NBS MONOGRAPH 107

Acceleration Due to Gravity at the

National Bureau of Standards

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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UNITED STATES DEPARTMENT OF COMMERCE

C. R. Smith, Secretary

NATIONAL BUREAU OF STANDARDS • A. V. Astin, Director

Acceleration Due to Gravity at the

National Bureau of Standards

D. R. Tate

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

Reprinted from the Journal of Research of the National

Bureau of Standards — C, Engineering and Instrumentation,

Vol. 72C, No. 1, January-March 1968

National Bureau of Standards Monograph 107

Issued June 1968

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 - Price 25 cents
Library of Congress Catalog Card Number: 68-60065
Foreword

The determination of the absolute value of the acceleration due to gravity contributes knowledge to several different disciplines of the physical sciences. It provides

a base which, together with the standard of mass, establishes the derived standard of

force. The standard of force, in turn, is a necessary quantity in the assignment of values

to the electrical units of current and voltage. The acceleration due to gravity is also an

important factor in the accurate pressure determinations needed for the thermodynamic

temperature scale and the establishment of the International Practical Temperature

Scale.

An absolute measurement of the acceleration due to gravity is of particular interest

to the science of geodesy. Rapid advances have recently been made toward the completion of a world gravity network which will establish reliable gravity values at a large

number of base points located strategically over the earth. To provide a new and more

accurate basis for the network, major laboratories in many nations have joined together

in a cooperative effort. The determination of the acceleration due to gravity at NBS,

which the present Monograph reports, is part of this effort.

A. V. ASTIN, Director.

Ill
Contents

Page

1. Introduction 1

1.1. Absolute gravity as the standard of force 1

1.2. Previous determinations 2

1.3. Accuracy and precision 2

1.4. Methods for gravity determinations 2

2. The NBS determination 3

2.1. General description 3

2.2. The falling objects 4

2.3. The vacuum chamber 6

2.4. Mechanical design 7

2.5. Optical system 9

2.6. Electronic system 9

3. Reference standards 10

3.1. Length 10

3.2. Time 10

4. Techniques of measurement 10

4.1. Length 10

4.2. Temperature 11

4.3. Compensation for guide curvature 11

5. Experimental procedure 12

5.1. Length measurements 12

5.2. Timer adjustments 12

5.3. Free-fall measurements 12

6. Results 13

6.1. Experimental data 13

7. Sources of bias 14

7.1. Longitudinal vibration 14

7.2. Electrostatic forces 15

7.3. Magnetic forces 15

7.4. Residua] air 15

7.5. Deflection of the falling object 16

7.6. Ground motion 16

7.7. Effects from optical windov^rs 16

7.8. Time constants of the circuits 17

7.9. Uncertainty in the length standard 17

7.10. Uncertainty in the length measurements 17

7.11. Uncertainty in the final result 18

8. Comparison with other gravity values 18

9. Conclusion 20

10. References 20

IV
Acceleration Due to Gravity at the National Bureau

of Standards

D. R. Tate

Institute for Basic Standards, National Bureau of Standards, Washington D.C. 20234

(September 26, 1 967)

A determination of the absolute value of the acceleration due to gravity was completed in June

1965 at the National Bureau of Standards near Gaithersburg, Maryland. The determination resulted

in a value of 980.1018 centimeters per second squared for a reference point on the first floor of the Engineering Mechanics Building. The result was pubhshed in The Journal of Research of the National

Bureau of Standards, Vol. 70C, No. 2, Engineering and Instrumentation, page 149, April-June 1966. The present paper describes in detail the apparatus and the techniques employed and presents the summarized data from which the value was derived.

Key Words: Absolute gravity, acceleration, free-fall, g, gravity, Potsdam.

1. Introduction

1.1. Absolute Gravity as the Standard of Force

Determinations of the acceleration due to the gravitational attraction of the earth have been made

many times since the early observations of Galileo.

Up to the end of the nineteenth century measurements

were made with the primary purpose of gaining knowledge about the figure of the earth and the effects of

the distribution of mass near its surface. In the years

that have followed, gravity values have not only con- tinued to be of great interest to the geodesist but also

have fulfilled an additional role in supplying the base

for the standard of force. Virtually every measurement

which involves force, such as electrical current or

pressure, depends in some manner on knowledge of

the gravitational field of the earth.

It is generally agreed that the unit of force is defined

by the second Newtonian law of motion and the primary standards of length, mass, and time. The determination of the acceleration due to gravity links the

dynamic relationship of the Newtonian law to a permanent reference in the gravitational field of the

earth. The gravitational field strength, multiplied by

an accurately determined mass value, then becomes

the working standard of force. From this point onward

the normal process by which a measurement system

is established, i.e., by the comparison of like quantities, can successfully include force as an element.

An essential part of the process, of course, is a means of extending the single point value of the field

intensity to other points as needed. This extension is

*It was necessary to rewrite this section, 17 lines, before the Monograph was reprinted.

accomplished through the use of comparators such

as geodetic pendulums and gravity meters. The results

comprise a unified network which will eventually cover the habitable portions of the earth.

The relationship between the acceleration due to gravity and the concept of the gravitational field is so

direct that it is frequently ignored in practical applications. However, distinction between the two concepts

is essential to a logical understanding of the basic

equations of mechanics. —

Consider a body of mass m at rest on the surface of

the earth. The reaction of the body against the earth

is the weight, W, expressed by

W= -mV<p-mwX(ajXR);

(f is the gravitational potential, w the angular rotational

velocity, and R the radius of the earth. Released in

free fall the body would experience an acceleration — ^<p, but it is observed from a point which has an

acceleration w X (co X R) on the surface of the rotating

earth. The relative acceleration, g, is then

g=- ^ (p-a)X((oXR).

Thus, multiplying by m,

W= mg.

The weight of an object at rest on the surface of the

earth is, therefore, equal to the product of the mass and

the local value of g. —

Although the validity of the assumption above, that

the inertial mass is equal to the gravitational mass,

1
has frequently been questioned, experiments such

as those with the Eotvos pendulum [1]' indicate that

any effects must be less than one part in 10". This

amount is several orders of magnitude less than the

sensitivity of any absolute measurement now under

consideration.

