Kater's original pendulum, showing
use, from Kater's 1818 paper
A
Kater's pendulum is a reversible freeswinging
pendulum invented by British physicist and
army captain
Henry Kater in
1817 for use as a
gravimeter
instrument to measure the local
acceleration of gravity. Its
advantage is that, unlike previous pendulum gravimetry methods, the
pendulum's
centre of gravity and
center of oscillation don't
have to be determined, allowing greater accuracy. For about a
century, until the 1930s, Kater's pendulum and its various
refinements remained the standard method for measuring the strength
of the Earth's gravity during
geodetic
surveys. It is now used only for demonstrating pendulum
principles.
Description
The pendulum consists of a rigid metal bar with two pivot points,
one near each end of the bar, by which it can be suspended and
swung, and either an adjustable weight that can be moved up and
down the bar, or one adjustable pivot, to adjust the periods of
swing. In use, it is swung from one pivot, and the
period timed, and then turned upside down and
swung from the other pivot, and the period timed. The weight (or
pivot) is adjusted until the two periods are equal. At this point
the period is equal to the period of an 'ideal' simple pendulum of
length equal to the distance between the pivots. From the period
and the measured distance between the pivots, the acceleration of
gravity can be calculated with great precision from the periodicity
equation for a simple pendulum, (1) below.
Gravity measurement with pendulums
A pendulum can be used to measure the
acceleration of gravity g
because its
period of swing
T
depends only on
g and its length
L:
- T = 2 \pi \sqrt { \frac{L}{g} } \qquad \qquad \qquad (1)\,
So by measuring the length
L and period
T of a
pendulum,
g can be calculated.
The first person to
discover that a pendulum was affected by gravity was Jean Richer, who in 1671 took a pendulum clock to Cayenne, British Guiana and discovered that the length
of a pendulum with a swing of one second there was 1 1/4 Paris
lines, or 2.6 mm, shorter than at Paris and realized this was
because the acceleration of gravity at Cayenne was less than at
Paris. Since that time pendulums began to be used as
precision
gravimeters, taken on voyages
to different parts of the world to measure the local gravitational
acceleration. The accumulation of geographical gravity data
eventually resulted in accurate models of the overall shape of the
Earth.
At that time, the local strength of gravity was usually expressed
not by the value of the acceleration
g which is now used,
but by the length at that location of the
seconds pendulum, a pendulum with a
period of two seconds, so each swing takes one second. It can be
seen from equation (1) that the length of the seconds pendulum is
simply proportional to
g:
- g = \pi^2 L_{seconds} \,
The problem
The period
T of pendulums could be measured very precisely
by timing them with precision clocks set by the passage of stars
overhead. Prior to Kater's discovery, the accuracy of
g
measurements was limited by the difficulty of measuring the other
factor
L, the length of the pendulum, accurately.
L in equation (1) above was the length of an ideal
mathematical 'simple pendulum' consisting of a point mass swinging
on the end of a massless cord. However the 'length' of a real
pendulum, a swinging rigid body, known in mechanics as a
compound pendulum, is more difficult to
define. In 1673 Dutch scientist
Christiaan Huygens in his mathematical
analysis of pendulums,
Horologium Oscillatorium, showed
that a real pendulum had the same period as a simple pendulum with
a length equal to the distance between the pivot point and a point
called the
center of
oscillation, which is located under the pendulum's
center of gravity and depends on
the mass distribution along the length of the pendulum. The problem
was there was no way to find the location of the center of
oscillation in a real pendulum accurately. It could theoretically
be calculated from the shape of the pendulum if the metal parts had
uniform density, but the metallurgical quality and mathematical
abilities of the time didn't allow the calculation to be made
accurately.
To get around this problem, most early gravity researchers, such as
Jean Picard (1669),
Charles Marie de la Condamine
(1735), and
Jean-Charles de
Borda (1792) approximated a simple pendulum by using a metal
sphere suspended by a light wire. If the wire had negligible mass,
the center of oscillation was close to the center of gravity of the
sphere. But even finding the center of gravity of the sphere
accurately was difficult. In addition, this type of pendulum
inherently wasn't very accurate. The sphere and wire didn't swing
back and forth as a rigid unit, because the sphere acquired a
slight
angular momentum during each
swing. Also the wire stretched elastically during the pendulum's
swing, changing
L slightly during the cycle.
Kater's solution
However, in
Horologium Oscillatorium, Huygens had also
proved that the pivot point and the center of oscillation were
interchangeable. That is, if any pendulum is suspended upside down
from its center of oscillation, it has the same period of swing,
and the new center of oscillation is the old pivot point. The
distance between these two conjugate points was equal to the length
of a simple pendulum with the same period.
As part of a committee appointed by the
Royal Society to reform British measures,
Kater had been contracted by the House of Commons to determine
accurately the length of the seconds pendulum in London. He
realized Huygens principle could be used to find the center of
oscillation, and so the length
L, of a rigid (compound)
pendulum. If a pendulum were hung upside down from a second pivot
point which could be adjusted up and down on the pendulum's rod,
and the second pivot were adjusted until the pendulum had the same
period as it did right side up, the second pivot would be at the
center of oscillation, and the distance between the two pivot
points would be
L.
Kater wasn't the first to have this idea.
French mathematician Gaspard de Prony first proposed a reversible pendulum in 1800, but his work was not published till 1889. In 1811 Friedrich Bohnenberger again discovered it, but Kater independently invented it and was first to put it in practice.
