Some ``Background''
Virtually everyone who has taken a physics course knows that a pendulum
can be used to measure the Earth's gravitational field; i.e.,
the acceleration, g » 9.8 m/s2. It is not widely known, however,
that a naive pendulum measurement will not show variations in
g from one place on the Earth to another. For example, the altitude
must increase from sea-level to 3000 m » 10,000 ft for g to decrease
by one part in a thousand. Similarly, the extreme sea-level global variation
with latitude and/or longitude is only about five parts per thousand.
Because gravity differences over the earth's surface are very small,
geophysicists have used the milligal (10-5 m/s2) to state
differences. In these units, going from the equator,
where the mean value of g = 978,049 to the poles, g increases
by 5,172 [1]. The
reference for these gravimeter (relative) based values
is the absolute g experiment that was performed with six Kater pendula
in Potsdam Germany
(latitude 52° N® 981,274) in the early part of this
century.
The pendulum has long been a favorite instrument for measuring the
acceleration of gravity; however, it is not a trivial matter to measure g
to better than about one percent with a simple pendulum-one in which a
concentrated mass swings at the end of a flexible cord. The biggest
challenge in this case is the accurate determination of the distance
from the support to the ``center'' of
the pendulum [2]. To obtain accuracies in the vicinity of
1 part in 104 has typically required a long pendulum. This has
disadvantages because,
as the length of a pendulum increases, it
is increasingly susceptible to noises, both from surrounding air and
also from the support which is never completely inertial. An
appreciation for the physics involved in the multiplicity of correction
terms required of such an experiment can be found in the paper by
Nelson and Olsson[3]. Fortunately, the corrections mentioned
in that paper, such as buoyancy and added mass of the air are not
required for present purposes.
The Kater Pendulum
Many of the difficulties of the simple pendulum are removed by working
with the pendulum first considered by Captain Henry Kater in 1815.
Being
``reversible'', the Kater pendulum oscillates about either of two axes.
In conventional
form (such as the unit sold by Cenco), there is either an adjustable
knife edge or a moving mass
which is positioned, ideally, so that the period
of the pendulum is the same about the two pivots.
When the periods are matched so that T1 = T2 = T, it
is easy to show [4] that T = 2pÖ{[(l)/ g]}, where l
is the spacing between the two axes. Thus the system is equivalent to
a simple pendulum of length l. Since l
can be measured with relative ease to a few parts in 104, the Kater
pendulum is thus attractive as a means to accurately measure g.
For those not familiar with the Kater pendulum, a pedagogical example using a meterstick is provided in the Appendix. The theoretical estimated period obtained there (T = 1.6 s) can be easily compared against a crude experiment. Simply hold the meterstick between the finger nails of thumb and middle finger and measure with a stopwatch. The large damping demonstrates that this ``system'' is not practical for a precise measurement of g. The same proves true when one tries to provide axes of rotation by using sharpened pins that barely press into the stick. Additionally, small circular rods inserted through the stick to serve as axes also were found to be unsatisfactory.
Present Instrument
The conventional Kater pendulum is difficult to use, partly because
the range of period adjustability is usually quite large. To design the
user-friendly pendulum shown in Figure 1, the computer was necessary.
The overall length of L = 37.40 cm was chosen so that the period would be
nominally 1 s, for rotation about each of axis 1 (an end), and
axis 2, a point » 0.67×37.4 cm away from axis 1. For a uniform rectangular physical pendulum of
length L, having small width and thickness compared to L, it is
shown in the appendix that T = 2pÖ{[2 L/ 3 g]}. Of course,
the practical
challenge is one of providing suitable axes. In the present case, this was
accomplished by means of two 1/4 in diameter holes whose centers were
drilled at the indicated positions, relative to the midpoint of the
pendulum. Because the introduction of these holes results in a more
complicated geometry, as compared to the idealized case just described;
their placement was determined via computer, using the parallel
axis theorem and recognizing
the hole as having negative mass.
The knife edge
The knife edge was produced from a small piece of carbon steel, of the
type used for cutting tools in a lathe. The starting stock was of square
cross section, 0.25 in × 0.25 in, about 1 in long. It was ground
on one end to an ``edge'' with the interior angle between planes
roughly 30°. To hold the pendulum, the unground end of the
piece was clamped between the jaws of vice-grip pliers, which was in turn
clamped to a conventional laboratory stand. For the brass unit
described above, the log-decrement of the motion was small enough that
the motion of the pendulum could still be visually perceived 30 min after
initiation of motion. It should be noted that a rigid support of the
knife edge is called for, since a flimsy one will both lengthen the
period and increase the log-decrement.
Period Adjustment
Typically, there are two ways that period adjustment is accomplished in
a Kater pendulum-either (i) moving the position of one of the pivots, or
(ii) changing the moment of inertia by altering the mass configuration.
For the present pendulum, the latter method has been used.
To change the period around the nominal 1 s value,
a pair of small binder clips were used, each of mass 1.2 g, which is
small compared to the pendulum mass of 129.3 g.
These clips are
the ordinary black spring metal type used to bind documents that are
too thick
for a paper clip. A pair was chosen, rather than a single clip,
for positioning on opposite sides of the pendulum. Thus the center of
mass of the system remains on the line of symmetry of the pendulum.
Operationally, the clips are moved in increments of 1 cm starting near one
end of the pendulum, and advancing toward the other end, measuring the period
about each of the two axes for a given position of the clips.