1 .2. Previous Determinations

The history of absolute gravity experiments published within the past 70 years has been reviewed by

several authorities. Cook [2], in particular, gives a very thorough treatment. Briefly summarized, an

absolute determination published in Potsdam, Germany, in 1906 by Kiihnen and Furtwangler [3] became

the base for the world network of relative gravity

observations. Two later determinations, one by Heyl

and Cook [4] at Washington, D.C. and the other by

Clark [5] at Teddington, England, disagreed with the

Potsdam result by significant amounts. Reexamination

of the Potsdam result by Dryden [6] and of Clark's

and Heyl's results by Jeffreys [7] improved the agreement but did not fully resolve the questions raised. In 1956 the results of three new de_terminations by

Agaletskii and Egorov [8] and Martsiniak [9] tended to support the conclusion that the original Potsdam

result is too high by some ten to twenty parts per

million. Three other absolute determinations appeared

within the next few years, Preston-Thomas et al. [10],

Thulin [11], and Faller [12]; all have shown reasonably

good agreement with each other and have ranged from

about 12.7 to 14.5 milligals ^ (mCal) below the Potsdam

result. More recently Cook [13] has reported the result

of a determination at the National Physical Laboratory

differing from the Potsdam result by approximately

13.7 mGal.

1 .3. Accuracy and Precision

The requirements for accuracy in absolute gravity

measurements are much the same for the geodesist

and the metrologist. An accuracy of one part in a million is critically needed and an accuracy of one

or two parts in ten million would be most desirable.

It has been suggested that some form of absolute

gravity apparatus which can readily be transported

and set up in different sites might resolve the apparent

bias between some measurements with geodetic pendulums. If such an approach is to be effective, a demonstrated stability in the portable absolute apparatus of one part in ten million is quite essential.

The results of absolute determinations in which

measurements of several non-related parameters are combined to obtain an estimate of the magnitude of a presumed constant quantity, the true value of which

is unknown, cannot realistically be assigned a level of

accuracy. One can and does derive an index, such as the standard deviation, of the precision of individual

determinations made over a short period of time. From

this index an estimate may be derived of the precision

which might be expected for the means of similar sized

' Figures in braclcets indicate the literature references at the end of this paper.

2 A milligal is a geodetic unit of gravity equal to 10^^ cm/s'-', approximately 1 ppm of the value of;.'.

groups of measurements drawn from the same source,

j If such groups of measurements are actually made, i separated by periods of a day or more, the standard j deviations of the individual measurements may be j much the same but the means of the groups will often i show more scatter than would have been predicted I from the internal consistency of any of the groups.

The conclusion is that the composition of the population from which the samples have been drawn has

changed with time. It is inferred that there must

| exist sources of bias which vary from day to day but j which do not change appreciably over the course of

j a single set of measurements.

| The nature and extent of variability of this bias i usually remain unknown. The demonstrated presence I of bias, of course, arouses the investigator to try to I analyse it by looking for correlation with other param- i eters. In general, though, this process has already ^ been carried out to the limits of practicality. This | residuum of bias is probably a major cause of the well- 1 known tendency of the scientist to continue his

j measurements indefinitely. I In a broad sense, therefore, one can consider that I an estimate of the accuracy of an experimental meas- ' urement can be made only when the resultant value I can be compared to a defined or assumed true norm,

j In the absence of such a true norm, a determination ' obtained from a series of measurements made with a 1 single piece of apparatus must be regarded as only a sample from an unknown population. Estimates of

the precision should be made, together with statements

of the sensitivity to variations in the component

j measurements in relation to the precision with which 1 those standards were established. Such statements I give limits to the error only from known sources. The | final appraisal of the limits of knowledge about the

j true value of the constant determined must come from | the results of a group of completely independent

j experiments differing in method, location, and in the

j personnel conducting them.

1.4. Methods for Gravity Determinations

Taken without regard for experimental difficulty, it would appear that the most direct method for determining the absolute value of the acceleration due to ' gravity would be to observe the motion of some object , moving without constraint under the action of the

gravitational field of the earth. Up to about 1950 the use of such a method was impracticable because of the ! difficulty in measuring the short lengths of time in- volved. Virtually all gravity measurements were made

with some form of pendulum.

Basically the pendulum method substitutes a system

with a maximum of constraint for a system with a ] minimum of constraint. From a purely practical view-

| point the pendulum method offers many advantages, i An uncertainty of one part in a million in the measure- [ ment of the pendulum length produces the same j, relative uncertainty in the determination of g. An

uncertainty of one part in a million in the measurement

of time results in an uncertainty of two parts in a million in g, but the length of the time interval can be i
increased to the point where timing errors become

negligible. The difficulties of the pendulum method

lie in the complex theoretical corrections for pendulum

flexure and the uncertainties contributed by motion

of the pendulum support and by the indefinite point

of rotation at the knife edge.

Experiments involving unconstrained motion under

gravitational forces have, with the perfection of fre- quency counting electronic circuitry, developed in

many diverse forms. The most elementary form of

such an experiment would be one in which an object

is released from rest and allowed to fall a known distance in a vacuum. If time is measured from the instant

of release to the point where the object has travelled

the measured distance, the acceleration may be computed from the simple relationship

_2d

Here the sensitivity in the determination of g is the

same as in the case of the pendulum and the efficiency

of measurement would be high. Unfortunately the ex- perimental difficulties in the measurement of time and

length and in the release of the object are so formidable

that no investigator up to the present has undertaken

such an experiment.

The restrictions of the elementary free-motion ex- periment can be eliminated by releasing the object

prior to the start of the time measurement. In this case the object acquires an initial velocity of unknown

magnitude which can be successfully eliminated by

the addition of one or more distances over which time

is measured. A parallel and equally effective approach

is to measure the distance the object travels during

two or more known intervals of time.

These techniques, while successfully eliminating

the experimental difficulties of the elementary experiment, are introduced at the cost of an increase in

sensitivity to errors in the measurement of length

and time. The method offers many advantages from

the experimental viewpoint and has been employed

in one form or another by several investigators.

In 1947 [14], M. Charles Volet, then Director of the

Bureau International des Poids et Mesures, suggested

that if an object were projected upwards, passing

two position sensing stations separated by a known

distance and then allowed to fall back, and the time

between successive passages at each level recorded,

a considerable increase in measurement efficiency

would be obtained. The method is indeed a most effective one since the sensitivity to length errors is the

same as for the elementary experiment and the

sensitivity to timing errors is very nearly equal to that in an elementary experiment with four times the height

of fall. The principal disadvantage is the increased

difficulty in the experimental technique required. The

method is currently being employed in experiments

at the National Physical Laboratory by Cook [2],

and at the Bureau International des Poids et Mesures

by Sakuma [15]. Sakuma's method embodies the con- cept of making the moving object one reflector of an

interferometer system illuminated by a gas laser light source. This approach has the great advantage of

making the length and time measurements occur

simultaneously and avoids the Hmitations of an intermediate length standard. Faller's experiment [12]

was the first published determination using the falhng

interferometer plate principle. He was limited to a very short length of drop by the necessity of using a conventional light source.

