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CHAPTER X
DETERMINATION OF " g " BY MEANS OF THE FREE
STINGING PENDULUM
C. H. SWICK
U. S. Coast and Geodetic Hi very
INT110DUCTION
The at.tractio~.~. oil the earth upon a unit mass on its surface varies from
place to place. This -['act was accidentally discovered in 1671 by the
French astronomer, Jean Richer, who found that his astronomical clocl<,
which had been care:t'ully regulated in Paris, lost 2~ minutes a day on
the Island of' Cayenne, South America, and required a shortening, of its
pendulum o-l' more than 2 millimeters to malice it keep correct time at
the latter place. Although this variation caused a great deal of comment
and speculation amon`, the scientists of that time, the first hint of the
correct explanation was given in Newton's Principia, published sixteen
years later. In that great work it was stated that the decrease in gravity
between Paris and Cayenne Night be due to the centrifugal force of tile
earth's rotation and to a slight bulging, of the earth at the equator.
At the time o:l' Richer's discovery the pendulum clock was a compara-
ti.vely new device. It had been inventecl only about fifteen years earlier
by Christians Huy~,ens, tl~e great Dutch astronomer and mathematician.
1-luygens in his astronomical observations had been confronted with the
need for more accurate timepieces than were then available He Recense
interested in the possibility of Usi.ll~, a pendulum as. a clock re~,ula.tor and
-finally invented the pendulr~ elm. It is quite remarkable that in nearly
three centuries since T-l:u~'s i.r,.vention no more accurate device for
regulating ~ timepiece leas ever been discoverecl. than the pendulum. It is
equally remarkable that during the two centuries since the beginning of
gravity determinations the pendulum has been the most accurate device
known for measuring the intensity of gravity.
While worl~in`~ on his invention Huy~ens. made an exhaustive mathe-
matical study of the pendulum and discovered many of the underlying,
principles which made possible its. use later :t'or gravity measurements.
He discovered, for example, that the period of a free swinging pendulum
varies as the square of its. length within certain limitations. However,.
:J'-or a compound pendulum (a simple pendulum is possible in theory only)
he found that. the length must be tolled as the distance between the center
off:' suspension and the ce~:~.ter oft' oscillation.
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1 ,32
FIGURE OF THE EARTH
The center of oscillation he found to be that point in a compound
pendulum where a, particle moves of its own accord at the rate forced
upon the other parts. of the pendulum. He found, in other words, that
the particle at the center of oscillation moves at exactly its own free rate
and so tends neither to retard nor accelerate the rest of' the pendulum.
Those parts of the pendulum which are near the center of suspension tend
to move faster than those farther away and so force the latter to move
more rapidly than they otherwise would, and vice versa. The distance
of this particle at the center of oscillation from the center of' suspension
is exactly equal to the length of a,theoret;.cal simple pe~:llllun~ ha,v-in',
tl~e same period of oscillation.
ABSOLUTE DETERMINATIONS OF GRAVITY
The first attempts to measure the intensity of' gravity were all made
with some i'orm of compound pendulum which approximated as closely
I, .,
as possible a theoretical s:~mpie penctu~um. Henry Inter was the first to
depart I'rom this principle. He applied to gravity measurements
Huygen's theorem that the centers of suspension and oscillation of a. con~-
pouncl pendulum are reciprocal if at unequal distances from the center of
`˘,ravitv. He made a pendulum which had two Loire edges and two sliding
weights on the stem. By swinging, this pendulum first in the usual man-
ner with the bob clown and then on the other l~nil:'e Aloe with the boll
up he was able to adjust the weights until the period of the pendulum
oaf orate l~nif'e edge was practically the same as its period on the other l~ni-fe
ecl~,e. The distance between the two l~ni-le eclges, after c.eriain corrections
had been appliecl, was then equivalent to the length of' a. theoretical simple
pendulum having, the since period, end the intensity of' :,ravitv could be
computed groin this le~:~,t,l-~ and the observed period of:' the penclulum.