Timing
Period measurements were made using a standard photogate with
the Precision Timer (Vernier software) system sold by Pasco, for their
CI-6510 Signal Interface. The beam of the photogate was positioned
vertically at the
midpoint of whichever hole was not being used as an axis. The horizontal
position was selected such that the beam was interrupted by the
small segment of brass between one side of the
pendulum and the near edge of the hole. For this placement of the photogate,
the period measured will be other than » 1 s if the amplitude
is unacceptably large or small.
Thus this arrangement proved useful in determining the onset of
non-isochronism (amplitude large enough to increase the period through
nonlinearity).
A 30 min comparison of the timer system against WWV showed that the times
displayed by the timer were uniformly slightly long, so all data were
corrected for this systematic error by dividing
by 1.00038.
Results
Data collected with the pendulum are shown in Figures 2 and 3.
Fitted to the data, which were plotted with Excel, are quadratic fits
for which the R2 values are close to unity.
For Figure 3, a linear least squares fit was used, and the matched
condition is one for which T = 0.9999 s.
The abscissa in both graphs corresponds to the clamping placement of the
binder clips, with zero being near the end of the pendulum which is opposite
axis 1. Position
36 (unused) corresponds to the clips being centered on axis 1 (top edge
of the top
hole).
Estimating g
Using the matched period value of 0.9999 s and the value of
l = 0.2481 cm, the acceleration of gravity
in Macon, Georgia was estimated to be
g = 4p2l/T2 = 9.797 m/s2.
Predicting the Uncertainty in g
Assuming random errors in the measurement of l and T, the
relative uncertainty in the acceleration of gravity is given by
dg/g = [(dl/l)2 + 2(dT/T)2]1/2.
For our case,
it was assumed that the machinest [5] produced the pendulum
(on a milling machine) with
all dimensions to the nearest 0.001 in. Thus the relative uncertainty in
l is dl/l = 1×10-4. The random uncertainty
in the period measurement (independent of the systematic error mentioned
earlier, that was corrected) was determined by taking 100 measurements
at a few points and calculating the standard deviation,
which yielded dT/T = 2×10-4. With these numbers,
the uncertainty
in g is determined to be 0.003 m/s2.
Leeways
A low-mass wooden (Oak) pendulum was built and evaluated, expecting
that it would be a less precise instrument.
It was fabricated ``crudely'' by cutting to the requisite
rectangular shape on a
table saw. The width
turned out to be 1.25 cm, the thickness
0.42 cm, and the length 37.4 cm. The holes for the knife edge support
were drilled with an ordinary 1/4 in bit.
As compared to the clip mass total
of 2.4 g, the mass of this pendulum was quite small at 13.2 g (an
order of magnitude less than the 129 g brass pendulum). As expected, the
timing errors proved larger, at approximately 3 ms-
about an order of
magnitude greater than those of the brass pendulum at 0.2 ms.
Additionally,
the length was not measured as precisely-the uncertainty being estimated
at 0.2 mm. The variation of period about the two axes proved similar
in trend to that of the brass pendulum, except the range of variation was
much greater, as expected. About axis 1, as the clips were moved
from 0 to 34 cm, the
period varied through a total range of 87 ms, compared to
9 ms for the brass; likewise, about axis 2,
the total range was 0.42 s,
compared to 36 ms for the brass.
With this wooden pendulum, the periods were found to match (from a
linear least squares fit) for the clips at 11.4 cm, yielding T = 1.003 s,
and an estimate for the acceleration of gravity, g = (9.77 +/- 0.05) m/s2.
It was thus demonstrated that a crude, low mass Kater pendulum can do as
well as a decent simple pendulum; although it is not recommended that
such a pendulum be built.
Based on these results, it is expected that a reasonably good pendulum might be made of soft aluminum, cut with shears. It is recommended, however, that a metal of higher density, such as brass, be used; and at the very least cut with a bandsaw and then filed to shape. Preferably, these operations should be performed on a milling machine. Not only will this yield an instrument whose dimensions are closer to the nominal values indicated in Figure 1, but a separate measurement of the distance between the holes is then also unnecessary, assuming that the translating stages of the machine are properly calibrated.
We now calculate the periods of the pair, starting with
the 1st axis.
The moment of inertia of the uniform stick, with respect to its
center,
is Ic = ML2/12 (assuming that both the width and the thickness are
much smaller than the length of the stick). For rotation about the 1st
axis at L/2 from
the center;
the parallel-axis theorem yields I1 = M(L/2)2 + Ic = ML2/3.
Thus, the period about the first axis is given by
T1 = 2p Ö{1/3ML2/(1/2MgL)} = 2p Ö{[2L/ 3g]} = 2p Ö{[2/ 3g]} » 2pÖ{[2/ 30]} = 1.6 s.
Now let the meterstick be turned upside down from the 1st position and
allowed to swing about an arbitrary 2nd axis at distance, x, from the
center toward `B'. Here the moment of inertia is given by
I2 = M x2 + Ic, and the period is
T2 = 2p Ö{[(x2 + [1/ 12])/ xg]}.
Setting T2 = T1 yields
the quadratic equation
x2-2/3x+[1/ 12] = 0, which has roots x = 1/2 (non-interesting) and
x = 1/6, which is the Kater pendulum case. In particular, note that
the distance between the two axes is l = (1/2+1/6)L = 2L/3. Thus, the meterstick Kater pendulum is described by
T = T1 = T2 = 2p Ö{[(l)/ g]} » 2p Ö{[(2/3)/ 10]} » 1.6 s.