2. The NBS Determination

2.1. General Description

Following a review and appraisal of several possible

approaches to absolute gravity measurements, it was

decided to employ a modification of the direct free-fall

method with the release of the object preceding the

timing sequence. The method makes use of a concept

introduced by Agaletskii and Egorov [8] in which the

released object falls within a chamber which is itself

falling. Agaletskii's technique was to determine the

acceleration of the object with respect to the chamber

and, separately, to determine the acceleration of the

chamber with respect to the earth. The measurements

were made by a multiple flash method at constant

frequency.

In the experiment at the National Bureau of Standards the acceleration of the inner falling object was

determined directly with respect to the earth from

measurements of the time required for two intervals

of length on a fused silica tube to pass an optical sens- ing station.

The apparatus for the experiment was first set up

at the National Bureau of Standards in Washington,

D.C., in 1961 in a pair of small rooms known as the

sub-subbasement of the East Building. This had been

the site of the absolute determination by Heyl and

Cook [4]. In February of 1964 the apparatus was

damaged in a flood which left the site in a nearly

unusable condition. The apparatus was reconstructed

in the summer of the same year and installed in room

129 of the Engineering Mechanics Building (Building

202) at the new site of the Bureau near Gaithersburg,

Maryland. The measurements leading to the final

value were taken during April, May, and June of 1965.

In the experiment a fused silica tube slightly more

than a meter in length rested on a seat in a vacuum

chamber a few centimeters longer than the tube. The

vacuum chamber was mounted on a carriage and held

by a latch at the top of a pair of vertical guide rods

secured in a rigid frame. The arrangement was such

that when the latch was released, the carriage was

free to descend vertically down the guide rods under

the influence of gravity. A schematic diagram of the

apparatus is shown in figure 1. Friction between car- riage and guide rods was reduced to a low level by

antifriction bearings. In the latched position the car- riage was under an additional downward force provided by an adjustable, short-travel, spring-loaded

plunger. When the latch was released the carriage

moved downward with an acceleration about ten per3
'A

h

Figure 1. Schematic view of gravity apparatus.

cent greater than the acceleration due to gravity for

the first 8 mm of its travel. The additional acceleration

of the car effectively separated the fused silica tube

from its seat inside the chamber, leaving the tube

floating without contact w^ith the chamber for the

duration of the fall of the carriage. By proper adjustment of the spring plunger travel, the relative displacement between the silica tube and the chamber

could be restricted to a millimeter or so after the

initial separation. After a fall of about IV4 m the

carriage was brought to a smooth stop by an air

dash-pot.

Only the force due to gravity acted upon the silica

tube inside the chamber. The tube, which was opaque,

carried three transparent slits, to be described later. The slits defined two lengths, one extending from the

slit near the lower terminus of the tube to the second

slit 30 cm above it, and the other extending from the

lower slit to the third slit 100 cm above it. The vacuum

chamber was provided with pairs of ports at locations

opposite the slits. A single position-sensing station

consisted of a light source, a long focus microscope,

and a photomultiplier cell. As each slit passed the

station, an illuminated image of it was projected on a plane containing a stationary slit within the microscope tube. Beyond this reference slit was the photomultiplier cell. The reference slit was dark until the

bright image of a moving slit swept across it, causing

the photomultiplier cell output to rise from nearly

zero to full anode potential. A two-channel electronic

Short length 30 centimeters

Long length 100 centimeters

Time for short length 156 milliseconds

Time for long length 348 milliseconds

Least count in measuring instruments

Length 0.1 micrometer

Time 0.1 microsecond

Temperature 0.02 degree Celsius

Sensitivity of the value of to variations equal to the least count

Short length 0.7 milligal

Long length 0.3 milligal

Time for short length 1.8 milligal

Time for long length 1.4 milligal

Temperature of silica tube 0.01 milligal

Temperature of invar standard 0.02 milligal

2.2. The Falling Objects

Four tubes were employed in the experiment, all constructed to the same nominal dimensions from

timer system was triggered at some predetermined

potential level near the midpoint of the linearly rising

photocell output curve. During the fall of the tube, both timer channels were started by the passage of the lower slit past the

position-sensing station. One channel was stopped by

the passage of the second slit and the other channel

was stopped by the passage of the third slit. They

thus registered the transit time for the 30-cm and

100-cm lengths, respectively.

The apparatus was arranged so that the fused silica I tube fell a distance of 6.8 cm before the first slit , initiated the timing sequence. Thus about 118 ms | elapsed between the release of the latch and the start i of the time measurements. I The lengths corresponding to the two distances on

the fused silica tubes were measured, using the same

apparatus employed for the free fall time measurements. The fused silica rod was secured firmly in its seat and a measurement made of the distance the

carriage moved between the locations where the

timers would have been triggered by the photocell

signals. The technique for the length measurements

is described in another part of this paper. The method

has the advantage of including effects of the vacuum

on the actual length of the rod as well as any optical

effects from the ports in the vacuum chamber walls.

More importantly, it is a direct measurement of the

quantity desired.

The following summary indicates the magnitudes

of the parameters measured and the sensitivity of

the result in terms of the least count in the measurements.

Magnitudes of measured parameters

i'
clear, fused silica tubing having an outside diameter

of 15 mm, and a wall thickness of 1 mm. Each tube

had a length of 107 cm. A metal sleeve with an outside

diameter of 19 mm fitted over one end of each tube

and was cemented in place with epoxy cement. The

external end of the sleeve was machined to a spherical

radius of 12 mm. The sleeve formed the lower support

tip for the silica tube.

At a point about 11 cm from the other end of the

silica tube a second sleeve was attached, fitted with

guide pins for alinement and rotation in the vacuum

chamber. Figure 2 shows a schematic view of an assembled tube.

Each sUica tube was fitted with three rectangular

clear apertures formed by pairs of bevel-edge metal

plates cemented to the outer surface of the tube wall.

The plate edges were perpendicular to the longitudinal

axis of the tube and spaced so that the aperture took

the form of a transverse slit about 1 mm wide and 7

mm long. In the experiment the dimensions of the

slit were of no particular consequence, as the timing

sequences were generated by the transition from

dark to Ught as the lower edges of the slits passed the

optical axis of the position sensing optical system.

The upper slit boundary was supplied to provide a sharp cut-off of the light beam in order to prevent

spurious triggering of the timers by photo-tube noise

in a slowly decaying light pattern.

\ /

I METER

30cm

Figure 2. Free-fall object showing slits and guide sleeves.

