Finding a better value for GJesse W. Beams

Citation: Physics Today 24, 5, 34 (1971); doi: 10.1063/1.3022732View online: https://doi.org/10.1063/1.3022732View Table of Contents: http://physicstoday.scitation.org/toc/pto/24/5Published by the American Institute of Physics

G-measuring device designed byBeams and his coworkers ,includes acircular rotating table and two masssystems. The tungsten spheres visible inthe photograph form the "large mass"system; the "small mass" system isconcealed in the vertical cylinder.Figure 1

Finding a better value for GSurprisingly, we know the gravitational constant onlyto within about half a percent. With this elegant method, we hopefor a precision of at least one part in ten thousand.

Jesse W. Beams

The gravitational proportionality con­stant G, although one of the first con­stants measured and perhaps the mostfundamental and universal constant inNature, is now the least accuratelyknown. G, which relates the force Fbetween two particles to their massesml and m2 and the distance d betweenthem

F~-Gm,mdd'

is now known only with an accuracy ofabout one half of one percent. Becausemany contemporary techniques, suchas in geophysics, require a good knowl­edge of G, as well as for purely theo~

retical interests, a group of us at theUniversity of Virginia, Charlottesville,have been developing the apparatusshown in figure 1 to measure the ab­solute value of G as well as its possiblevariation with time or other factors.

Significance of GOur own interest III measuring G is

motivated by its very fundamental­ness; the gravitational constant shouldreally be known as accurately as pos­sible. But there are more "practical"reasons as well. Finding values forthe densities and density distributionsin the interiors of the earth, moon, outer

Jesse Beams is professor emeritus andsenior research scholar at the Universityof Virginia in Charlottesville.

planets and stars, for example, haverecently become important, and thesedeterminations require precise knowl­edge of G. Precise measurements ofthe orbits of celestial bodies give usvalues of GM, where M is the massof the body, rather than values of G.

Seismological exploration, along withother precise geophysical measure­ments, may soon be able to give usequations of state for the interior of theearth that are accurate to about O.l.to 1.0 percent. Comparison of theseresults with laboratory experiments onthe equation of state should greatly in­crease our knowledge of the earth'sinterior, as well as provide a sensitivecheck on the extrapolation of the labstudies. But to get at these values,we shall probably need to know G to atleast one part in 10\ about 100 timesmore precisely than any of the valuesso far determined.

Cosmological theory, according toRobert Dicke, 1 predicts a possibleannual change in G of one part in lOic10 10• Testing these predictions experi­mentally would of course be very ex­citing.

Past attempts to measure GThe gravitational constant is difficult

to measure precisely because of theextreme weakness and great univer­sality of gravitational interactions.The gravitational attraction between

a proton and an electron, for example,is roughly 10-40 times the electrostaticattraction, and careful studies showno variation of G with the magnitudeof the masses, the distance betweenthem, their temperature and so on. IS

Consequently it is necessary to measurevery small quantities under conditionssuch that one can not isolate the desiredvariable; it is not possible to shield thetest masses from the gravitational fieldsand field gradients produced by all theother matter in the universe.

Both large-scale and laboratory-scaleexperiments have "been used to mea­sure G. In one large-scale experimentin Peru, Pierre Bouguer:J estimated therelative masses of an isolated mountainand the earth from their volumes anddensities, and then found a value forG from the deflection of a plumb linenear the mountain. George Airy andhis collaborators,:l in another experi­ment, estimated the relative masses ofa spherical shell of the earth and of theentire earth. By recording the oscilla­tion periods of a pendul urn at groundlevel and one at the bottom of a mineshaft, they effectively measured theacceleration of gravity at the earth'ssurface and at the bottom of the shaft,and so determined a value for G. Theresult is valid only if the earth is spher­ical and homogeneous throughout, orif its density distribution is known.