Drawing of Kater's pendulum
(a) opposing knife edge pivots
from which pendulum is suspended
(b) fine adjustment weight moved
by adjusting screw
(c) coarse adjustment
weight
(d) bob
(e) pointers for reading
The pendulum
Kater built a pendulum consisting of a brass rod about 2 meters
long, 1 1/2 inches wide and 1/8 of an inch thick, with a weight
(d) on one end. For a low
friction pivot he used a pair of short triangular 'knife' blades
attached to the rod. In use the pendulum was hung from a bracket on
the wall, supported by the edges of the knife blades resting on
flat agate plates. The pendulum had two of these knife blade pivots
(a), facing one another, about a
meter (39.4 inches) apart, so that a swing of the pendulum took
approximately one second when hung from each pivot.
Kater found that making one of the pivots adjustable caused
inaccuracies, making it hard to keep the axis of both pivots
precisely parallel. Instead he permanently attached the knife
blades to the rod, and adjusted the periods of the pendulum by a
small moveable weight
(b,c) on
the pendulum shaft. Since gravity only varies by a maximum of 0.5%
over the Earth, and in most locations much less than that, the
weight only had to be adjusted slightly. Moving the weight toward
one of the pivots decreased the period when hung from that pivot,
and increased the period when hung from the other pivot. This also
had the advantage that the precision measurement of the separation
between the pivots only had to be made once.
Experimental procedure
To use, the pendulum was hung in front of a precision pendulum
clock to time the period. It was swung first from one pivot, and
the oscillations timed, then turned upside down and swung from the
other pivot, and the oscillations timed again. The small weight
(c) was adjusted with the
adjusting screw, and the process repeated until the pendulum had
the same period when swung from each pivot. By putting the measured
period
T, and the measured distance between the pivot
blades
L, into the period equation (1),
g could
be calculated very accurately.
Kater performed 12 trials. He measured the distance between the
pivot blades with a microscope comparator, to an accuracy of
10
^{-4} in. (2.5 μm). As with other pendulum gravity
measurements, he had to apply small corrections to the result for a
number of variable factors:
- the finite width of the pendulum's swing, which affected the
period
- temperature, which caused the length of the rod to vary
- atmospheric pressure, which reduced the effective mass of the
pendulum by the buoyancy of the displaced air, increasing the
period
- altitude, which reduced the gravitational force with distance
from the center of the Earth.
He gave his result as the length of the
seconds pendulum. After corrections, he
found that the mean length of the solar seconds pendulum at London,
at sea level, at 62° F, swinging in vacuum, was 39.1386 inches.
This is equivalent to a gravitational acceleration of
g =
9.81158 m/s
^{2}. The largest variation of his results from
the mean was 0.00028 in. This represented a precision of gravity
measurement of 7(10
^{-6}) (7
milligals).
In 1824, the British Parliament made Kater's measurement of the
seconds pendulum the official standard of length for defining the
yard.
Use
The large increase in gravity measurement accuracy made possible by
Kater's pendulum established
gravimetry
as a regular part of
geodesy. To be useful,
it was necessary to find the exact location (latitude and
longitude) of the 'station' where a gravity measurement was taken,
so pendulum measurements became part of
surveying. Kater's pendulums were taken on the
great historic
geodetic surveys
of much of the world that were being done during the 19th century.
In particular, Kater's pendulums were used in the
Great Trigonometric Survey of
India.
Reversible pendulums remained the standard method used for absolute
gravity measurements until they were superseded by free-fall
gravimeters in the 1950s.
Repsold-Bessel pendulum
Repsold pendulum.
Repeatedly timing each period of a Kater pendulum, and adjusting
the weights until they were equal, was time consuming and
error-prone.
Friedrich Bessel
showed in 1826 that this was unnecessary. As long as the periods
measured from each pivot, T
_{1} and T
_{2}, are
close in value, the period
T of the equivalent simple
pendulum can be calculated from them:
- T^2 = \frac{T_1^2 + T_2^2}{2} + \frac{T_1^2 - T_2^2}{2} \left (
\frac {h_1 + h_2}{h_1-h_2} \right ) \, \qquad \qquad \qquad
(2)
Here h_1\, and h_2\, are the distances of the two pivots from the
pendulum's center of gravity, so h_1 + h_2 = L\,, the separation of
the pivots, which can be measured with great accuracy. h_1 - h_2\,
cannot be measured with comparable accuracy. It is found by
balancing the pendulum on a knife edge to find its center of
gravity, measuring the distances of each of the pivots from the
center of gravity, and subtracting. However, because T_1^2 -
T_2^2\, is so much smaller than T_1^2 + T_2^2\,, the second term on
the right in the above equation is small compared to the first, so
h_1 - h_2\, doesn't have to be determined with high accuracy, and
the balancing procedure described above is sufficient to give
accurate results.
Therefore the pendulum doesn't have to be adjustable at all, it can
simply be a rod with two pivots. As long as each pivot is close to
the
center of oscillation of
the other, so the two periods are close, the period
T of
the equivalent simple pendulum can be calculated with equation (2),
and the gravity can be calculated from
T and
L
with (1).
In addition, Bessel showed that if the pendulum was made with a
symmetrical shape, but internally weighted on one end, the error
caused by effects of air resistance would cancel out. Also, another
error caused by the finite diameter of the pivot knife edges could
be made to cancel out by interchanging the knife edges.
Bessel didn't construct such a pendulum, but in 1864 Adolf Repsold,
under contract to the Swiss Geodetic Commission, made a symmetric
pendulum with interchangeable pivot blades 56 cm long, with a
period of about 3/4 second. The Repsold pendulum was used
extensively by the Swiss and Russian Geodetic agencies, and in the
Survey of India. Other
widely used pendulums of this design were made by
Charles Peirce and C.
Defforges.
References
- Lenzen & Multauf 1964, p.315
- Poynting & Thompson 1907, p.12
- Poynting & Thompson 1907, p.15
External links