The experiment required that the critical edges of

all three slits be mutually parallel, coplanar, and

perpendicular to the axis of the fused silica tube. To

this end, a special jig was made to hold the silica tube

and aperture edges during the operation of cementing

the edges to the tube.

The jig consisted of a channel-shaped frame of

steel having two vee-blocks attached on the centerline of the channel web at positions corresponding

to the lower support tip and the upper guide sleeve

of the silica tube. Three pairs of rectangular notches

were machined into the flange-edges of the channel

at locations corresponding to the 0-, 30-cm, and 1-m

aperture locations on the silica tube. The parallelism

of the jig notches was checked with an autocollimator

and the notches hand-finished to be parallel within

7 s of arc. This amount corresponds to an error of

0.03 ^tm in 1 mm.

In using the jig, cut and finished aperture edges

were attached to special holders. These were flat bars which could be set into the jig notches and

secured in place with hold-down screws to maintain

alinement during the setting of the cement. The

holders were constructed with flats lapped on the

leading edges at the points where they contacted the

forward edges of the jig notches. The center portion

of the holder edges was relieved. An aperture edge

was attached to each of three such holders by metal

fingers which gripped the aperture plates at the sides,

the defining edges of the aperture plates being made

coplanar with the lapped flats on the holders by a

light lapping operation. The finished assemblies were

then checked in a comparator.

A silica tube, complete with supporting tip and

guide sleeve, was clamped in the vee-blocks which

had been shimmed to provide a clearance of about

0.07 mm between the surface of the sUica tube and the

under face of the aperture plate and holder assemblies.

A dot of epoxy cement was then placed on the center

of each aperture plate mounted in its holder. The

three holders were inverted and each carefuUy placed

in a jig notch and secured in place with the lapped

edges forced against the leading edge of each notch.

After the cement had set for 24 hr, the gripping

fingers coupling the holders to the aperture edges

were released and the holders removed from the jig.

The bond between the silica rod and the aperture

plates was then reinforced with additional cement.

Three opposing aperture plates to complete the framing of the apertures were then installed by a similar

technique. The bond formed between the metal aperture edge plates and the sUica tube by the epoxy

cement was quite strong. Even the bond formed by

the initial small dot of cement had sufficient strength

after 3 or 4 days to require considerable force to separate the two parts.

As a final step each silica tube was given several

coats of silver conductive paint to prevent the formation of localized electrostatic charges. Preliminary

experiments with silica rods or tubes in vacuum had

shown that under certain conditions a substantial

electrostatic charge was generated by the separation

5
of the supporting tip of the tube from the seat in the

vacuum chamber. Even though the falling object

was surrounded by the grounded metal vacuum chamber, the internal field was not necessarily zero and the

silica tube tended to be drawn out of the true vertical

line as it fell. In some cases the effect was so severe

as to cause the timers to malfunction. The conductive

paint, which reduced the surface resistance of the

silica tube of less than 10 fi, eliminated this difficulty.

Only the clear apertures and corresponding areas on

the opposite face of the tube were not coated.

2.3. The Vacuum Chamber

The vacuum chamber was fabricated in three sections joined by flanges with O-ring seals. It was

constructed of brass tubing having an outside diameter

of 4.76 cm and a wall thickness of 0.32 cm. The outside

of the chamber was covered by two layers of 3-mm

cork sheet and finally by a layer of heavy aluminum

foil to reduce the effects of ambient temperature

fluctuations. The vacuum chamber assembled on

the carriage of the machine is shown in figure 3. The lower section of the chamber contained the

support cone for the silica tube, the vacuum gage,

the vacuum pump connection, and the optical windows

for the zero and 30 cm apertures of the silica tube.

The middle section of the vacuum chamber housed

the upper guide bearing for positioning the silica

Figure 3. Upper portion of gravity machine showing guide rods,

carriage, and vacuum chamber.

tube preparatory to release. In the upper section of ' the vacuum chamber were the optical windows for j the 1-m aperture of the silica tube. i The vacuum chamber was so arranged that it was j necessary to disassemble only the upper section in

order to remove or insert a silica tube.

' The support cone formed an integral part of the i removable base closure for the lower section of the

vacuum chamber. It was made by machining a base

with a recess in the form of a 90° vertex angle cone.

| The major portion of the cup formed in this way was ! then milled out, leaving only three ribs, 120° apart i and about 7 mm wide. The shape was a compromise | between the desirable 3-point support and a full

j cone seat. It was considered that a 3-point support

would tend to produce localized deformations of the

silica tube support tip which would in time make the

contact unstable. Earlier experiments with a support

seat which more nearly approximated a fuU cone had

likewise resulted in some undesirable characteristics.

The upper guide bearing mechanism was built as a unit into the middle section of the vacuum cham- '

ber. It consisted of two end flange plates with O-ring

| seals each connected by a metal bellows to a central

| sliding unit. A section of brass tubing of the same , dimensions as the other sections of the vacuum | chamber was attached between the end flange plates . providing structural support and also guidance for

the central sliding unit. The central sliding unit

| terminated in a tube having an inside diameter of I 19 mm which projected above the end flange plate

j into the upper portion of the vacuum chamber. To i this tube was attached a bushing having an inside

j diameter just slightly larger than the upper sleeve j of the silica tube. The upper surface of the bushing

was a double-vee cam surface to engage the guide

pins of the silica tube sleeve as shown in figure 4. When a silica tube is installed in the assembled i vacuum chamber, the arrangement is such that when j the central sliding unit is moved to the upper limit of

its travel, the silica tube is picked up by the guide

pins in the vee-shaped notches and rotated into the

proper orientation with respect to the optical axis of

the position sensing station. Then, as the central

sliding unit is lowered, the silica tube is seated in the ' support cone at the bottom of the vacuum chamber but , is precisely alined by the guide bushing surrounding

the sleeve on the silica tube. When the central sliding

unit reaches the lower limit of travel where it can be \ locked in place, the guide bushing is completely , disengaged from the silica tube sleeve and the silica

tube stands supported only by the support cone. It

is then ready for a free-fall measurement.

During length measurements on the silica tube the I central sliding unit was left in the center of its travel ! where the silica tube received lateral guidance from , the bushing but rested on the support cone. The six optical windows in the vacuum chamber

were Pyrex glass discs, 4.5 mm thick and 19 mm in

diameter. They were optically polished to be flat within one-half wave length of the green mercury line.

The discs were inserted in machined wells in the cham- !

6
Figure 4. Vacuum chamber with upper section removed, showing

fused silica tube and alignment mechanism.

ber wall on O-ring seals. The clear apertures were

reduced to areas about 13 mm high by 8 mm wide by

masks.

