Measurement of Gravitational Constant G using a Cavendish
Balance
H. Potter
(Completed 9 April 2006)
A computerized Cavendish balance was used to calculate G, the gravitational
constant. It was determined to be 4.35 x 10-11, precise to within about .01 x 10-11,
but only accurate to within an order of magnitude of the accepted value1 of
6.672041 x 10-11.
I. Introduction
Isaac Newton first postulated the existence of gravity, a universal attractive force
between all massive bodies, in the 18th Century. Critical to making quantitative
predictions using his theory, however, was the measurement of the gravitational constant
G in his fundamental equation
F
GmM
.
r2
(1)
One solution to this problem of measuring the gravitational constant
experimentally is to use a Cavendish balance. A Cavendish balance is a device in which
two less massive balls are placed on opposite ends of a lever such that the lever balances
when upheld by a thin wire. Two more massive balls are placed on a rotating lever that
has an axis of rotation that passes through the wire upholding the balancing smaller
masses. These heavier balls are then brought close to the smaller balls in an alternating
manner so as to induce oscillatory motion. Since this oscillatory motion is driven by the
force of gravity between the balls, G can then be calculated by measuring various
quantities related to these oscillations. A computerized Cavendish balance was used in
this experiment to execute this approach.
II. Experiment
The computerized Cavendish balance used in this experiment relayed the beam
angle to a computer as a voltage. This voltage, however, was not zeroed at the steady
state position of the beam. Thus only changes in angle could be calculated because only
changes in voltage could be measured. In order to convert changes in voltage to changes
in beam angle, a conversion factor was calculated. This was done by inserting a thin rod
into a small groove in the end of the beam as it was swinging. This restricted the angle
through which the beam could pass. The size of the groove into which the rod was
inserted was known, and thus the angle through which the beam was rotating was known
from the dimensions of the beam. The maximum and minimum voltages that were
observed over this period allowed the specific change in the beam angle to be associated
with a specific change in the voltage reading. By performing the same calculation on the
other end of the beam in order to take into account any asymmetry as much as possible,
and then taking the mean ratio of change in angle to change in voltage, a conversion
factor was determined that could be used to convert a change in voltage into a change in
beam angle.
Before the collection of the primary data could begin, the angular displacement of
the beam had to be allowed to decay to as stable a state as possible from a rather large
initial displacement. This preliminary decay data was recorded in order to provide
additional data for calculating the decay parameter of the apparatus.
Once the beam was relatively stable, the beam with the large masses on it was
rotated in one direction so that the large masses were brought close to the small masses.
This provided a driving force, and the beam began to rotate towards the larger masses.
As this motion began to reach a turning point, the beam with large masses was rotated in
the other direction. This provided a driving force in the other direction. Successive
applications of this process led to a progressively greater angular displacement. After
sufficiently many turning points of this driven oscillatory motion were observed
(approximately 10), the beam with the large masses was brought to a perpendicular
position. This allowed the motion of the balance to decay. The entire process was
recorded on the computer for later analysis. The process was then repeated in order to
acquire a second data set.
III. Results and Data Analysis
A conversion factor was calculated for each end of the beam by dividing the mean
change in angle by the mean change in voltage. The best total value for the conversion
factor was then taken to be the mean of these two values. All data used in calculating this
conversion factor is provided in Table 1.
Small Gap:
ΔVS (V)
1.877
ΔθS (rad)
0.0214
Highs (V)
2.682
2.688
2.685
2.697
2.752
2.752
2.743
2.714
Means:
Large Gap:
ΔVL (V)
Highs (V)
2.797
4.034
ΔθL (rad)
4.043
0.032
Means:
4.039
Conversion Factor:
Lows (V)
0.838
0.823
0.853
0.826
0.844
0.841
0.835
0.837
Lows (V)
1.253
1.244
1.229
1.242
1.14E-02
Table 1: Data used in determining the conversion factor.
