UMEÅ UNIVERSITY
Department of Physics
2004-03-23
Agnieszka Iwasiewicz
DETERMINATION OF
THE GRAVITATIONAL CONSTANT
WITH
THE CAVENDISH TORSIONAL BALANCE
Aims of the lab
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•
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To measure G. The constant G that appears in Newton's universal law of
gravitation is one of the most important quantities in cosmological theories
(which predict the future of the universe as a whole), yet it is one of the least
accurately known of the fundamental constants. In this experiment you can
measure G using the same technique as was used by Cavendish in 1798 - oddly
enough this basic technique has not been improved much in the intervening two
centuries.
Work sharing and cooperation. It is not easy to perform the necessary
measurements working alone. You will therefore have a chance to exercise the
organization of group work.
Experimental know-how. You will be asked to analize the experimental data and
present your results, therefore you can make use of your knowledge obtained in
the experimental methods course.
Let computers do the calculation for you. The use of computers for data analysis
is a very important skill for modern experimental physicists, and not only them.
The lab gives you an opportunity to learn how to use the program Origin for
nonlinear curve fitting, finding parameters and calculating uncertainties.
1
Introduction
Gravitational interaction arises between any two massive bodies in the universe.
The magnitude of the attractive gravitational force is proportional to the masses of the
bodies (m1 and m2) and inversely proportional to their relative distance (r) squared:
mm
F = G 12 2 .
r
The gravitational constant G is one of the fundamental constants. It is not easy, however,
to measure it with a high accuracy.
The first experiment leading to the determination of the gravitational constant was
performed by Henry Cavendish1 in 1798. The value obtained by him with the use of a
torsional balance was 6.754(41)⋅10-11[m3/(s2⋅kg)]. The experiment had a great scientific
significance, also because the measurement of G allows to estimate the mass of the Earth.
The Cavendish experimental value’s accuracy was not improved for almost a century.
Many people after Cavendish tried to measure the gravitational constant, also by other
methods, but the torsional balace principle still proves to be a good idea.
The most recent result, obtained by Armstrong and Fitzgerald (published in November
2003), is G = (6.67387±0.00027)⋅10-11[m3/(s2⋅kg)]. It is in well agreement with the other
recent results, and therefore you can safely compare your experimental result – soon to be
obtained – with G = 6.674⋅⋅10-11[m3/(s2⋅kg)].
The reference list given below is the right place to start if you want to know a bit
more about the recent measurements of G:
1. T. R. Armstrong and M.P. Fitzgerald, Phys. Rev. Lett. 91, 201101 (2003)
2. S. Schlamminger, E. Holzschuh and W. Kündig, Phys. Rev. Lett. 89, 161102
(2002)
3. T. J. Quinn, C. C. Speake, S. J. Richman, R. S. Davis and A. Picard, Phys. Rev.
Lett. 87, 111101 (2001)
4. J. Moore, Physics Worlds, August 2001, p. 5
5. T. Quinn, Nature 408, 9190 (2000)
6. P. J. Mohr and B. N. Taylor, Rev. Mod. Phys. 72, 351 (2000)
7. J. H. Gundlach and S. M. Merkowitz, Phys. Rev. Lett. 85, 2869 (2000)
8. G. G. Luther and W. R. Towler, Phys. Rev. Lett. 48, 121 (1982)
1
Henry Cavendish (1731-1810) had very broad scientific interests. He was the first to develop the methods
of gas manipulation, established the accurate composition of the atmosphere, studied the chemical
composition of water and recognized hydrogen as a separate substance. The mean density of the Earth,
calculated by Cavendish to be 5.448 larger than the density of water, now is estimated for about 5.5. His
experiments with electricity were confirmed many years later by Maxwell.
2
Experimental setup
The Cavendish torsional balance2 consists of a rectangular aluminum pendulum
case with a glass window through which one can see the torsional pendulum (see figure
1). A 25 cm long tube on the top side of the pendulum case contains a torsional band on
the bottom of which the pendulum’s horizontal arm is hanged. Near that connection a
concave mirror is mounted. It is a part of the optical system for monitoring the
pendulum’s position during its motion. The other two elements of this system are a lamp
and a ruler on which one can measure the position of the light spot reflected by the mirror
on the pendulum.