The vacuum connections to the vacuum chamber

were made through a special flange and O-ring seal near the midheight of the chamber. A copper tube

connected to the flange led to a 19-mm valve supported by the chamber. A detachable connection from

the valve permitted coupling the system to a mechanical pump by a flexible hose. The system was designed

to present a minimum throat diameter of 19 mm. The

vacuum pump was a two-stage mechanical pump

with a free air capacity of 140 liters per min.

The vacuum chamber was equipped with a thermocouple-type vacuum gage inserted in the chamber

wall at a point about 24 cm below the connection to the vacuum pump. Once the vacuum chamber had

been outgassed it was possible to attain a pressure of

about 0.13 N/m^ (1 fxm of mercury). The vacuum pump

was operated continuously except during the length

or free-fall measurements. Generally the leakage rate

for the system ranged from about 0.4 to 0.6 N/m^/hr

at this pressure.

A mercury-in-glass thermometer was imbedded in

the thermal insulation of the vacuum chamber near

the vacuum gage. The thermometer bulb was kept

in thermal contact with the brass wall of the chamber

by a pad of compacted aluminum foil. The thermometer

was covered by the cork insulation except for a slit exposing a portion of the engraved scale. The thermometer scale was graduated to 0.2 deg Celsius and

read by estimation to 0.02 deg. This thermometer was

identical to the thermometers used to measure the

temperature of the length standard.

2.4. Mechanical Design

The frame of the apparatus for the determination

of the acceleration due to gravity consisted of a base

plate of steel 120 cm square and 3.8 cm thick, grouted to the floor of the building and held down by four 2.5-cm

anchor bolts. At each corner of the plate there was

a hoUow square-section steel column 3.7 m high.

Bridging the tops of the columns was a plate of steel

120 cm on each side and 5 cm thick, cut out along the

edges to reduce the weight. Two guide rods for the

carriage were supported in the frame gripped in rings

bolted to the base and upper plates. The internal

diameter of each ring was 7.5 cm; each ring had four

centering screws. The guide rods were 6.35 cm in

diam with a tolerance of 0.1 mm. They were selected

to be straight to within 0.08 mm per m of length.

The spacing between the guide rods was nominally

40 cm. The carriage of the apparatus was about 117 cm long

and was formed of two aluminum alloy plates separated

by vertical tubes of aluminum alloy as shown in figure

3. The two guide rods passed through holes in the

carriage plates and through the interior of the two

aluminum alloy tubes.

At both ends of one tube two pairs of ball bearings were mounted so as to center the steel guide rod

within the aluminum alloy tube. A single pair of ball

bearings at the upper end of the opposite tube constrained the carriage against rotation about the

opposite guide rod. The arrangement of the ball

bearing guides is shown in figure 5. The five pairs of

ball bearings were adjusted against the guide rod with

a slight negative clearance to eliminate any possible

Figure 5. Upper end of carriage showing guide bearings and latch connection.

7
lateral freedom of motion. The motion of the carriage

was found to be extremely linear and yet with only

about 100 g friction in the direction of free fall. Since it was necessary to counterbalance the weight

of the carriage and vacuum chamber about 22 kg

during length measurements, a pair of ball bearing

pulley wheels were attached to the upper main frame

plate near the vertical extension of the ends of the

carriage frame. Weight hangers were connected to two spring steel ribbons 12.7 mm by 0.18 mm which

passed over the pulleys and terminated in fittings

that could be coupled to the frame of the carriage.

A view of the assembled frame, carriage and vacuum

chamber is shown in figure 6. The wood platform for

the observer was supported from the building floor

and did not touch the gravity apparatus.

The main guide rods were alined parallel to the

direction of gravity by the following procedure. Near

each guide rod a fine wire was suspended from the

upper plate of the frame. A mass of sufficient weight

to stress the wire moderately was attached to the lower

end of the wire. The lower end of the wire and the mass

were immersed in a pot of oil to damp out oscillations.

A microscope with reticule graduated to 0.01 mm was

mounted on the counterbalanced carriage and focussed on the wire, first in a plane perpendicular to the

face of the carriage and later in a plane parallel to the face. Microscope readings were taken on the wire

at the upper and lower limits of travel of the carriage

and the position of the guide rod corrected by adjustFlGURE 6. General view of gravity machine.

ing the centering screws in the retaining rings until | the motion of the carriage was parallel to the plumb j line to within 0.02 mm in one m. ; The release latch and initial accelerating spring i were mounted on a subassembly securely attached i to the upper frame plate of the gravity apparatus. The ! arrangement of the subassembly is shown in figure 7. i The latch was designed in such a manner that the

| carriage received no lateral torque or impulse when I the latch was released The latch parts were ground i and finally lapped in place to provide a smooth, clean ! separation at release. | The portion of the latch attached to the carriage | was coupled to it through a bridge frame anchored i at the upper ends of the two aluminum alloy tubes that

' formed the vertical members of the carriage (fig. 5). i This method of mounting insured that the upper plate | of the carriage would not be subjected to bending , due to suspension at its center. Consequently there I would be no sudden release of stored energy to be

| transmitted directly to the vacuum chamber. | When the carriage was suspended from the latch, i

it was also in contact with the plunger of the initial

j accelerating spring. The accelerating spring assembly i consisted of a precompressed helical spring contained ! in an adjustable barrel with provision to adjust both I the force exerted by the spring and the vertical travel

'

over which the plunger remained in contact with the : carriage. These parameters controlled the amount of i separation between the silica tube and the vacuum | chamber support cone. As adjusted in practice, the

spring exerted a force equal to approximately ten

percent of the weight of the carriage and followed

the carriage for about 8 mm in its downward travel

after release of the latch. The spring assembly appears i as the knurled and threaded barrel located just above

j the latch in figure 7. | The range of the acceleration spring adjustment was | such that it was possible to adjust from a condition i where the silica tube separated only momentarily i

Figure 7. Latch assembly with accelerating spring.

8
from the supporting cone to a condition where the

last aperture on the silica tube was no longer in line

with the vacuum chamber optical window at the time

it passed the sensing station. The actual operating

condition was chosen at an intermediate value where

the observed transit times were independent of the

accelerating spring adjustment.

The vacuum chamber was attached to the carriage

of the apparatus by centering screws so arranged

that the upper and lower flanges of the chamber each

rested against two adjustable screw stops on the

longitudinal and transverse axes of the carriage. At

each end a locking screw acting along a line at 45° to the screw stops held the flange in firm contact. This

arrangement permitted the entire vacuum chamber

to be removed from the carriage and then replaced

without disturbing the alinement.

