We measured the value of the gravitational constant G at 8.3969e-11 ± 10.418e-10 m​3⋅​ kg-​ 1⋅​ s-​ 2​, using a computerized Cavendish balance

1.
Departments of Astronomy and Physics
https://www.astro.umass.edu/
https://www.physics.umass.edu/
April 30, 2019
Scientific Editor
Astrophysical Journal
Dear Professor Hertel,
Please find enclosed a copy of our manuscript, “Measuring Big G Using a Computerized Cavendish
Balance”, for your consideration for publication in the Astrophysical Journal. This manuscript is a
combined effort of David Weidmann, Adam Redfern, and myself, in which we attempt to calculate
the universal gravitational constant G.
In this manuscript, we investigate the value for the gravitational constant G through use of a
computerized Cavendish balance. We familiarize ourselves with the background physics, the setup of
the apparatus, the software used, and the equations necessary to do the calculations and error
propagation.
We believe our work is an important investigation into the understanding of the gravitational
constant, which is a key aspect of much of astrophysics, and as such, is a relevant work to be included
in your journal.
Thank you for your consideration.
Sincerely,
Madeline Boyce
University of Massachusetts Amherst
Department of Astronomy and Physics

2.
Measuring The Gravitational Constant G Using a Computerized
Cavendish Balance
Madeline Boyce, David Weidmann, and Adam Redfern
Under guidance of Professor Scott Hertel?
28 April 2019
Abstract
We measured the value of the gravitational constant G at 8.3969e-11 ± 10.418e-10 m​3​
⋅kg​-1​
⋅s​-2​
, using a
computerized Cavendish balance. Although this value is within one standard deviation from the best
known calculation of ​6.674484e−11 ​m​3​
⋅kg​-1​
⋅s​-2
(with relative standard uncertainty of 11.61 parts per
million​[1]​
), our standard deviation is so large as to make this measurement unreliable. Our setup was
comprised of a Computerized Cavendish Balance set atop a stabilizing table. Our datataking was
mostly focused on tracking the angular oscillations of an aluminium bar hanging by a 25 micrometer
tungsten thread (fig. 1). We used these oscillations to estimate the equilibrium angles of the bar at
different distances from two large lead balls. We then used this information, together with
characteristics of the setup such as mass of the lead balls, length of the bar, etc, to calculate G.
I. INTRODUCTION
Our experiment was first performed in 1798 by
Henry Cavendish, our apparatus’ namesake​[2]​
.
Although not the inventor of the apparatus, he
was the first to use it to calculate the universal
gravitational constant “Big G”. Our version of
the apparatus differs from his in multiple ways.
His was on the order of 2 meters tall, ours can
fit on a tabletop. Our wire is on the order of
micrometers while that would have been
unavailable to him at the time. Crucially, we
take measurements using a laser motion
tracker, while Cavendish was forced to use a
telescope and his own eyes to track the
oscillations​[3]​
. “Big G”, the proportionality
constant used in the calculation of the
gravitational force between two bodies, is a
very small measurement. The force of gravity
is the weakest of the known forces, which is
why this particular experiment is difficult to
perform. G is an important value for many
calculations, such as orbital patterns of
astronomical bodies, the theory of relativity,
and many more. Therefore, obtaining an
accurate value of G with small error is
important.
II. APPARATUS
This experiment used a TEL-RP2111
Computerized Cavendish Balance​[4]​
. This
balance consists of a metal frame with
removable glass front and back panels. Inside
the frame, a thin tungsten wire (on the scale of
micrometers) supports an aluminum boom at
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3. the center pivot point. Two small masses, each
with a mass that we measured of
14.93g±0.005g, sit on either end of the boom.
This frame sits on a larger external boom,
which can also pivot at the center point. Two
large masses, each with a measured mass of
1.5kg±0.05kg, sit on either end of this external
boom, with the boom positioned so that one
mass is on either side of the frame. This whole
setup sits on top of a stabilizing table, and is
connected to a computer with the Cavendish
software.
We tracked the oscillations of an aluminium
bar, on which the two small lead balls acted as
the weights of the dumbbell, in two different
orientations relative to the a slightly larger bar,
on which the two large lead balls acted as the
weights of the larger dumbbell. At different
distances from the large balls, the small balls
(and thus the aluminum boom to which they
were connected) felt different gravitational
forces. As a result, when set into motion, the
two orientations each had a unique
equilibrium angle.
The internal beam is suspended between the
capacitor plates of an SDC transducer​[4]​
. The
output of the transducer is proportional to the
angular movement of the internal beam​[4]​
.
The computer reads in the transducer output
and then reads out on the screen the angle of
the beam, with a resolution on the order of 25
microradians​[4]​
. This is how the angle of the
internal beam was measured.
III. METHODS
The balance was first calibrated by aligning the
beam all the way counterclockwise and setting
that position as the left border in the software
and then turning the beam all the way
clockwise and setting the position as the right
border. The angle at these extreme positions
was set to +/- 70 mrad, a value found
experimentally through repeated calibrations
in conjunction with an optical lever​[4]​
. The
internal beam was then positioned to hang
freely in the center of the balance. This step
was both very important and very difficult.
Any adjustment to the position of the beam
caused the beam to oscillate, and it was
necessary to wait for the movement to steady
before we could tell if the beam was positioned
correctly.
The large masses were placed on the outside
boom and once the movement caused by that
action settled, data was recorded. After an
hour, the boom was turned to the opposite
position, and data was recorded for another
hour, for a total of two hours worth of data.
IV. ANALYSIS
A damped sine curve was fit to the data for
each boom position. The y​0 value of each fit
was taken to be the equilibrium angle of each
position, as it was the mean of the positive and
negative amplitudes. Table 1 shows
measurements taken of the apparatus, while
table 2 shows values that were given in the
Cavendish balance manual.
2