Improved types of the reversible pendulum were later developed as, -for
example, the Repsolcl pe~dulu~n Lehigh by means oil-' a hollow shell at the
opposite end from the bob was made syn~netrical at the two ends so that
the air resistance woulcl be the singe wl:~eu the penclulu~n was swells on
either knife edge.
Although the reversible pendulum is the most precise means Anon
for determining, the absolute i.ntensit.v of ~,ravitv, it has certain inherent
limitations. that impair its accuracy. For example, the distance between
the l~nil'e ecl~t,es is very Difficult to measure with sufficient exactness. By
ma.ki~l~ this measurement with the }pendulum hanging, on a. l~n;i'e edge
the stretching deice to its own ve;~ht is included in the remeasured length.
To talkie account o-l' the stretching, clue to tile centrifu~,a.1 force of its vibra-
tory motion, however, is much more clif~cult and uncertain. )Tan,v other
small inaccuracies enter into the absolute determination of' gravity with
the reversible pendulum.
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DETER,lII.VATION OF " 9 " B Y 1WEA,YS OF PENDVLU,llI 1.33
REL NTIVE DETERMINATIONS OF GRAVITY
In contrast to absolute determinations it is comparatively easy to mea-
sure the difference ire gravity between. two places with very great preci-
sion. It is only necessary to swing an invariable pendulum under the
same conditions at both places. and to determine its. change of period.
I-i' one of the places at which the pendulum is swung is a base station or
some other station at which the intensity of gravity has been previously
determined, the value of gravity at the new station can be readily com-
putecl by the formula
Pal, A,,
.~8 = p2
in which g is the intensity of gravity and P is the period of the pendulum
the subscripts s and b referring to the new and base stations, respectively.
Since it is difficult, or practically impossible, to make the conditions
under which a pendulum is swung, exactly the same at two different sta-
tions, certain corrections must be applied to the observations in order to
talkie care o-! the differences. What is usually done is. to reduce the obser-
vations at both the base station and the new station to certain standard
conditions., corrections being applied for temperature, pressure, a.~npli-
tude of vibration, and flexure.
All modern precise gravity detern~;nations are nude by the relative
method, and practi.callv all o:L' them are based on the absolute value of
gravity at Potsdam, Germ..~n>. The absolute value at Potsdam is' the
result of a lone, series of' very care.-l'ul n~ea.surements extending over manic
years and is probably as accurate as it is possible to obtain. The preci-
sio~ of' this determination, however, is. not so important as it Piglet
seem. The important thing :is to have a.llthe ~-ra.vity measurements in
the world leased on the same system, tl:~at is, referred to the ogle base sta.-
tics-. '1'1~is -is especially essential. in fi-u-~:e-oi'-the-ea.rtl-~ investiga.tio~-~s
where the variation of' ˘,r..~vitv ~.i.tl~ hate of latitude is tile ~->rincipal
rector to lie considerecl in clerivin~, the shape of' the eartl~.
Nearly every country engaged in measuring the intensity of ~t,ravitv
has its own base station whose value has been cleterm;~ed with more or
less care by the re]a.tive method :t'ro~n Potsclan~. or groin the previously
established base station of tonne other coun.tr\r. All relative gravity ~neas-
~rements in ~ country are usually referred. to its own base station. The
periocls of the pendulums are detern~;ne~l there T~el'ore anal after each
Held season. not only to obtain the l',, for use in the l~receLlin~, I'orn~ula
but to make sure that. the pe~clulu~ns super no cha.~',e o-t le~-~th d-uri~:,
their use ifs the -held.
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Representative terms from entire chapter:
knife edge
~4
FIGURE OF THE EARTH
The first pendulums used for relative determinations were patterned
more or less a.f'ter the pendulums used for absolute determinations. They
were a meter or more in length, heavy and cumbersome to handle, and
were much inferior to the modern apparatus in accuracy.