In the process of alining the vacuum chamber in

the carriage, rotational orientation was established

by observing the superposition of the images of a point light source reflected from the upper optical

window of the chamber and from a glass flat in contact with the upper cross plate of the carriage. The

optical axis of the sensing station was alined to be

perpendicular to the same reference plane.

The apparatus was provided with a pneumatic

receiving cylinder on the lower plate of the frame

and a plunger attached to the lower crossplate of the

carriage to bring the carriage to a stop at the end of

its fall. The plunger was machined to fit the cylinder

with a radial clearance of about 0.5 mm. A lubricated

O-ring in a groove in the plunger completed the air

seal between cylinder and plunger. The receiving

cylinder was 56 cm long and 10 cm in diameter. A

5-m length of high-pressure type rubber tubing was

connected to the cylinder through a port at the lower

end. The other end of the tubing was connected to a closed-end section of larger diameter tubing. This

combination provided damping to reduce the tendency

of the carriage to rebound after being brought to a stop.

2.5. Optical System

The optical system of the position sensing aparatus

is shown in figure 8 with the photomultiplier cell

housing slightly retracted. This assembly was mounted

on a steel bracket attached to a concrete pier directly

behind the gravity apparatus (see figs. 3 and 6). The

lens, a 100-mm microscope objective lens, focussed an image of the aperture plate edge on a slit mounted

in the focal plane of a microscope eyepiece in the

viewing system housing. This arrangement permitted an observer to focus the instrument by eye and to aline the slit with respect to the image. Once the

alinement was completed, the optical system was

terminated by the photomultiplier cell mounted in

the sliding housing and making a light-tight seal to the viewing housing. The slit in the viewing system

was approximately 4.5 mm long and 0.075 mm wide.

The width of the slit was not critical and could have

been much narrower since the output of the photomultiplier reached a maximum value in a 10-/LAm travel

of the silica tube. As the magnification of the viewing

Figure 8. Optical system with photomultiplier assembly.

lens was approximately unity, a slit 10 to 12 /xm wide

would have been adequate. The wider slit was chosen

in preference, however, since during the preliminary

adjustment by eye, a narrow slit produces diffraction

patterns making alinement and focussing of the

images diflicult.

The photomultiplier tube used was a 10-stage,

head-on tube having a maximum response in the spectral region of 400 to 500 nm. The tube was operated

at a cathode potential of 900 V. The photomultiplier

tube anode was connected directly into a coupling

amplifier. The coupling amplifier provided a lowcapacitance, 51,000 Cl load on the photomultiplier

anode. The anode potential was 6 V. The output of

the coupling amplifier was connected to a 90-Q, ter- minated coaxial cable leading to the timing system.

The light source was an incandescent filament,

galvanometer lamp operated at 6 V, direct current.

The lamp had a straight, helical coil filament. The lamp

was housed in a case having a lens on a focussing

mount. The lens projected a roughly parallel beam of

light which, with a vacuum chamber window in line

with the axis, passed through the chamber window,

through the transparent portion of the fused silica

tube and provided a uniform illumination to the aperture edge plates mounted on the opposite side of the

silica tube. The light source was mounted on a special

telescope stand to facilitate levelling and positioning.

2.6. Electronic System

The power supplies employed for the apparatus

included a low voltage direct current power supply

for the light source and a high voltage direct current

power supply for the photomultiplier tube. The

signals from the photomultiplier tube were coupled

by a coaxial cable into the input amplifier of a fastrise cathode ray oscilloscope. The triggering circuit

of the oscilloscope was adjusted to start the sweep

circuit at some potential level of the linearly rising

(negative) signal from the photomultiplier tube. The

9
timers were then triggered from the sawtooth sweep

circuit of the oscilloscope. This arrangement offered a high degree of flexibility in adjustment plus very

good isolation between the timers and the photomultiplier tube.

The timers were especially designed for the apparatus and sequence of signals employed. A master unit

contained all the gating circuits required, a 1-MHz

thermostatted quartz crystal oscillator coupled to a 10 X frequency multiplier, the shaper, and a seven- place decimal counting unit. A second slave unit

carried another seven-place decimal counting unit.

Both counting units had a least count of one-tenth

of a microsecond.

The timers were so coupled that on receipt of a

signal from the oscilloscope the gates to both the

master unit and the slave unit were opened, starting

both decimal counting units. Upon receipt of a second

signal, the gate to the decimal counting unit in the

master timer was closed, leaving the gate to the slave

unit open. The receipt of a third signal closed the

gate controlling the slave unit.

The timers were provided with an optional circuit

such that an interval displayed on the indicator panel

could be retained while the gates were reset for a new timing sequence. Subsequent intervals were then

added to the intervals already displayed. This circuit

was not employed in the measurements on the falling

silica tube but was utilized in assessing the errors of

the gating circuits as described later.

3. Reference Standards

3.1. Length

The standard of length for the determination was

an invar, 1-m line standard having an H-shaped crosssection. The scale was fabricated and ruled by the

Societe Genevoise d'lnstruments de Physique of

Geneva, Switzerland, in 1959. The scale was ruled

in 1-mm intervals with line widths of about 5 fim. The

coefficient of expansion of the invar was determined

as 0.00000123 per degree Celsius.

The lengths of 0-300 mm and 0-1000 mm intervals on the scale have been compared to the standards

of the United States periodically since the scale was

received in February 1960. The results are shown in

table 1. It was considered that the scale was very

stable for an invar standard and that the results

indicated no progressive drift in the lengths. Accordingly, the means of all of the reported corrections

have been applied to the measurements of the fused

silica tubes in the gravity determinations.

3.2. Time

The standard of time for the determination was

based on the National Bureau of Standards frequency

standards, since only time intervals were measured.

The crystal controlled oscillator of the master timer

was compared to a 1,000-Hz signal derived from the

Bureau's quartz oscillator, which, in turn was comTable 1. Calibrations of invar line standard No. 5727

Date of

Corrections to nominal lengths Estimated limits calibration of error 0 to 300- mm

interval 0 to 1000-mminterval

September 1960 October 1961 November 1962

April 1964 December 1965

m X 10"«-0.37

1") -.22

-I-.03 -.20

mX 10-"+ 1.16 + 1.10 + 1.55 + 1.55 + 1.29

m X 10-« ±0.20

±.20

±.40

±.30

±.30

Mean correction Standard deviation -0.19

.17 + 1.33

.21

^ No determination made of 0 to 300-mm interval.

pared on a daily basis to the WWV frequency standard.

