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.

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

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 time every 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

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

Fall 2012 Constant Accel. Data The measurements from Series 1 resulted in G = 5.23x10-11 m3/kg/s2—22% less than

the accepted value—and the measurements from Series 2 resulted in G = 6.16x10-11 m3/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

Error DiscussionThere 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

Compare With Coulomb ExperimentForces 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.

A Note on Previous Presentations…Previous versions of this PowerPoint contained the following

slide:

This inserts a correction factor with no explanation of its derivation.

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 θTorquegrav-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])

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.

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

Procedure IIMaking 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 ()

10•It may be helpful to write down the time every few minutes,

to keep track of data when calculating

Procedure IIICalculating 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

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

Data for Equilibrium Method

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0 500 1000 1500 2000 2500 3000

Time (s)

Dis

pla

cem

ent

(m)

First Data Set

S1

S2

Second Data Set

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

Group B - Data for Constant Acceleration Method

y = 1.58E-05x + 9.94E-03

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

0.080

0 500 1000 1500 2000 2500 3000 3500 4000

Time (s^2)

Po

soti

on

(m

)

AccelerationMethod

Linear(AccelerationMethod)

Error Discussion INeither 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

Error Discussion IIUncertainty 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 (~3.9

o) 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

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

Procedure IIMaking 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 ()

17•It may be helpful to write down the time every few minutes,

to keep track of data when calculating

Procedure IIICalculating 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

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

Data for Equilibrium Method

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0 500 1000 1500 2000 2500 3000

Time (s)

Dis

pla

cem

ent

(m)

First Data Set

S1

S2

Second Data Set

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

Group B - Data for Constant Acceleration Method

y = 1.58E-05x + 9.94E-03

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

0.080

0 500 1000 1500 2000 2500 3000 3500 4000

Time (s^2)

Po

soti

on

(m

)

AccelerationMethod

Linear(AccelerationMethod)

Error Discussion INeither 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

Error Discussion IIUncertainty 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 (~3.9

o) 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

Sourceshttp://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