2
Measuring the decay of the beam angle can be reduced to finding a single
constant X, which is taken to be the average of two calculated quantities x and x’. These
quantities x and x’ are defined in terms of changes in beam angle at an odd number N of
consecutive turning points during free decay as follows.
x  1
1   N 
1   2    3   4      N 4   N 3    N 2   N 1 
(2)
x  1 
 2   N 1 
 2   3    4   5      N 5   N 4    N 3   N 2 
(3)
These definitions were used to calculate a value for X for each of the three decay data
sets. These three values for X were then averaged in order to determine a best value for
X. The specific data is given below in Table 2, 3, 4, and 5.
Preliminary Decay Data:
Vn (V)
n
ΔV (V)
1
-2.920
-4.384
2
1.464
3
-2.187
-3.120
4
0.933
5
-1.717
-2.271
6
0.554
7
-1.387
-1.651
8
0.264
9
-1.137
-1.184
10
0.047
11
-0.984
-0.879
12
-0.105
13
-0.859
-0.644
14
-0.215
15
-0.762
NA
-0.292
First and Last:
-2.158
Δθ (rad)
-0.0501
ΔV' (V)
3.651
-0.0356
2.650
-0.0259
1.941
-0.0189
1.401
-0.0135
1.031
-0.0100
0.754
-0.0074
-0.0246
1.679
Δθ' (rad) Time (min)
15.40
0.0417
17.30
19.20
0.0303
21.06
22.91
0.0222
24.76
26.60
0.0160
28.46
30.31
0.0118
32.10
33.98
0.0086
35.90
37.70
39.55
41.30
43.26
0.0192
Table 2: Data for preliminary decay.
3
First Run Decay Data:
Vn (V)
n
1
-0.795
2
-0.301
3
-0.737
4
-0.356
5
-0.679
6
-0.401
7
-0.639
8
-0.435
9
-0.615
10
-0.459
11
-0.588
12
-0.484
13
-0.588
NA
-0.499
NA
-0.575
First and Last:
ΔV (V)
Δθ (rad)
-0.494
-0.0056
ΔV' (V)
0.436
-0.381
-0.0044
-0.278
-0.0032
-0.204
-0.0023
-0.156
-0.0018
-0.104
-0.0012
-0.207
-0.0024
0.323
0.238
0.180
0.129
0.183
Δθ' (rad) Time (min)
23.10
0.0050
24.80
26.65
0.0037
28.55
30.58
0.0027
32.30
34.15
0.0021
35.90
37.85
0.0015
39.65
41.55
43.20
45.10
47.20
49.00
0.0021
Table 3: Data for decay in first run.
Second Run Decay Data:
Vn (V)
n
ΔV (V)
Δθ (rad)
1
-0.179
0.650
0.0074
2
-0.829
3
-0.298
0.461
0.0053
4
-0.759
5
-0.371
0.339
0.0039
6
-0.710
7
-0.423
0.238
0.0027
8
-0.661
9
-0.462
0.171
0.0020
10
-0.633
11
-0.481
NA
-0.612
NA
-0.496
NA
-0.609
NA
-0.517
First and Last:
0.302
0.0034
ΔV' (V)
-0.531
-0.388
-0.287
-0.199
-0.196
Δθ' (rad) Time (min)
22.88
-0.0061
24.80
26.55
-0.0044
28.58
30.15
-0.0033
32.30
33.90
-0.0023
35.93
37.75
39.50
41.43
43.35
45.10
47.08
48.60
-0.0022
Table 4: Data for decay in second run.
4
Decay Data:
Preliminary:
x
x'
0.847
0.853
X
0.850
First Set:
x
x'
0.872
0.860
X
0.866
Second Set:
x
x'
0.838
0.860
X
0.849
Best X
0.855
Table 5: Calculated x values for each data set, as well as the overall mean value.