Henry Cavendish did the first (1798)4

PHYSICS TODAY / MAY 1971 35

laboratory-scale determination of Gwith apparatus designed several yearsearlier by John Michell, who inventedthe torsion balance for this experiment.Cavendish's experiment appears to havebeen the first measurement of any of theso-called "fundamental constants."He mounted two solid homogeneousspheres (for which the total mass maybe considered as at the center), eachwith mass m on the ends of a light stiffrod that is suspended in the horizontalplane by a thin fiber. If two massesm' are now brought near the spheres(see figure 2), the gravitational at-traction between the masses m and m'produces a torque that twists the sus-pension fiber. When the masses m'are moved into symmetrical positions,

the deflection is in the opposite direc-tion. If the restoring constant of thefiber is known, the value of G can bedetermined.

Other experimenters, includingCharles Vernon Boys and Roland vonEotvos, have repeated Cavendish'sexperiments with some variations.' Inmany of these experiments the twomasses m were placed at differentheights on the torsion balance instead ofin the same horizontal plane; this ar-rangement allows m and m to bebrought closer together, increasing thegravitational torque between them andsimplifying the calculations. It hasthe disadvantage, however, of sensitivityto gravitational gradients and is in facta method of measuring these gradients.

Still other experimenters, such as Phil-lip Johann Gustav von Jolly and JohnHenry Poynting, used a chemicalbeam balance to measure G.B With thiskind of balance, equal masses are placedon each of the balance pans, and thepointer deflection is noted when a largemass is placed first under one pan andthen under the other.

The most reliable laboratory experi-ments until now were probably the workof Paul R. Heyl and his coworkers7 atthe US National Bureau of Standards.In their apparatus (see figure 2) twopendulum masses are mounted on a tor-sion balance, the attracting masses areplaced on the imaginary line that passesthrough them, and the period of thependulum is measured. The two at-

Experimenter

CavendishWilsingvon JollyPoyntingBoysBraunvon EotvosCremieuHeylHeyl and Chrzanowski

Table 1.

Year

1798188918811894189518961896190919301942

Past Results

Method

Torsion-balance deflectionMetronome balanceCommon balanceCommon balanceTorsion-balance deflectionTorsion-balance deflectionTorsion-balance deflectionTorsion-balance deflectionTorsion-balance periodTorsion-balance period

G(NmVkg2)

6.7546.5966.4656.6986.6586.6586.656.676.6706.673

PHYSICS TODAY / MAY 1971

Torsional pendulum Attracting mass in"near" position

Pendulum mass

Attracting mass in "far" position

Torsion balances as used by HenryCavendish and Paul Heyl in theirmeasurements of G. Cavendish (1798)measured the force exerted by two12-inch diameter lead spheres (W) ontwo 2-inch lead spheres (x), as shown inthe engraving at left. The deflection wasobserved through telescopes (T) outsidethe room. In Heyl's experiment (1930's),the period of a torsional pendulum ismeasured when the attracting masses arein the "near" and "far" positions.Figure 2

tracting masses are next placed in a linethat is perpendicular to the line joiningthe masses and passes through its cen-ter, and the period is measured again.The advantage of this method is thatperiods of the pendulum can bemeasured with greater accuracy thanthe small deflections in the previousmethods.

A. H. Cook of the University ofEdinburgh8 and Antonio Marussi of theUniversity of Trieste are now planningan experiment that is essentially basedon Heyl's. The study, to be carried outat the Grotta Gigante in Italy, will havea torsion-wire suspension of about 90meters with mass m equal to 10 kg andm' equal to 500 kg.

The accuracies of reported values forG are, except for Heyl's work, difficultto judge. Table 1 is a partial list of re-ported values, and, although an analy-sis9 of Heyl's work gives (6.670 ± 0.015)X 10"" Nm2kg^ for G, it is doubtfulthat any of the data in the table are reli-able to better than half a percent. Arecent analysis of Heyl's data by L. M.Stephenson10 shows an annual variationof G that is larger than the statisticalerror, with a maximum at the vernalequinox and a minimum at the autum-nal equinox.

