepted value of b • Not a significant source of error 22

23. Sources http://en.wikipedia.org/wiki/Cavendish_experiment http://www.nhn.ou.edu/~johnson/Education/Juniorlab/Cavendish/Pasco8215.pdf http://physics.nist.gov/cuu/Constants/codata.pdf http://www.physik.uni-wuerzburg.de/~rkritzer/grav.pdf http://www.npl.washington.edu/eotwash/publications/pdf/prl85-2869.pdf 23