In order to calculate G the driving data for the first and second data sets must be
analyzed. This is done by calculating a quantity denoted  D for each data set. It is
defined in terms of changes in beam angle at an odd number N of consecutive turning
points as the beam is being driven as follows, where x is understood to be the value for
Best X, as calculated above.
D 
  2   3      N 1   N   x1   2      N  2   N 1 
N  11  x 
(4)
This definition was used in combination with the previously calculated value for Best X
in order to find  D for each run independently. The magnitudes of these values were
then averaged in order to determine a value for Best  D . The data are shown in Tables 6
and 7.
First Run Driving Data:
Vn (V)
n
ΔV (V)
Δθ (rad) Time (min)
N
1
-0.395
0.299
0.0034
1.65
11
2
-0.694
-0.360
-0.0041
3.60
3
-0.334
0.425
0.0049
5.45 Best X
4
-0.759
-0.480
-0.0055
7.58
0.855
5
-0.279
0.504
0.0058
9.53
6
-0.783
-0.553
-0.0063
11.60 θD (rad)
7
-0.230
0.611
0.0070
13.45 3.53E-04
8
-0.841
-0.632
-0.0072
15.45
9
-0.209
0.666
0.0076
17.30
10
-0.875
-0.690
-0.0079
19.20
11
-0.185
21.10
Table 6: Driving data for the first data set and the resulting value for  D .
5
Second Run Driving Data:
Vn (V)
n
ΔV (V)
Δθ (rad) Time (min)
N
1
-0.728
-0.382
-0.0044
5.45
9
2
-0.346
0.413
0.0047
7.50
3
-0.759
-0.471
-0.0054
9.35 Best X
4
-0.288
0.532
0.0061
11.28
0.855
5
-0.820
-0.571
-0.0065
13.40
6
-0.249
0.601
0.0069
15.23 θD (rad)
7
-0.850
-0.632
-0.0072
17.25 -3.54E-04
8
-0.218
0.672
0.0077
19.05
9
-0.890
21.00
Table 7: Driving data for the second data set and the resulting value for  D .
This is the bulk of the experimental data needed in order to calculate the
gravitational constant G. A few additional measurements, however, must be taken of the
Cavendish balance used in the experiment. Using a caliper and a tri-beam balance all of
the necessary data can easily be taken. When used in combination with a few additional
parameters that are typically provided with the instrument, and can be calculated if they
are not, a value for G can be calculated.
The measurements that must be made are as follow. The length L of the tungsten
string supporting the balancing beam; the width W of the glass enclosure that surrounds
the balancing beam; the masses M1, M2, m1, and m2 of the large and small masses being
used; the diameters D1, D2, d1, and d2 of the large and small spherical masses being used;
and the gaps G1 and G2 between the glass enclosure and the large mass that is not
touching the glass enclosure in each of the two rotational positions.
Those measurements that do not typically have to be made and those quantities
that can be calculated from other measured quantities are as follow. The mass mb, width
wb, and length lb of the balancing beam; the missing mass mh due to the holes in the beam
that allow the small masses to rest without rolling; the distance d of the small masses
from the axis of rotation; the diameter Dfiber of the tungsten fiber; the wire constant μ; the
torsion constant K; the distance R from the center of one of the larger masses to the
center of one of the smaller masses when they are in close proximity; a correction factor
fb that takes into account the gravitational attraction of the large masses for the beam; and
a correction factor fd that takes into account the attraction of each large sphere for the
distant small sphere. These last four quantities are defined by the following equations.
fb 
1  1  d l   d  1  d l   d  

,

2
2l d   1  l   d 2
1  l   d  

(5)
where
d
,
R
l
l  b ,
2R
d 
(6)
(7)
and
6
R
W D1  D2 G1  G2
.


2
4
2
fd 
K
R
R3
2
 2d 
D 4fiber
32 L

2 3/ 2
.
,
(8)
(9)
(10)
In the data analysis for this experiment, mb, lb, wb, μ, Dfiber, mh, and d were
provided in the lab manual that was provided with the Cavendish balance. All other
quantities were either measured or calculated.