MODERN GRAVITY APPARATUS
There are two general types of modern gravity apparatus for use on
land. The older of these two types is 1
0\ O~ ~ ~y t~ # 0F ~155
flab ~p~~US
Tntedcrometer
Two chronometers
Thermometers; manometers; etc.
[IG. 1.~0DiulUulS of the Coast and Geodetic Survey.
A ~ the dummy pendulum gab ~ thermometer ~tt~cbed to as ~em. ~ is one
of the principal pendulums us it appears Ruben Ida OD the knife edge. C
small ~uxiL~ry pendulum carrying ~ levy vial Ahab is used in 1CVeliD/tbC knife
edge. D add ~ are the other priLcip~1 pendulums.
Tbc pendulums are of Ample Construction. (See Digure 1.) Each con-
~sts of ~ thin stem With ~ heavy tot at one end and an inverted stTr~l~
at the otbe~. ~ piece of agate carefully ground to sat optic~1 pl~ne.on its
lower side is secured in the stirrup in surb ~ position that file plane. is at
right Andes to the vertic~1 axis of the peninlum. A1I pats of Me pen-
dUln~ ate rigidly ~tt~cbed to egg other t~ means of closely fitted joints
FI GLiRE OF THE EAR TH
aIld rickets to prevent ally rha.r~',e ;~ the length of' the pendulum under
ordinary usable, except the change due to temperature variations.
The placing oft' a Platte instead. of' the usual l~nit'e echo in the head of the
pendu~un~ ma: seem at first thought to be a.n unnecessary reversal of
position of these i.mporta.~t parts of' the apparatus. As originally con-
structed~ each pendulum bad a. L:nit'e edge in the head and swung on. a
plane attacher!. to a shelf' in the receiver. After the apparatus had been
usecl. a short twinge in the Delco it was feared that the knife edges might
eventually wear appreciably and cause a. lengthening of the pendulums
and so the planes and kDi.-i'e eclges were reversed in position. A:lter more
than thirty years encl. considerable use the l~r!,i1'e edges now show no indi.-
c~a,tion of wear anal there is some doubt as to whether this change was
~,ecessarv or clesirable. NVi.th the present arrange~ner~t. it is quite essen-
tial to center the pendulum carefully ol1 the lignite edge, but if the Loire
ecl~,e were ifs the pearl of'' the pendulum the position of the peIlclulum on
the support woulcl slot be so important.
The crummy pendulum its similar to the other pendulums in all parts
but the head which lies a. different form-to prevent the dummy from
talking, up sympathetic vibrations with the swinging pendulum. The
purpose of the dummy pendulum is to carry a thermometer in contact
with its stem and thus malice possible a close approximation to the tem-
perature of the s,wingi.n~ penclulum. The latter, of course, could not be
made to carry a thermometer without causing a. change of virtual length
which would destroy its ac~cura.c~r. As the dummy is made of the saline
material and l~a.s. pract;.cally- the same shape as the swinging pendulum
there is every reason tc believe that it acquires almost exactly the same
temperature as tl.~e latter cluring the observations, especially as the two
are only a i'ew centimeters apart in the receiver and are usually placed ir
position several hours. before observations are begun.
The knife edge on which the pendulums are swung is a most impor-
tant part of the gravity apparatus L;,l~e the plane in the head of the
pendulum, it is made of' agate. It is rigidly attached to a, brass plate
which is secured by a. screw to a, shelf in the receiver. The knife edge is
surround to give as close an approximation as possible to an ideal kni:f'e
edge, a geometrical line. The most nearly perfect knife edge that can
be made will be found when tested to have what has been called the miss-
ing triangle, that is, the two planes forming the edge will not quite meet.
The base of this missing triangle on the knife edges of the Coast and
Geodetic Survey'~,ravity apparatus has been found by optical means to
be less than one micron in length.
As the pendulums swing through a very small arc, the two planes which
Fleet to form the l~n;i'e edge are ground at an angle of 13(~° with each
DETERlIINATION OF ",J'' BY MEANS OF PENDULUM 106
other. This Dies an ecl'~,e strolls a,.~.cl rigid a.~cl not Basil: i~-~jurecl. In
order to prevent any tendency for the pendulum to rock ice a. d;.recti.o~
parallel to the l~n.il'e edge, the ~niclclle third of the edge is cut away.