Our methodThe method that we are developing at

the University of Virginia" has twonovel features that give it a potential for

improved accuracy. A rotationalacceleration, rather than a deflection,indicates the interaction force of themasses, so that the effect of the inter-action is cumulative and integrable overa long period of time. And the masssystems rotate about an axis manytimes during a measurement, cancellingall but the higher-order effects ofgravitational fields or field gradientsfrom extraneous masses.

In our arrangement (see figure 3), twolarge tungsten spheres (the "large masssystem") are mounted on a table thatcan be rotated. Mounted from thesame rotary table is a gas-tight chamberin which a horizontal cylinder (the"small mass system") is suspendedfrom a quartz fiber. The gravitationalinteraction between the two mass sys-tems tends to deflect the small masssystem in the direction that brings itsaxial center line in alignment with theline connecting the centers of the twolarge spheres, changing the angle 8 be-tween the two lines. A beam from alight source mounted on the rotary tableis reflected from a mirror mounted onthe small mass system, near the axis ofthe quartz fiber, and falls on a photo-diode that is also mounted on the rotarytable, so that there is an angle /3 be-tween the incident and the reflectedlight beams.

When the gravitational interactionbetween the large and small systemsbegins to change, /3 also begins to

change. Even very minute changes in/} are sensed by the photodiode, whichsends an "error" signal to the motorthat drives the table. The table thenrotates to maintain /3, and therefore d,constant. Because d remains con-stant, the small mass system ex-periences a constant torque, whichcauses a constant angular accelerationof the rotary table. By measuring theperiod of the table, we can determinethe acceleration very accurately, andthis acceleration is a direct measure ofG.

Theoretical analysis of the gravita-tional torque system gives

G = a±<i>oA(l+B + C.)

where a = <i> ± «o is the measuredangular acceleration, w is the angularacceleration of the torsion pendulumwith the large spherical weights on thetable and u0 is the angular accelerationwith the weights removed. A, B and Cand functions of the length L and theradius a of the suspended cylinder, thedistance 2R between the centers of thelarge spheres, the moment of inertia/s of the system suspended from thequartz fiber, the masses m and M of thesmall and large mass systems and theangle 8.

The experimental problem then is tomeasure a, 6>0, R, M, a, L, /B, m and 8as accurately as possible. Five of theseparameters (a, L, /s. M, m) are con-stants of the apparatus and so measur-

PHYSICS TODAY / MAY 1971 37

In the present experiments a helium-filledchamber, supported on a rotary table andprecision air bearing, encloses atorsional pendulum. Two tungstenspheres are placed on the table. Anysmall change in the angles /3 (andtherefore e) is sensed by the photodiodesand rectified by rotation of the table.The acceleration of the table is ameasure of the gravitational interaction.Figure 3

Quartz fiberN

Small mass system ,

Airtight chamber

Tungsten spheres

Tracking optical lever

Incident light

Lightsource

Photodiodes

Reflected light

Precision bearing

Rotor ~-^^^

Vlercury \

Mercury \slip rings \ _ _ * \ i

^_—— Stator

^^^- Copper probes

able independently of the actual experi-ment; the other four are determined inthe course of the experiment.

Our procedure is first to level thetable and check the vertical alignmentof the cylindrical chamber. We adjustthe quartz fiber to coincide with the axisof rotation, fill the chamber with he-lium, and assure ourselves that the ef-fective twist in the fiber is as close tozero as possible. The tungsten spheres,already at room temperature, are put inplace and their heights adjusted so thattheir centers of mass are in the samehorizontal plane as the axis of the cylin-drical rod. The spheres are placedso that their centers of mass are on animaginary line through the perpendicu-lar to the axis of rotation of the table,with the radial distances of the twospheres from this axis equal to within± 10"5 cm, and R is measured.