The horizontal arm of the pendulum carries two lead balls, weighting 15g each.
The torsional pendulum can turn (oscillate) in the horizontal plane around the
equilibrium position. Elastic properties and dimensions of a torsional band determine
the size of the elastic forces, and therefore the length of the pendulum’s period of
oscillation. The band is made of bronze and has the cross section of 0,01 - 0,16 mm2.
There are also two big lead balls which can be placed on the holders next to the
pendulums case. The big and small balls interact by the gravitational forces, equal in
magnitude but of opposite directions. These forces also influence the motion of the
torsional pendulum.
Figure 1.
Cavendish
torsional
balance.
The torsional pendulum is released by the two magnets, placed on the back side of the
pendulum case.
2
The terms printed in this font are explained/translated to Swedish in “Appendix 1 – word list”, at the
end of these instructions.
3
Figure 2 shows a view of the experimental setup (a torsional pendulum and a ruler) from
above in the two cases, without and with the big masses placed next to the pendulum
case.
• Point A corresponds to the equilibrium position of the torsional pendulum if
there is no gravitational interaction (left part of the figure).
• The equilibrium position is shifted if the two heavy balls interact by gravitational
forces with the pendulum (right part of the figure). In this case, the light spot on
the ruler will be oscillating around the point B, which corresponds to the new
equilibrium position of the pendulum.
Figure 2.
A view of the experimental setup from above. On top of the picture for each case there is
a sketch of a light spot position on the ruler versus the time of observation.
Notice: the two equilibrium positions of the torsion pendulum differ by the angle θ0 (in
the horizontal plane), but due to the reflection law the distance from B to A is related to
the angle 2θ
θ0. The distances S, L, a and the mass of the big balls M can be measured
directly. Our experimental setup does not allow the direct measurement of either the
distance d or the mass m of the pendulum’s small balls (given in lab instructions), nor of
the distance s (which you have to determine on the basis of what can be measured).
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Calculation of the pendulum’s moment of inertia I
The moment of inertia of a rigid body is calculated as a sum of all the mass
"pieces" multiplied by their distance from the rotational axis squared.
The moment of inertia I of the torsional pendulum is a sum of moments of
inertia of all its components. The moment of inertia of the two small lead balls is the
main contribution, since the other components of the pendulum are much lighter and
located closer to the rotational axis (and therefore contribute much less to I).
Approximation 1.
We will calculate the total moment of inertia of a pendulum as the sum of
moments of inertia of only the two lead balls. What do we neglect and why?
The moment of inertia of a solid sphere with the radius r and mass m is given by:
2
I k = mr 2 .
(1)
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Steiner's theorem allows us to use the above result to calculate the moment of inertia of
a ball, which is placed at a distance d from the rotational axis3:
2
I k ' = I k + md 2 = mr 2 + md 2 .
(2)
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If we neglect the less significant terms (after making approximation 1), the total moment
of inertia of the pendulum is just the sum of Ik' for the two small lead balls:
4
I = 2 I k ' = mr 2 + 2md 2 .
(3)
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Approximation 2.
One can make an approximation to eq. (3) in order to keep the following
calculations simpler. Try to compare the size of two terms given on the right
hand side of eq. (3): the term which is much smaller can be neglected.
I ≈ ...................
This approximation is not obligatory. If you decide to make the
approximation, you have to keep in mind that it introduces a small error to
the results obtained later.
3
In our case d is the distance from the ball's mass center to the middle of the pendulum's arm. See figure 2.
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Free motion of the pendulum
Let us now consider the case when the torsional pendulum is oscillating freely.
Its movement is then described by Newton's second law of dynamics for rotation:
d 2θ
Mθ = I 2 ,
(4)
dt
where Mθ is the magnitude of the torque (force moment) exerted by the bronze band on
the pendulum when it is twisted by the angle θ from its equilibrium position.