The flash apparatus consists of' a box oil. a stanfl with an obse.rv;.~,
telescope mounted above it. The box contains an electro-mag~et. which
is connected to a, shutter ire the side of the box towarcl, the per~dulum
receiver. It also contains a small electric light which is di.rectl~ bestial
the shutter. NYhen the electro-magnet is placed in circuit with ~ break-
circuit chronometer it will open the shutter :E'or an instant at each
break of the chronometer and thus pern~.it a beam of li.gr,ht to emerge al;
one-second intervals. This beam of' light is clirected through a, windows
in the :t'ront of the receiver onto; two small mirrors, one attached to the
l~nife-edge plate a,ncl the other to the swinging pendulum. From these
mirrors the beam is reflected in two parts baffle into the observing tele-
scope. The coincidence observations, used in obtaining the period of' the
pendulum in relation to the chronometer, are made by noting the times
when the two reHect.ed parts of the beam are in the same horizontal ]i-~e
in the telescope, as explained in more detail later.
The interferometer (see :Figure 2) needs no particular description. It
is simply a, special form of the Michelson interferometer adapted-to the
purpose of measuring, the flexure or horizontal movement of the kni:t'e
edge caused by the horizontal component of the stresses induced by the
swinging, pendulum. It hats a strong source of light of Knowles wave-len`~th
a.ncl two accurately ground plane-glass reflectors, one on the interferometer
itself' and the other attached by means of a, light aluminum arm to the
l~nife-edge plate in the receiver. When the two reflecting surfaces have
been placed close together and a.djusl;ed to a- slight angle to give inter-
ference fringes, any movement of the knife edge will cause a, siclewise
shifting, of the fringes. By determining the amount of this, shifting in
terms of the width of one fringe it is possible to obtain the corresponding
movement of the knife edge within ~ few thousandths of a micron, and
-T'rom this the correction to be applied to the period of the penclulun~.
GRAVITY ])]3TERMINATIONS
Ire all modern cletermir~atio~ls of' ˘,ra.vity on la.ncl dI1 accuracy of' one
or two parts ill a nonillion is desired and usually attained. As the average
intensity or acceleration of `~,ra.vit~: is about 980 cm/see" this accuracy
corresponds to an error of about 0.001 or 0.002 cm/see" in the total in-
tensity. To attain this accuracy it is necessary to determine the a,vera~e~
Period o:l' the pendulum within less than a millionth of a. second of time.
It is :t'ai.rly cl.itficult to malice any physical measurement, within one part
ice a million lint an interval of' time is one of:' the most difficult of all to
~ ~-
~e as
100
, .
FIC~ 0F r~ ~6 T~
~S~:ES~:E~ESE:E.E:ESE:ESESES~:E:ESE:ESE:ESESESESE:E.E.ESESE.E:E:ESE:E.ESE.E.E.E:ESE
FIG. 2.-[eDdulum receiver bud interferometer, Coast and Geodetic
Survey upp~utus.
Tbe extremely small unmount by ~bicb 1be receiver is swayed by the
osculations of the pendldum is measured limb the interferometer in terms of
Me ~aved~D~ of barium light.
DETERMINATION OF "I" BY MEANS OF PENDULUM 159
measure with such a high degree of' precision. Fortunately, a free swing-
ing pendulum will maintain a constant period of oscillation, except as it
is affected by variations of temperature and pressure and by a. decrease in
its amplitude of oscillation. If corrections are applied for these variations
it is possible to obtain the period of the pendulum by dividing a. fa,irl~,
long interval of time by the number of' its vibrations during that interval
and thus. secure considerable precision.