The dimensions of the small masssystem are a compromise between keep-ing the distance between the spheres assmall as possible (to maximize w) andusing feasible-size spheres. (The pre-cise dimensions of the system are intable 2). Because 6 must be not onlymeasured but also held constant, it

should be about 45 deg, the angle forwhich the torque on the small masssystem, and therefore the angular ac-celeration &> of the table, is a maximum.The error-angle /J is held constant bymeans of a servo system (see figure 4).

Determining the angular accelera-

tions OJ and wo requires determiningthe period of the rotary table. Theoptical arrangement is such that the ro-tation period is equal to the time be-tween two voltage pulses, and ismeasured in microseconds with a pre-cision of about one part in 10'. After

Table 2. Mass Systems

Large masses (high-density tungsten spheres)

Mass (kg)Diameter (cm)Distance between center of mass and

geometrical center (cm)Sphericity (cm)

Sphere 1

10.489980 ±0.0000710.165072

4.610 X 10"4

12 X 10 "6

Small mass system (copper cylinder)

Length L (cm)Diameter a (cm)Mass m (gm)Moment of inertia of aluminum-alloy

stem (gm cm2)Angle between the axis of copper cylin-

der and the perpendicular to track-ing mirror

3.9649 ±0.00040.19824 ±0.00044.0512 ± 0.0001

0.0408 ±0.0001

44 deg 43 min ± 0.5 min

10.10

712

Sphere 2

490250 ±0.00007.165108

.569X10-"X 10 ~6

38 PHYSICS TODAY / MAY 1971

Servo and timing systems. Servo motorrotates table to ensure the constancy ofthe angle between the small and largemass systems. Timing arrangementmeasures the period of rotation bycounting pulses from the photodiodeand is accurate to at least 1 in 10 7

Figure 4

fTracking optical lever

\J

Servosystem /

/

o^

Torsional pendulum

Mirror

•o— ] / Timing optical lever

Rotary table ' \ /

\Photodiodes

Two-phase drive motor

Electronictimingsystem

successive rotation periods aremeasured, the constant acceleration ofthe rotary table can be computed.

Once our apparatus is going properly,the angular acceleration of the tableis constant; the constancy of this accel-eration is in fact a good measure of sys-tem noise. After the acceleration goeson for several hours (between 4 and5 X 10"B rad/sec-), and the table isrotating at about one or two revolu-tions per minute, we carefully removethe tungsten spheres and continuetracking for several hours longer. Thissecond acceleration wu is a measure ofthe gravitational interaction of smallmass system with the table and withother fixed masses turning with it, aswell as a measure of fiber twist andother perturbations.

We have done experiments with thetable decelerating as well as accelerat-ing, but in all experiments we let thetable rotate a whole number of turns,so that the effect of ever-present massasymmetry surrounding the apparatusis negligible.

Our preliminary values for G haverevealed several sources of uncertain-ties that limit the precision of our

measurements, and consequently we aretrying to improve the various parts ofthe apparatus and to develop betterways to do the delicate measurements.In our earlier experiments the majorerror was in measuring co and resultedfrom variations in spindle friction, in-stabilities in the electronic circuits andtemperature gradients produced by themotor and bearing. We substituteda gas bearing for the precision ballbearing, improved and stabilized theelectronic circuitry and thermally iso-lated the motor. Our results improved(see figure 5); the acceleration now re-mains constant to the order of one partin 104, and we believe this precision cansoon be improved by at least a factor often. Before the recent improvements,our best value was G = (6.674 ± 0.012)X 10"" Nm-/kg-, where 0.012 is threestandard deviations. This result agreesvery well with Heyl's value of (6.670 ±0.015) X 10-"Nm-/kg-.

The precision with which the center ofmass of the two tungsten spheres can bedetermined is also a limiting factor, andwe are encouraged to believe1- that bysubstituting for the tungsten spheres wecan measure G much more precisely.