With all of these quantities defined, it is now possible to calculate G from them,
using either of the following equations. The first equation doesn’t take into account any
of the correction factors, whereas the second one takes these correction factors into
account. Both are calculated for comparative purposes, but the corrected G is taken to be
the better experimental value for G.
K D R 2
.
G
2Mmd
Gcorr. 
K D R 2
.
2Md m  mh 1  f d   mb f b 
(11)
(12)
The data resulting from all of these calculations are organized below in Tables 8 and 9.
7
All in SI Units
Given:
mh
3.4E-04
d
mb
6.656E-02
7.174E-03
μ 1.57E+11
wb
1.273E-02
lb
0.145
0.0589
D2
0.0543
0.0010
Dfiber
Measured:
L
2.5E-05
W
M1
0.0399
G1
0.93295
G2
0.0000
M2
0.9354
m1
0.01445
D1
0.0544
Calculated:
R 4.74E-02
M 0.934175
m 0.014475
fd
0.0377
m2
0.01450
fb
K
d'
l'
0.194
1.02E-07
1.404
1.530
Table 8: All quantities relevant to the calculation of G in SI units, categorized by how
they were found.
Actual Value1 of G:
First Set:
Second Set:
Best:
θD
3.53E-04
3.54E-04
3.53E-04
6.672041E-11
G
4.50E-11
4.52E-11
4.51E-11
Gcorr.
4.35E-11
4.36E-11
4.35E-11
Table 9: A summary of the final results regarding the calculated value for G.
IV. Conclusion
The experimental value for G was 4.35 x 10-11 Nm2/kg2, in comparison with the
accepted value of 6.672041 x 10-11 Nm2/kg2. A precise determination of the expected
uncertainty in the value of x and  D could be carried out using rather complicated
formulas provided with the lab manual; however, these values would then have to be
further propagated, along with further uncertainties from all of the other variables
involved, in order to determine the uncertainty in G. A much easier and simpler way to
estimate the precision of the calculation of G is to look at the difference between the
values of G for the two data sets. This analysis yields an uncertainty of .01 x 10-11
Nm2/kg2. The discrepancy between the experimental value for G and the accepted value
for G, however, is much greater than this expected uncertainty. This suggests that the
experimental measurements taken were very precise, but not very accurate, and thus that
systematic errors caused the experimental value to be significantly different from the
accepted value, but that random errors were minimal due to the reasonably large data sets
and the various precautions taken to minimize such errors. Essentially the procedure was
very carefully implemented, but certain aspects of the procedure tended to cause the
experimental value for G to be somewhat smaller than it should have been. Given the
preponderance of variables, this could have been caused by any number of variables
8
being somewhat different than they should have been; however, only a few
measurements were taken, and a few likely sources of systematic error can be postulated.
If the large masses were not brought to rest directly against the glass enclosure,
the actual distance R would have been greater than that used in the calculations. This
would lead to a smaller calculated value for G, as can be deduced from Equation 1.
Friction would also lead to the beam turning less due to the gravitational attraction of the
large balls than it should have theoretically, thus leading to a smaller experimental value
for G. The calculation of K may have also been somewhat inaccurate, which would
directly affect the experimental value of G. In particular, if K was somewhat higher, then
G would also be somewhat higher. Also, the method used for determining the conversion
factor may not have been as accurate as it was assumed to be, and if the alternate method
of determining the conversion factor was used it may have affected the value calculated
for the conversion factor. This change would then have propagated through the rest of
the calculations, thereby affecting the value calculated for G. Any of these sources of
error would have equally affected the calculations done on both data sets, and thus have
been a source of systematic error, accounting for some of the inaccuracy of the
experimental value for G.
1
CRC Handbook of Chemistry and Physics: A Ready-Reference Book of Chemical and
Physical Data, edited by Robert C. Weast (CRC Press, Cleveland, OH, 1975), p.F-234.
9