The gravity pendulum swings in a. partial vacuum, under a pressure
of less than a tenth of ad atmosphere, and is supported on a very accu-
ratelv ground knife edge. It is, therefore, subject to very little friction
of any kind and, if started with a, total arc of' vibration of 1~°, it will
swing for 12 hours and still have a. total arc of nearly 2S', which is large
enough to permit of' accurate ~oinc.idence observations. In 12 hours, the
pendulum oscillates about 86,000 times and this males possible a very
exact determination of its period.
The period of the pendulum is obtained in terms of some timepiece.
Due to its portability, a. chronometer is usually employed for this pur-
pose. There. is at present no timepiece in the world of the portable kind
that will maintain a constant rate within one part in a million, the a.ecu-
rac.y required for gravity world. It is. necessary, therefore, to determine
the rate of the chronometer or other timepiece used in making the
gravity observations and to apply a correction for this rate. A fairly sim-
ple and entirely satisfactory way of obtaining the chronometer rate is to
make chronographic comparisons between the second beats of the chro-
nometer and the time signals, of the U. S. Naval Observatory as sent out
by radio from Annapolis, Arlington, or Bellevue, using an automatic re-
cording device for this'' purpose. It is necessary even then to apply a cor-
rection for the variation in the error of' the sending clock at the Naval
Observatory from day to day, although this variation usually amounts to
only a few hundredths of a. second.
The relative precision required in making the various observations :for
a determination o:t' gravity is well worth considering. An error o-t'
0.000 0~00 So second in the period o:l' the pendulum corresponds, to 0.00]
c~n/sec" in the derived value of the intensity of gravity and may be caused
by any one of the -~:'ollowing errors:
1) A total relative error of 10 seconds in the coincidence readings at
the beginning and end of' a 12-hour swing.
2) An error of' 0.04 second in the observed daily rate of' the chronometer.
3) A variation of 0.()2 second in the time required for the armature
to open the shutter ifs the flash box between the beginning and ending of
1.2-1~our swing.
11
160
FIGURE OF THE EAR TH
4) An error of 0.06 ° C. in the mean temperature of' the pendulum if
it is made of bronze or 0.9 ° C. if it is made of i.nvar.
b) An error of 3 mm. of mercury in the observed mean pressure within
the receiver.
6) Frrors of 10' and b' in the observed arc of ~-ibra.tion at the begin-
ning and end, respectively, of a 12-hour swing.
7) An error of 0.014 wave-length of sodium or helium light in the
observed flexure of the pendulum support.
Item No. 4 in the preceding list of errors shows very conclusively one
advantage of invar over bronze pendulums. Bronze pendulums, must be
swung in a constant temperature room as it is impossible to obtain the
mean temperature of the pendulum with sufficient accuracy over an inter-
va.l of several hours if there is very much fluctuation of the surrounding
temperature. The invar pendulums, on the other hand, can, ordinarily,
be swung in a. tent as it is only necessary to obtain their mean tempera-
tures within a. few tenths of a degree Centigrade.
Although i.nvar metal is rather peculiar in its behavior and seems to
be subject to molecular and other uncertain changes, the pendulums of
this metal, which have been used for all recent gravity determinations of
the Coast and Geodetic Survey, have given no particular trouble in this
respect.. Inva,r pendulums are magnetic, however, and must be tested for
magnetism at each station and kept as nearly demagnetized as possible.
In regard to item ~ above, it may seem to the reader that the specified
accuracy would be impossible to obtain. It is accomplished by varying the
conditions and taking the mean of a large number of independent obser-
vations. The size of' the flexure correction depends to a, large extent, of
course, on the kind of support used for the receiver. In order to avoid
the delay incident to building a pier the receiver is, often mounted directly
on a concrete floor or on stone brooks. or bricks which rest on a, concrete
floor. If the floor is thin the flexure is apt to be large in such cases.