Approximation 3.
If the pendulum oscillates with small angles of deflection from equilibrium we
can assume that the motion of the pendulum is harmonic.
For small angles the magnitude of torque is linearly dependent on the value of the angle:
M θ = −κθ
(5)
The proportionality constant κ is called a torsional constant and it depends on the elastic
properties of the bronze band. Eq.(5) is actually a rotational analogue of Hooke's law,
which describes for example the force exerted on a mass connected to a spring.
Comparison of eqs. (4) and (5) leads to the differential equation
d 2θ κ
+ θ = 0.
(6)
dt 2 I
A complete solution to the differential equation (6) will not be presented here. For our lab
it is useful to note that the solution implies the pendulum's oscillation period:
I
T = 2π
.
(7)
κ
Approximation 4.
In reality the motion of our torsional pendulum is damped. The damping term
is not included in eq. (6). The period of the damped oscillations is in reality
slightly longer than the one given by eq. (7). However, in this experiment the
damping correction is not necessary, therefore we will assume that eq. (7) is
applicable.
The equilibrium position of the pendulum is the position where all the force
moments acting on the pendulum add up to zero. The pendulum oscillates around its
equilibrium position.
The light from the lamp is directed onto the mirror in such a way that the reflected light
spot appears on the ruler. Therefore, in our experiment the equilibrium position
corresponds to a certain light spot’s position (which we will determine experimentally
and call A, as shown in the left part of figure 2).
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Gravitational interaction
On placing the big balls next to the pendulum, a new interaction arises. Additional
attractive gravitational forces change the force balance in the system. The force
moments acting on the pendulum therefore add up to zero in a different place, and the
equilibrium position of the pendulum is shifted by an angle θ0 with respect to the free
case. It means that the small balls in equilibrium are moved closer to the big balls by a
distance s (see figure 2, part on the right). The pendulum will now oscillate around the
new equilibrium, and the light spot will be moving in a different region on the ruler.
Again, we will determine the new equilibrium position and label the corresponding light
spot’s position by B.
Finding the formula for G
In order to calculate the gravitational constant G from our experimental data, we need
to derive an appropriate formula. You can base this on the formulas presented above.
Here are some hints about how to do that:
1. Express the value of θ0 in terms of S and L.
Approximation 5.
In case of small angles we can make the trigonometric approximation
sin α ≈ tan α ≈ α . In our case the angles are small enough. Use this in the
above problem.
2. Determine the analytical formula for κ for our pendulum.
3. Use the results from 1. and 2. to calculate the torque acting on the pendulum
twisted by an angle θ (due to the elastic properties of the bronze band).
4. How strong is the gravitational force acting between the two small and the two
big balls?
Approximation 6.
At this point you can take into account only the interaction of the small balls
with their neighbouring big balls and neglect the interaction between the
small ball and the big ball placed on the other side of the pendulum. If you
choose to do so, estimate how large an error you introduce.
5. How big is the total torque (caused by the gravitational force) acting on the
pendulum if we consider all four balls?
6. Compare the results of 3. and 5. and write down the final formula for G. Check
the units.
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Estimation of errors in the final value of G caused by
the approximations
Approx. 1 and 2 – affect the final result by about 0.5%.
Approx. 3 – error is negligible, because the swinging angles are really very small and
the motion is almost harmonic.
Approx. 4 – in comparison with the exact solution with the damping term in the
differential equation we can notice an error of about 1% of the final result.
Approx. 5 – as the angles are small, this approximation is almost exact, with an error
less than 0.1% in the final value of G.
Approx. 6 – estimate the error: ……………………………….
There is a lot of heavy furniture and people in the lab. Why does our presence not
introduce visible changes in the experimental conditions, even though we are much
heavier than the big lead balls?
………………………………………………………………………………………………
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Is there anything else that should be included in the discussion of errors?
