On the use of a Torsion Pendulum to measure
Newton’s Gravitational Constant
Jia Kang Chai
Shane Duane
ID: 08764522
SF Theoretical Physics
Fergal Haran
13th December, 2009
Abstract
The aim of this experiment was to measure Newton’s gravitational
constant, G using a torsion pendulum. A value of G = 5.5 ± 1.3 ×
10−11 N · m2 /kg2 was obtained using one method, which agrees inside experimental error with the accepted value of G = 6.67 ± 1.3 ×
10−11 N · m2 /kg2 given by Kleppner & Kolenkow [1]. Another method
was tried and data were taken, yet the author did not understand the
theory sufficiently to complete the second method of the experiment.
Data taken in method one of the experiment also had to be discounted
due to poor plotting, which meant the error was larger than it should
be.
Introduction & Theory
Method (i)
If we have an arrangement like that in Fig. 1, where the small spheres
are in equilibrium and have ceased oscillating, then we may analyze the
situation as follows. The large spheres of mass m2 exert an attractive force
Fg on the smaller ones of mass m1 by Newton’s law of gravity
m1 m2
(1)
b2
where b is the distance between the centres of mass of the small and large
sphere, and G is Newton’s constant. This force creates a total torque τg on
the small spheres by
Fg = G
τg = 2Fg d = 2G
m1 m2 d
b2
(2)
where d is the distance to each small mass from the centre of the rod joining
them.
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The torsion wire holding up the small spheres acts to restore the small
spheres against the gravitational attraction of the large spheres with a torque
τr , which we assume to be linearly proportional to the angle α of deflection;
τr = kα,
(3)
where k is the constant of torsion of the wire. The equation for the period
T of oscillations is that of a rigid pendulum,
4π 2
.
(4)
k
Here I is the moment of inertia of two masses on the ends of a light rod
about the centre of the rod, given by
T2 = I
I = m2 d2 + m2 d2 = 2m2 d2 .
(5)
Now, setting the two torques equal at equilibrium gives
b2
kα
2m1 m2 d
b2
2m2 d2 · 4π 2
=
α
2m1 m2 d
T2
2π 2 b2 d
=
· 2α
m1 T 2
G=
(6)
(7)
(8)
Now, since α is small, we may linearlise it and use the fact that
α=
S
2L
to obtain
2π 2 b2 Sd
,
(9)
m1 T 2 L
which is our final equation. The factor of 2 is a mistake, yet the flaw in the
derivation of (9) could not be found.
G=
Method (ii) In Graph 2 S is plotted against t2 given by
1
S = a∗0 t2 ,
2
where σ = 21 a∗0 is the slope of the graph and a∗0 is the linear acceleration.
Now the angular acceleration a0 is related to a∗0 by
a∗0 =
2L
a0 ,
d
hence
a0 =
2
σd
.
L
(10)
It was unclear if the steps taken thus far were correct or not, and the full
derivation could not be worked out. A basic understanding of the theory as
laid out in the guide was beyond the author despite many hours spent at
the task.
Experimental Method
Method (i)
The apparatus was wet up as in Fig. 1, with the rod connecting the two
small spheres suspended by the torsion wire. A mirror was situated at the
centre of the rod so that an incident laser beam would reflect and make clear
the angle α on a screen. A computer and detector were used to record the
displacement of the laser beam about the centre of the screen, this is plotted
in graphs 1 and 2.
Firstly, the spheres were left to gain equilibrium at position I and then
the large spheres were quickly rotated, causing the small ones to experience
a torque and undergo damped oscillations. Here S vs t is plotted in Graph 1.
This was repeated for the apparatus in the other position, namely position
II. The part of Graph 1 corresponding to position II is somehow shifted
down well below the centre-line of the graph, thus making it impossible to
glean data from it. No satisfactory explanation for this could be found, and
the data from this section were discounted.
Method (ii) For this section of the experiment the apparatus was left
to return to equilibrium and then balls were moved to the other position.
Recording of S vs t2 was then performed, the slope σ representing the linear
acceleration of the reflected beam.
Results & Analysis
Values for the various parameters in the experiment are given in Table 1
below.
Method (i) The graph for the full deflection method has S plotted versus t. The first three peaks and troughs were taken to give values of
S = 0.019 ± 0.003 m and T = 627 ± 30 s. By using (9) a value of
G = 5.5 ± 1.3 × 10−11 N · m2 /kg2 was obtained.
Method (ii) Graph 2 of S vs t2 was plotted for the acceleration method
and a slope of 1.18 ± 0.05 × 10−3 m/s2 was found.
Discussion & Conclusions
The value for G found by the full deflection method was in good agreement
with the accepted value in [1], yet the poor plotting of position II led to
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a waste of experimental data, which should have been avoided. Also, the
theory was not well understood, as the experimental guide was unclear and
inconsistent in its definition of the relevant variables.
References
1. An Introduction to Mechanics, Kleppner & Kolenkow, McGraw-Hill
2. Cavendish Experiment, wikipedia.org
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1
Appendix: Tables & Figures
Figure 1: Experimental Setup
Quantity
b
d
L
g
m1
Value
0.047 m
0.050 m
0.665 m
9.802 m/s2
1.500 kg
Error
0.004 m
0.001 m
0.002 m
0.001 m/s2
0.005 kg
Table 1: Experimental Data
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