Certain metals with intermediate tohigh densities have density variations ofless than one part in 10' per centimeter.These metals can not be formed intoprecise spheres, because they will dis-tort under their own weight, but they areusable as rectangular blocks. The useof blocks as the large mass system com-plicates the theory but not beyond thecapabilities of computer analysis.

In our original conception of thisexperiment, magnetic rather thanquartz-fiber suspension of the smallmass system was planned. We still be-lieve this method will ultimately givegreater accuracy, primarily because therestoring torsion of the small-masssuspension system would be very signifi-cantly reduced.

Although I hesitate to estimate theultimate precision with which thismethod can measure G, our experiencehas shown that many important im-provements can be made. The pos-sibilities include a completely new de-sign of the rotary table and chamberthat gives special attention to thermaland vibrational isolation; certainchanges in the servo-motor drive sys-tem; improved metrology; substitution

PHYSICS TODAY / MAY 1971 39

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Constancy of measured acceleration results f rom refining experimental methods.Data with ball bearing and no temperature control (a) form erratic curve. Better re-sults were found (b) when the room temperature was controlled, the photodiodes andother electronics were improved, and a gas bearing replaced the ball bearing. Whenthe motor temperature was also controlled (c), Beams found more improvement in thedata; changing the rotation direction caused the two points to be missed. Figure 5

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of the metal blocks for the tungstenspheres, and an accurately groundquartz cylinder in the small mass sys-tem. Measuring G to one part in 106

is not beyond conception, once these im-provements are made, nor is detectingvariations in G of one part in lO6.

References1. R. H. Dicke, The Theoretical Signifi-

cance of Experimental Relativity, Gor-don and Breach, New York (1964).

2. F. C. Champion, N. Davy, Properties ofMatter, Blackie, Glasgow (1959); R. vonEotvos, D. Pekar, E. Fekete, Ann.Physik68, 11 (1922).

3. C. V. Boys in Dictionary of AppliedPhysics (R. Glazebrook, ed.) 1, 3 (1923).

4. H. Cavendish, Phil. Trans. Roy. Soc.88, 469(1798).

5. C. V. Boys, Phil. Trans. Roy. Soc. A, 1(1895); R. von Eotvos, Wied. Ann. 59,392(1896).

6. P. von Jolly, Wied. Ann. 14, 331, (1881);J. H. Poynting, Phil. Trans. Roy Soc.A 182, 565 (1891); J. H. Poynting in

Encyclopedia Brittanica, 11th ed., Vol.12, page384, New York, 1910.

7. P. R. Heyl, J. Res. Natl. Bur. Std. (US)5, 1243 (1930); P. R. Heyl, P. Chrzanow-ski, J. Res. Natl. Bur. Std. (US) 29, 1(1942).

8. A. H. Cook, Contemporary Physics 9,227 (1968); Proceedings of the Int.Conf. on Precision Meas. and Fund.Const., 3-7 Aug. 1970.

9. "Recommended Prefixes; Defined Val-ues and Conversion Factors; GeneralPhysical Constants," US Natl. Bur. Std.Misc. Pub. US Government PrintingOffice, Washington, D. C. (1963).

10. L. M. Stephenson, Proc. Phys. Soc.(London) 90, 601 (1967).

11. J. W. Beams, A. R. Kuhlthau, R. A.Lowry, H. M. Parker, Bull. Am. Phys.Soc. 10, 249 (1965); R. D. Rose, H. M.Parker, R. A. Lowry, A. R. Kuhlthau,J. W. Beams, Phys. Rev. Lett. 23, 655(1969); W. R. Towler, H. M. Parker, A.R. Kuhlthau, J. W. Beams, Proceedingsof Int. Conf. Precision Meas. and Fund.Const., 3-7 Aug. 1970.

12. J. W. Beams, Bull. Am. Phys. Soc. Series2, 11, 526(1966). a

PHYSICS TODAY / MAY 1971