The comparison of the master timer with the

1,000-Hz reference signal was accomplished by start- f ing the timer with a single sweep of the oscilloscope \ triggered from the reference signal. After an interval [ of approximately 100 seconds the oscilloscope sweep t was reset and again triggered by the 1,000-Hz signal. |' Even though the interval displayed was not exactly f 100 seconds, it was some integral multiple of 1,000 ^is. ? Since the diff^erence between the crystal oscillator of F the timer and the frequency standard never exceeded |i 50 jLts in 100 s, there was no ambiguity in determining ! the relative error of the timer frequency base. * The crystal oscillator was checked prior to each f free-fall determination and again after it. Although, |f as indicated above, the oscillator was always main- f tained within 5 parts in 10^, the observed corrections t were applied to aU time interval measurements. i

I 4. Techniques of Measurement ,

4.1. Length .

Measurements of the two lengths, Li and L2, were

carried out using the same apparatus employed in

the free-fall determinations. The weight of the carriage

was balanced by means of the steel tapes running over

| the frame pulleys with appropriate weights on the | other ends of the tapes. The invar length standard

j was attached in holders beside the vacuum chamber

in such a position that the rulings on the speculum

surface could be viewed by a reading microscope

attached to the support table for the position sensing

apparatus. A fine motion drive was attached to a guide

rod and served to advance the carriage by the small

motions required.

Basically the procedure consisted of moving the

carriage to a position where the lower aperture on

the silica rod was in a position with respect to the

sensing station such that the output signal from the

photomultiplier cell was at the same potential as that

which would trigger the oscilloscope-timer combination. At this point a reading was taken on the zero

ruling of the invar scale with a filar micrometer on

the reading microscope. The carriage was then

moved on to bring the second aperture into line with

the sensing station and a microscope reading taken

10
on the 300-mm graduation of the invar scale. The

process was repeated again for the upper aperture

and the 1,000-mm graduation.

It will be seen that this process is one of measuring

j how far the silica tube must be moved to produce

, the triggering signals rather than of measuring its length. This method of measurement has the advantage

of including the effects of vacuum on the silica scale

as well as any possible displacement of the images

by the optical windows of the vacuum chamber. More

important, since there are complex diffraction effects

at a shadow edge, the measurement is related directly

to the conditions present in the experiment and

measures directly the quantity sought.

The reading microscope used for the observations on the invar scale was mounted on the base plate for

' the optical system. Provision was made for the microscope base to be attached on either side of the position

sensing system. The microscope was mounted on a special base with a rack and pinion focussing control

and a fine adjustment screw. The microscope tube

length was 215 mm with a 22.5-mm objective and a

12.5x-filar micrometer eyepiece. The filar micrometer

eyepiece was equipped with double cross-hairs spaced

the equivalent of 12 /xm in the object plane. The

micrometer drum was graduated into 100 divisions

I per revolution. The reading microscope was calibrated

j on an auxiliary scale and had a factor of 1.112 /xm per

division. During observations the micrometer drum

was read to one tenth of a division. In no case was it necessary to measure differences exceeding 50 divisions with the filar micrometer eyepiece.

During length measurements it was necessary to locate the carriage in a position where the output

from the phototube amplifier was at the potential

required to trigger the timing system. A direct current

voltmeter with a trimmer was connected in parallel

to the output of the amplifier and adjusted so that it indicated full scale at the maximum output of the

amplifier. The triggering potential was then set to be

at the half scale reading. The indication was such that

the small graduations on the voltmeter corresponded

to movements of about 0.2 jxin at the carriage. The

position could be set to a small fraction of a voltmeter

division with the aid of the fine-motion drive. The

meter was slow enough in response to mask the effects

of photomultiplier tube noise and building vibration.

The fine-motion drive which made possible the

adjustment of the carriage to the final position for

reading the length standard consisted of a hinged

brass clamp which could be opened, slipped around

a steel guide rod and locked in place. A bronze spring

provided sufficient frictional drag on the smooth rod

to hold the clamp against the 100- or 200-g unbalanced

weight of the carriage, yet permitted the clamp to be

slid by hand as a coarse adjustment. Attached to the

clamp was a flat wedge, driven by a fine thread screw,

actuating a small push rod to move the carriage.

4.2. Temperature

The invar length standard was equipped with two

mercury-in-glass thermometers, one mounted near

the center of the 0 to 30-cm length and the other near

the upper end of the scale. The thermometers were

graduated to 0.2 deg Celsius and had errors less than

0.02 °C at the ambient temperature of the room

(22 °C). The thermometers were mounted against the

back surface of the web of the H-section of the length

standard; the bulbs were packed in aluminum foil compressed against the surface of the invar and

covered by 6 mm of cork sheet up to the graduated

portions of the stems. The exposed portions of the

cork pads were covered with a layer of aluminum foil. A similar thermometer was mounted in contact with

the lower section of the vacuum chamber. The chamber

was covered by a 6-mm layer of cork sheet topped by

a double layer of aluminum foil. While it was recognized that the temperature of the silica tube was not

necessarily the same as the temperature of the

chamber wall, it was assumed that the tube temperature maintained a reasonably constant relationship

to it, once thermal equilibrium was established. The

temperature of the vacuum chamber walls during

length measurements averaged about 0.2 °C less than

that during the free fall measurements. This difference

probably was caused by the larger number of pieces

of electronic equipment operating during the free

fall measurements.

4.3. Compensation for Guide Curvature

The two guide rods, previously described, on which

the carriage ran were of a special case-hardened steel,

centerless ground and selected for straightness. The

diameters were 63.47 ±0.020 mm and the rods were

straight within 80 /i-m per meter. These tolerances were quite adequate for operation in the free-fall

phase of the experiment but the slight curvature of

the rods was sufficient to make an appreciable error

in the length measurement. The error is proportionate

to the product of the angle of rotation of the carriage

and the distance between the axis of the invar scale

and the silica tube.

If the position of the silica tube and the invar scale are interchanged with respect to the direction of rotation, the algebraic sign of the error is reversed. Thus

the mean of two measurements with the scale interchanged between them should be free of the error due to curvation of the guide rod, provided that the

rotation of the carriage is the same in both sets of

measurements.

The carriage of the gravity apparatus was provided

with duplicate sets of holders for the invar scale, one set on each side of the vacuum chamber equidistant from the centerline of the silica tube. A length

measurement consisted of two sets of measurements,

one for each location of the invar scale. The final

computed lengths from each set were averaged to obtain a working value. Ordinarily the first set of meas- urements was made at the start of a day. Upon completion of the first set, the invar scale was transferred

to the opposite set of holders; about five hours later

the second set of measurements was made. The order

in which the two positions were occupied was reversed on alternate days of length measurement.
Analysis of the length measurements made on four

different fused silica rods indicated that the length

deviation due to curvature of the guide rod vv^as about

1.6 fxm at the 30-cm length and about 0.7 />tm with

opposite sign at the 1-m length.