COINCIDENCE OBSERVATIONS
The gravity pendulum is a free-swinging pendulum. It has no, directly
connected mechanism to drive it or count its oscillations, as that would
seriously affect its accuracy. The exact number of its oscillations over
an interval of several hours must be obtained, however, for use in comput-
ing its period. It would be almost a hopeless task for an observer to watch
the pendulum for several hours and count the half-second oscillations. It
would be even more difficult i'or him to determine by direct observation
the exact chronometer times within a hundredth of a second at which the
pendulum passes through its neutral position, or some other definite point
in its oscillation, at the beginning and end of the interval.
DETERMINATION OF "I" BY MEANS OF PENDULUM 161
The method of observation by coincidences is a. very simple way of
overcoming these difficulties. A coincidence observation. consists in noting
the chronometer time when the pendulum and the chronometer are beat-
ing exactly together. They are considered as beating together if the pen-
dulum passes through its neutral position, corresponding to its. position
of' rest, at exactly the same instant as the shutter in the flash box is
opened by the break-circuit title ok the chronometer. At that time the two
images of the illuminated slit in the flash box, one reflected from the
stationary mirror in the receiver and the other from the moving mirror
on the pendulum, appear in the telescope in the same horizontal line.
The pendulum is. constructed of such a length that it has. a period
slightly greater than one-half of a sidereal second. Therefore, when the
shutter is opened by the break-circuit ticl: of the chronometer next fol-
lowi~g a coincidence it hats not quite reached its neutral position arid the
two indulges of the slit no longer appear opposite each other in the tele-
sc.ope. With each succeecli~:~g sec.o~d the movable image gets farther and
i:'arther away front the stationary one until it passes out of the held of
view. Al'ter a short tilde it reappears in the telescope butt moving in the
opposite direction. l~ina.lly the two! images appear again in a horizontal
line and give another coincidence.
In the interval between the two coincidences the pendulum has made
ens 1~s oscillation than twice the number oŁ seconds ticked off by the
chronometer. The number of' its oscillations is tl~erel'ore easily obtained.
The interval between two coincidences is known as. the coincidence in-
terval. If the chronometer has a steady rate this interval is fairly con-
stant at a station I'or any one of' the pendulums. Its average value can
be determined by observing 3 or 4 coincidences when the pendulum is
first started to oscillate, and 3 or 4 more after 10 or 12 hours. The num-
l~er of coincidence intervals, or the number of times the pendulum has
lost a beat on the chronometer, can then be obtained by dividing the total
time between the first and last coincidences by the' average coincidence
interval. With those data the number of oscillations of the pendulum
during the 10 or 12 hours can be readily found and.. finally, flee peri.ocl of
one oscillation ;: terns ol' the chronometer second.
~ .^v .~
OTHER OBSERVATIONS
At the time the coincidence observations are made, the temperature
and pressure within the receiver and the length of the are through which
the pendulum is swinging must also be read. These observations are made
through windows in the receiver. To facilitate the arc readings, a small
observing telescope is secured to the receiver which can be moved by a
race`; alla pillion to a position such that a cross wire in the telescope co;~-
.
~2
FIGURE OF THE EARTH
cides with the index line on the bob of the pendulum at first one extreme
off its motion and then at the other. Just below the pendulum bob there
is a slightly curved scale, graduated to millimeters, attached to' the bottom
of the receiver. The amplitude of the swing is obtained by reading on this
scale the positions of the index line of the pendulum at the two extremes of
-its vibration.
The flexure observations at a station are usually made after the pen-
dulum observations have been completed. The receiver is left exactly as
it was during the other observations except that the air is let into it and
a plug is removed near the top to permit the insertion of the aluminum
arm which is attached to the knife edge plate and which supports the sepa-
rate reflecting plate of the interferometer. The observations consist. in
reading the width of one fringe and the movement of the fringes on a
decimal scale located at the focal point of the telescope. After five obser-
vations of each quantity have been made the observing conditions are
varied by cha.ngin~, slightly the angle between the reflector plates and thus
varying the width of the fringes or bit changing the amplitude of vibration
of' the pendulum. Altogether, four different sets of conditions are used.
