Determination of the Gravitational Constant G
Raghuveer Dodda, Physics 425
Newton’s law of gravitation states that the force of attraction between two point objects
is proportional to the mass of each of the objects and is inversely proportional to the
square of the distance between the objects. The constant of proportionality is designated
as G, and it is measured in this experiment by using an apparatus called a Cavendish
balance.
INTRODUCTION
Newton’s law of gravitation is one of the
basic laws of physics. Though it
describes the interaction between two
point masses, it is extremely powerful in
its application because one can
understand all masses as collections of
point masses. Two most common
examples of application of this law are
to explain the tides, and the motion of
celestial objects. Historically G, the
constant of proportionality, has been
very difficult to measure accurately. It is
known to have the value 6.67300 × 10-11
m3 kg-1 s-2.
smaller mass m is also l. The larger
masses M also form a dumbbell and are
allowed to move in the same plane as the
small ones.
MATERIALS AND METHODS
Figure 1 is a schematic picture of the
Cavendish balance. A Cavendish
Balance consists of a dumbbell
suspended freely by a wire. The mass at
each end of the dumbbell is designated
as m. The dumbbell oscillates in the
horizontal plane perpendicular to the
vertically
suspended
wire.
The
Cavendish balance also provides a
mechanism to place the two large masses
M at desired distance from the small
masses m. However, the two large
masses are constrained to move in
unison – if one of the Masses M is at a
certain distance l away from a smaller
mass m closest to it, the distance
between the other mass M and the other
Figure 1 - Schematic diagram of a Cavendish
balance
If the masses M are not present, the
dumbbell made out of small m oscillates
around an equilibrium position and will
come to rest in this position. When the
pendulum is nearly at rest, we start the
experiment. The experiment involved
the following phases:
1. Set-up and Calibration phase
2. Free decay phase
3. Resonant Drive phase
4. Calculation of K (the torsion
constant of the wire)
5. Calculating G
Set-up and Calibration phase:
The Cavendish balance was set up on a
level surface. The large masses were
taken far away from the balance so that
they could not affect the small masses.
The pendulum was allowed to come to
equilibrium position by not disturbing it
for about an hour before the calibration
was done.
The Cavendish balance is connected to a
data acquisition apparatus that acquires
the angular displacement of the
dumbbell-m about the suspended wire as
a voltage. The purpose of calibration is
to know how the voltage at any given
time is related to the angular
displacement of the dumbbell-m at that
time. It is a linear scaling factor.
Note: The dumbbell-m was actually a
rectangular piece if metal with 2 holes at
each end into which the masses m were
placed. Hereafter, this dumbbell-m will
be referred to as the boom.
Free Decay phase:
The boom was allowed to move about its
equilibrium position. As the motion of
the boom damped out, the angular
displacement of the boom was recorded
as a function of time. This data was used
to find the time period T and the
coefficient of damping b for the boom’s
motion.
Resonant Drive phase
After the boom’s oscillations were
completely damped out, the large masses
M were placed near the boom such that
the gravitational force they exerted on
the small masses created a torque. Figure
2 indicates the positions of the masses in
this phase. The boom will find a new
equilibrium position that makes an angle
D ( in Figure 2) to its equilibrium
position from the free decay phase.
Figure 2 - Positions of the Masses M in the
Resonant Drive phase
As soon as the boom started moving
away from masses M after the initial
movement towards them, the masses M
are moved into the complementary
position shown as the dotted position in
Figure 2. Now, the boom tries to settle
into a new equilibrium position on the
other side i.e., at an angle of -D to its
equilibrium position from the free decay
phase. Moving the masses M ensures
that the boom is always moving towards
the large masses M, so that the torque
applied on the boom always does
positive work increasing the energy of
the boom. This is why this phase is
called the Resonant Drive phase.
The angular displacement of the boom is
recorded as a function of time. This data
can be used to determine the angle D.
Calculating K
K is given by the following formula:
right angles to the boom and in the
plane perpendicular to the torsion
fiber (TEL-RP 2010).
4
(1)
 b 2 )I
T
I is the total moment of inertia of the
boom about the suspension point, and
can be calculated from the physical
properties of the system (i.e., from
distances of separation, and masses).