4.
Measured values for the apparatus:
mass of big balls, M 1.5 ± 0.05 kg
mass of small balls,
m
0.01493 ± 5e-5 kg
distance from
rotation axis to the
center of small ball, d
0.06736 ± 10e-5 m
radius of small ball, r 0.00684 ± 10e-5 m
mass of inside boom,
m​b
0.010375 ± 10e-5 m
length of inside
boom, l​b
0.14928 ± 10e-5 m
width of inside
boom, w​b
0.01258 ± 10e-5 m
Table 1
Given values of the apparatus:
distance between the center of
small and large balls, R
0.046m
mass correction for hole in
boom, m​h
0.00034 kg
correction for attraction of big
ball on distant small ball, f​d
0.035
correction for gravitational
torque exerted on boom, f​b
0.19
Table 2
The method used in this experiment involves
many corrections to the original equation for
G. The original equation for G is found by
equating the gravitational torque to the
restorative torque of the tungsten wire. The
gravitational torque is:
(1)τG = R2
2GMmd
The restorative torque of the wire is:
(2)θτR = K D
Equating these two torques and solving for G
gives us:
(3)G = 2Mmd
KΘ RD
2
K is the torsion constant, found from the
equation for the oscillation frequency:
(4)IK = 4π2
T +b2 2
In this apparatus, the minor axis, b​2
, is small
enough compared to the major axis to be
ignored. The total moment of inertia I is the
sum of the moment of inertia for the small
balls I​s and the moment of inertia for the inside
boom I​b​:
(5)2(md mr ) , IIs = 2
+ 5
2 2
b = 12
m (l +w )b
2
b
2
b
The corrections are long and complicated. For
further detail on them, please refer to the
Cavendish balance manual. The final
corrected equation for G is:
(6)G =
KΘ RD
2
2M[(m−m )(1−f )+m f ]dh d b b
V. RESULTS
Plots of the angle of the internal beam at the
two equilibrium angle are shown in Figure 2.
Table 3, below, shows the angles found from
the plots.
Position Angle (degrees)
Θ​1 1.7485 ± 0.00294
Θ​2 2.4621 ± 0.00187
= Θ​D2
Θ − Θ2 1
0.3568 ± 0.00348
0.006227 ± 6.073e-5 (rad)
Table 3
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5.
Using equation (4), we find that our value for
G is:
G=8.3969e-10 ± 10.418e-10 m​3​
⋅kg​-1​
⋅s​-2
VI. SYSTEMATIC UNCERTAINTIES
The Cavendish balance apparatus is extremely
sensitive to outside vibrations. Data was taken
late on a Saturday night so as to reduce
interference from many students walking
through the building. It was also noticed upon
breaking down the apparatus that the tungsten
wire that suspended the internal boom had a
slight twist in it, which caused the boom to sit
a little lopsided. This likely caused much of the
uncertainty and the difference from the
expected value.
The fact that the error was larger than our
measured value was unexpected. The error on
each value that went into G was relatively
small. However, as more and more
measurements with error are used, the error
propagates through larger than the individual
components. The error calculation was done
twice, once by hand, and once through
MatLab, to be sure the calculation was correct.
VII. CONCLUSIONS
Our final value for G was found to be:
G=8.3969e-10 ± 10.418e-10 m​3​
⋅kg​-1​
⋅s​-2
This value is off from the best known value by
an order of magnitude, and the error is
relatively large. This is most likely due to the
uncertainties discussed in the previous section.
For future experiments, it would be beneficial
to correct the positioning of the Tungsten wire
to keep internal boom arm parallel with the
two SDC transducer plates. We could also
refine our data analysis methods to more
accurately encapsulate necessary information.
We would like to have spent more time
understanding the Cavendish Torsion Balance
apparatus and the accompanying software to
improve collection of reliable data, as well as
efficiently budget necessary time for multiple
nights of data collection.
VIII. ACKNOWLEDGMENTs
Thank you to Professor Scott Hertel and
Buqin Wang for all of their help in this
experiment. Thank you also to the physics
department of the University of Massachusetts
Amherst.
IX. REFERENCES
[1] ​
Li, Qing, et al. “Measurements of the Gravitational
Constant Using Two Independent Methods.” ​Nature​, vol.
560, no. 7720, 29 Aug. 2018, pp. 582–588.,
doi:10.1038/s41586-018-0431-5.
[2]​
“Cavendish Experiment.” ​Cavendish Experiment​,
sciencedemonstrations.fas.harvard.edu/presentations/Cavendi
sh-experiment.
[3]​
“Cavendish Experiment.” ​Wikipedia​, Wikimedia
Foundation, 7 Mar. 2019,
en.wikipedia.org/wiki/Cavendish_experiment.
[4] ​
“Cavendish Balance.” (n.d.). Retrieved from
https://www.telatomic.com/all-produts/cavendish-balance
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6.
X. FIGURES
fig 1:​​(left) a labeled picture of the Cavendish balance. (right) a top down diagram of the internal boom and masses.
fig 2:​​plots of the angle of the internal boom as a function of time at the two external boom positions
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