The movement. of the l~ni.fe edge plate when the receiver is supported
on an ordinary pier and the pendulum. is swinging through an arc' of five
millimeters amounts to about one-fifteenth of a fringe width of sodium
or helium li.~.t. This corresponds to about' 0.02 micron, or less than on
ore-millionth of an inch. If' the pier is too small, or if the soil around it
its loose, the fle,xure may be two or three times as much as this. Its stile
depends also on how far the foot screws project below the receiver and on
how tight the lock nuts are set. It has been found by careful tests that
on a good pier the greater part of the flexure is. in the receiver and its
foot screws. Probably this is due to ~ large extent to the elastic compres-
sion of the metal at the lower ends of the foot screws.
VON- STERNECK PENDULUM APPARATES
The VOll Sternecl: apparatus is quite different in many of its details from
Blat of' the Coast and Geodetic Survey, although similar in several essen-
tial respects, such as in the length of its pendulums. The pendulum sup-
port consists of a well-braced frame secured to a heavy circular base plate
which rests on foot screws. All levers for manipula,tin˘, the pendulums
and for making any of the acljustments, required when cha,ngin`~, from
one pendulum to another are located in the base plate to permit the use
of a bell-jar type of' receiver.
There are four principal pendulums and ea.cl:~ has a liaise edge secured
in its head. Each hats its own place iffy the receiver and each. its own sup-
portin', plane. All four pendulums are placed on tl~.is France at the same
time. This is clone before the receiver is placed ;~ position. A system
OFTERMINA LION OF 9 B Y MEA NS OF PEND ULU.M 1 (; .3
of' mirrors and prisms controlled -i'rom the outside by means of levers
working through pac.kin~, boxes in the base makes it possible to change
the observations from one pendulum to another without opening, the re-
ceiver or changing the position of the observing, telescope.
During the observations on any one of the pendulums the other three
are prevented from talking up sympathetic vibrations by means of
clutches controlled frown the outside of the receiver which
164
FI CURE OF THE EAR TH
GRAVITY DETERMINATIONS AT SE! WITH PENDULUM APPARATUS
Geophysical investigations have heels greatly impeded in the past by
the lack of accurate gravity determinations over the oceans. Many at-
tempts were made, by a number of scientific men o-t' different countries
using several different kinds of specially designed apparatus to meet this
lack but the results were of little geophysical value. IJntil recently it
seemed impossible to obtain an accuracy at all comparable with that of
land determinations. The problem was finally solved very successfully
by Dr. Meinesz who adapted the von Sterr~eck apparatus I'or use on a, sub-
merged submarine.
The method used by Dr. Meines% depends for its success upon com-
bining the simultaneous oscillations of two pendulums in the same re-
ceiver swinging in nearly opposite phases in the same plane. When the
two pendulums are thus combined, there are obtained the oscillations of
a hypothetical pendulum which is free from the effect of any small l~ori-
zontal accelerations of the ship in the direction in which the pendulums
swing. The apparatus is used in a submerged submarine as the motion
of a surface ship, even when the sea. is almost calm, is. ordinarily too great
to be corrected for in this manner.
In order to be able to combine the oscillations of each pair of pendu-
lums swinging in the same plane Dr.: Meinesz designed a, photographic
apparatus which ~,a.ve a continuous record on a moving film of the oscilla,-
tions of' the pendulums. By an analytical process. the simultaneous oscil-
la.tions of each pair were combined to give the oscillations of the cor-
res~ondin~ hypothetical pendulum. Both pairs of' pendulums of the
- Or ~ = --a 1~ ~
von Sterneck apparatus were thus recorded and combined and in this
way the effects of both the endwise and s.idewise motions. of the ship were
eliminated from the results.
After making a series ol about thirty determinations between Holland
and Java through the Mediterranean Sea, Dr. Meanest found that the
computations involved in combining, the records of' the pairs of pendu-
lums by the analytical method were discouragingly difficult and laborious.