The following data were used to
calculate the value of G:
Calculating G
Table 1 - The Gravitational Constant, G
K (
2
Since the following equation is obeyed
at the new equilibrium position for the
boom, we can use it to calculate G:
2G 






M md
 K  D
R2
(2)
M is the mass of one large masses
m is the mass of one small mass
d is the lever arm
K is the torsion constant
D is the new equilibrium angle
R is the distance of the center of the
large sphere to the center of the
small sphere.
The following assumptions are implicit
in EQ2 if it is used as sole equation to
calculate G:



The balance is symmetrical about the
axis of rotation: the separation
between the large and small masses
is the same for both arms and for
both equilibrium positions; the two
large masses are identical; the two
small masses are identical.
The mass of the boom without the
masses m is small and can be
ignored.
The position of the large masses in
both extreme positions is such that
the gravitational force of the large
masses on the small spheres is at
RESULTS
K
Value
1.0385
0.014559
0.066653
2.71816E-7
Error
0.001
0.00001
3.71E-05
2.04717E-7
Units
Kg
Kg
Meter
N-m
D
0.00091
3.71E-5
Radian
R
0.0461025
0.000158
Meter
M
m
d
Table 2 - The Torsion Constant, K
<T>
b
I
Value
1.0385
0.014559
0.066653
Error
0.001
0.00001
3.71E-05
Units
Seconds
Second-1
Kg-m2
Table 3 – The Time Period T, and
Damping Coefficient b
Time in Seconds
( 5)
77
164
260.5
380.5
500.5
617
757
867
993.5
1113.5
1237
The Voltage at the turning points in
Volts ( 0.1)
3.364
-0.161
2.222
-0.142
1.831
0.142
1.579
0.332
1.343
0.474
1.211
Table 4 -The angular displacement, D
We can use the fact, that the time taken
to travel from one maxima to the
adjacent minima is T/2, to calculate T.
The first column of Table 3 can be used
to calculate the <T>, the average value
of T.
Time in
The Voltage at the turning
Seconds ( 5)
points in Volts ( 0.1)
554
0.981
677
0.801
805.5
1.001
922.5
0.786
10415
1.021
1170
0.781
1290.5
1.03
1407.5
0.757
1524
1.05
1641.5
0.752
1764.4
1.055
If we define X  e-bT/2, it will be the
factor that when multiplied to one
extremum will yield the next one (the
correct sign must be taken into account).
We can use this property of X to
calculate X using the following formula
where n is the n-th exteremum (in volts)
with the first extremum at time zero
(data from Table 3 is used):
The value of G was calculated as:
2.60842E-10  1.96453E-10 m3 kg-1 s-2
This value is about one order of
magnitude off.
DISCUSSION
The following is a graph of the motion
of the boom under free decay:
X  1
( 1   N )
( 1   2   3   4  ...   N 1 )
(4)
Using that property of X, we can also
calculate D from the Resonance Drive
phase as follows (data from Table 4 is
used):
D 
(1  X )( 1   2   3   4  ...   N )
(5)
( N  1)(1  X )
The following is a graph for the motion
of the boom under Resonance Drive:
Figure 3 - The boom under free decay
The graph obeys the equation where the
variables have usual meanings (e is the
equilibrium angle in free decay):
 (t )   e  Ae bt cos( wt   )
(3)
Figure 4 - The boom under Resonant Drive
Once we know X, we also know b. So,
we have all the parameters to calculate
G.
However, the value of G is off by one
order of magnitude (The Mathematica
code and Excel file used to calculate G
are available). This is probably due to
poor calibration because all the other
parameters appear correct and are of the
expected magnitude.
The calibration constant was calculated
to from the geometry of the slot as
follows (using the small angle
approximation):
  tan( ) 
.156  .062
 0.32 radians.
2.938
From the following graph the change in
voltage, when the shift was  = 0.32
radians, is just the voltage difference
between two sharp peaks, which is V =
0.7 volts. So, 0.7 volts correspond to
0.32 radians
Figure 5 - Calibration plot that relates 0.7 V
to 0.32 radians
LITERATURE CITED
TEL-RP2010. Tel-Atomic manual for
Computerized
Cavendish
Balance.
Jackson, Michigan: TEL-Atomic, 1994.