………………………………………………………………………………………………
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………………………………………………………………………………………………
………………………………………………………………………………………………
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Experimental task
PREPARATIONS AND MEASUREMENTS:
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Read the following part of the instructions and make sure that you understand the
tasks before starting the experiment.
Prepare a plan of your experimental activities.
Since not all of the values appearing in the formula for G can be measured
directly, without dismounting the experimental setup, some of them are given
below:
Radius of the small ball:
r = 6.0 (1) ⋅10−3 [ m ]
Mass of the small ball:
m = 15.0 (1) ⋅10−3 [ kg ]
Distance from the small ball’s mass center to the middle of the pendulum’s arm:
d = 5.0 (1) ⋅10−2 [ m]
•
You have to measure (or determine from other measurements) the following
values:
LENGTHS:
a = ………………….
L = …………………
S = …………………. (Calculate from the A and B positions obtained)
MASS:
M = ………………….
PERIOD (from the time measurements):
T = ………………….
CALCULATE THE LENGTH s:
s = ………………….
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There are two alternative approaches to the experiment. You can choose which
version you want to try in the lab. In order to determine the points A and B one
can:
o either:
study the free pendulum’s motion first (to determine A), and then the
motion with the two big lead balls placed next to the “våghus” (to
determine B)
o or:
study the movement of the pendulum with the two big balls placed as on
the right side of figure 2 (to determine B) and then change the position of
the heavy balls to the other side – obtaining a mirrored situation with
respect to the right side of figure 2 (to determine B’). Position A is then
taken to be half way between B and B’.
Choose one of the above versions and argue why do you think it is better.
Turn the lamp on and adjust the position of the reflected spot on the ruler.
Depending on which version of experiment you want to do – place or remove the
heavy lead balls onto/from their holders
Start the pendulum oscillation using a magnet.
Ensure that the oscillations are not too small or too big – to get the best results the
pendulum should be swinging as much as possible, but should not collide with the
walls of the pendulum case. Amplitude of 3 – 4 cm is usually optimal.
Record the position of the light spot every 10 – 30 seconds, until the spot has
reached the turning points at least three times. Write your data in a table prepared
before the measurement.
Run the second series of measurement according to your plan.
ANALYSIS AND CALCULATIONS
There are two ways of extracting the period of the pendulum and the equilibriumcorresponding points A and B (or B and B’) from the experimental data. Pick one of the
methods, or use both and compare the results.
Each way, we need to plot the data points for the two measurement series on the separate
plots.
1. “By hand” – on each plot find the position of the three consecutive maxima and
label them n1, n2 and n3. For the harmonic movement it is possible to determine
the approximate equilibrium position n0 from a simple formula:
1n +n

n0 ≈  1 3 + n2  .
2 2

The period is also determined directly, by checking for example the time between
the pendulum reaching the same maximum twice.
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2. One can also use the computer program Origin to find the value of T and the
equilibrium points (B and A or B and B’). We need to find the curve that would
fit the experimental data. We use a model with five fitting parameters Pi:
y = P1 + P2 e− P3 x sin ( P4 x + P5 ) .
2π
.
P4
For more detailed instructions about curve fitting in Origin refer to the instruction
“Appendix 2 – Origin” available in the lab.
From here the equilibrium point is simply the value of P1 and T =
Now you are able to determine the value of G.
Compare the experimental result with the table value and discuss the difference. What
caused the errors?
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Appendix 1 – word list
Here is a small “dictionary” of some words and expressions appearing in the
instructions. Some of the translations are not exact form the linguistic point of view, but
the aim was to keep the physical nomenclature correct.
bronze = brons
Cavendish torsional balance = Cavendish gravitationsvåg
concave mirror = konkav spegel
equilibrium position = jämviktsläge
force balance = kraftbalans
lead balls = blykulor
moment of inertia = tröghetsmoment
optical system = ljusvisaranordningen
pendulum case = våghuset
rigid body = stel kropp
torque (force moment) = torsionsmoment (kraftmoment)
torsional band = torsionsband
torsional pendulum = torsionspendel
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