As soon as he returned to Holland, therefore, he set about modifying the
apparatus in such a way as to shorten the computations. In this he was
very successful. lIe virtually designed a new apparatus, although a fen
of the details were copied from the Volt Sterneck apparatus.
The new Meinesz apparatus has three principal pendulums, all of
which swing in the sane plane. The two outside pendulums are started
oscillating at the same tinge by means of a device which automatically
starts them out of' phase with earls other. The middle pendulum does not
oscillate except as it is started by the motion of' the ship. A complicated
but ingenious systems of prisms and mirrors combines the oscillations of
DETERMINATION OF " g " BY MEANS OF PENDULUM 165
the middle pendulum with those of each of the other two directly on the
photographic film. In this way the record is made to show the oscillations
of two hypothetical pendulums which, as previously explained, are un-
affected by the horizontal acceleration of' the ship in the direction in
which they swing, and a great amount of' exacting computations is, thus
eliminated. In order to keep the apparatus as nearly as possible in a
level position, it is supported in a cradle hung in gimbals. The pen.du-
lums swing at right angles to the axis of the ship. Besides the three
principal pendulums there are two short, heavily damped pendulums,
one of which records the tilt of the apparatus, at right angles to the plane
of' oscillation and thus takes account of the endwise acceleration of the
ship, and the other serves as err object of reference in recording the a.mpli-
tude of oscillation of the middle one of the principal pendulums. These
data are needed in making the computations.
Dr. Meinesz first tested out his new apparatus by making a few deter-
minations in 1920 between Holland and Port Said. Then in 1926 he used
it for an extended series of about one hundred and fifty determinations
between Holland and Java by way of the Panama Canal. In the fall of
1,'328 he brought the apparatus to the United States and in cooperation
with the United States Navy and the Carnegie Institution of Wa.shi.ng-
ton he made about fifty determinations, in the Gulf of Mexico and in. the
waters surrounding the West Indies.
This thorough trial has proved beyond any doubt that the problem of
determining
~ 6~;
FIGURE OF THE EAI?TII
error which may be introduced from this cause may be inferred from the
fact that an east or west velocity of one knot at the equator causes a.
change of about ~ part in 130,000 of the total intensity of gravity.
SUGGESTIONS FOR ADDITIONAL READING
It is suggested. that those interested in making a more detailed study of the
different kinds of pendulum apparatus and methods used in determining gravity,
consult the following books and articles:
Fran~,ois, Ch. Note sur l'appareil gravimetrique de l'Observatoire Royal de
Belgique. Anna,les de l'Observatoire Royal de Belgique. Brussels. p. 1-36
(1920).
Galbis y Rodriguez, Jose. Determinacion relative de lo intensidad de lit fueler
de ,gravedad. Memorias del Instituto Geografico y Estad~st,ico. Madrid.
NIII: 7~D (1905).
Haasemann, L. Bestimmung der Intens,itat der Schwerkraft auf 35 Stationen
..... Veroffentlich. Kg. Preen. Geodat. Ins,t., No. 71. Berlin. p. 1-32
(1916).
Meinesz, F. A. Vening. Observations de pendule sur la Mer Pub. commis
sion geodesique neerlandaise. Delft. p. 1-16 (1923).
-- . Projet d'un nouvel appareil pendulaire. Bulletin geodesique No. '5. Paris.
p. 1-10 (1925).
. The determination of gravity at sea in a submarine. Geog. J. London,
65: 501-521 (1925).
Mendenhall, T. C. Determinations of gravity with the new half-second pendulums
of the Coast and Geodetic Survey Appendix 15, U. S. Coast and
Geodetic Survey Report for 1891. Washington. p. 503-531 (1892).
Sterneck, Robert von. Der neue Pendelapparat des kaiser!. konigl. militar-geo-
~r,,raphisch. Institutes. Mitteilungen k. k. militar-geogr. Institutes. Vienna
7:83-116(1887).
Swick, C. H. Modern methods for measuring the intensity of gravity. U. S. Coast
and Geodetic Survey Special Publication No. 69. Washington. p. 1-96 (1921).