5. Experimental Procedure

5.1. Length Measurements

In the plan of the experiment, length measurements

and free-fall measurements were made on alternate

days. A set of length measurements consisted of ten

independent determinations of the two lengths, the

silica tube being reseated and realigned and the

reading microscope refocussed between determinations. At the conclusion of the set the invar length

standard was transferred to the opposite side of the

carriage and, after a lapse of about five hours, a second

set of ten measurements was made.

The procedure in each determination was always

the same. After reseating the tube and focussing the

microscope, the thermometers were read and the

temperature recorded. The carriage was then moved

to a position where the first or lower aperture was at the reference level as indicated by the meter connected

to the output of the photomultiplier tube amplifier.

The zero graduation of the invar scale was centered

between the cross-hairs of the filar eyepiece and the

filar drum reading recorded to the nearest tenth of a

division. The carriage was then moved to the second

aperture position and the corresponding filar reading

recorded. The process was repeated for the third

aperture and the carriage returned to the first aperture

position where the initial reading was repeated.

For the calculation of the two lengths, any change

between the first and final lower aperture readings

was assumed to be due to thermal drift, and values

corresponding to the second and third aperture readings were obtained by interpolation with respect to the average time required for the operation. Any run

in which the first and final lower aperture readings

changed by more than 2 /xm was discarded as such

changes were apt to be caused by actual movement of

the silica tube in its seat. The lengths of the 30-cm and 1-m aperture distances were computed from the observed filar readings, the

calibrated lengths of the invar standard intervals and

the temperature correction for the invar standard

(1.23 X 10"^ per degree Celsius). The measured lengths

of the silica tube aperture intervals were corrected to 22 °C for comparison using an experimentally determined coefficient of 0.64X10"^ per degree Celsius.

Since the temperature at which the free-fall measure- ments were made was close (within 0.2 °C) to that

of the length measurements, coefficients of expansion were not determined separately for each of the four

silica tubes used in the experiment.

5.2. Timer Adjustments

During the free-fall determinations, the times for

the passage of the 30-cm length interval and for the

1-m interval were obtained from the two seven-decadel

timers already described. The smallest unit of time sol displayed was one-tenth microsecond. Two corrections! were applied to the time measurements. A correction I for frequency of the crystal oscillator was applied,!

although it was negligible compared to other sourcesli

of error. A correction for differential delay in opening|)

and closing of the gate circuits was also necessary.!]

The indication in the unit which displayed the timeji

interval for the 1-m length was consistently short by l nearly one-tenth of a microsecond due to an electronics

delay in opening the counting gate for this bank. ] The gating errors for time interval meters were

assessed by presenting the system with a series ofj

pulses generated by a decimal dividing network and ai

crystal controlled test oscillator. The pulses were!/

generated at intervals of 10 ms and the timers meas-8

ured the time for a single and double pulse interval, i A total of one hundred such measurements were sum- med in the counter banks of the timers. Since the

mean of such a group of measurements tends to eliminate random effects, the result should reflect

i

the systematic error in the time measurement system, i It is assumed that the pulse source contained only I random variations since it was a continuous system I and the test oscillator was synchronized with the

time interval crystal frequency to better than one part

in ten million.

5.3. Free-Fall Measurements

Free-fall measurements were made on days alter- nating with length measurements. The free-fall meas- urements normally consisted of two sets of 16 determinations, one set being made at the start of the day

and one set at the end of the day.

The first steps in the free-fall measurements en-, tailed determinations of the crystal frequency error and the mean gating errors for both channels. The

timing system was connected to the 1,000-Hz reference

signal and the time for approximately 100,000 cycles

measured. The input and triggering circuit used was

the same as that used for the photomultiplier cell. The departure from the nominal multiple of 1,000 fxs indicated the crystal frequency error in 100 s. Single

cycle measurements of the period of the 1,000-Hz

standard signal, as received, all fell within a band of

±0.2 /LIS, about the average value of the period. A

group of measurements over 100 s (approximately 10^

cycles) showed a standard deviation of 0.9 ^is, approximately one part in 10^. After the crystal frequency

error was measured, single and double individual

period measurements were made on a decade divider

pulse signal at a frequency of 100 Hz. The timers were operated to total 100 such measurements and

the mean gating errors computed from the two sums

indicated by the timers.

After these values had been recorded, the valve

in the vacuum pump line was closed and the pump

disconnected. The vacuum pressure was read, the

gage indicator disconnected, and the timing system

connected to the cable of the photomultiplier cell.

12
At this point the system was ready for the time

measurements on the falling silica tube. The silica

j tube was alined in the vacuum chamber by means of

the upper guide bearing mechanism and the guide

bearing was retracted and locked in place. The

vacuum chamber wall temperature was recorded and

the release latch opened, permitting the carriage to

fall. At the conclusion of the fall the carriage was

immediately returned to the latched position and the

! two time intervals recorded. The silica tube was re-

[ ahned in the chamber and the apparatus was ready

for the next determination.

During the set of measurements, the wall temperature of the vacuum chamber was recorded for every

fifth drop of the carriage. At the conclusion the

vacuum pressure was read and the frequency and gate

error measurements repeated. The entire set of meas- Iurements required about three-quarters of an hour.

6. Results

6.1. Experimental Data

Four different fused silica tubes were included in

the experiment. There were five sets of length measure- ments for each tube, a set consisting of two sub-sets

made with the invar length standard on alternate sides

I of the vacuum chamber as outlined in the section on

' length measurements. Bracketted between each pair

of sets of length measurements were two sets of freefall measurements. A set of free-fall measurements

consisted of about sixteen individual free falls of the

apparatus. A few sets contained only fifteen observaI tions and some sets contained as many as twenty-one.

i Since five sets of length measurements provide four

! adjoining pairs, there were normally eight sets of

free-fall measurements for each tube.

There were two exceptions to this general rule. In the case of tube No. V, four free-fall measurements

were included between the first two pairs of sets of

length measurements, changing the total to twelve.

In the case of tube No. VI, a readjustment of the optical

system of the position sensing station became necessary after the conclusion of the third set of length

measurements. The adjustment was made and a new

set of length measurements inserted to go with the

succeeding free-fall measurements.

A summary of the length measurements for the four

silica tubes is given in table 2. Each entry in the table

represents the mean of two sub-sets of 10 independent

measurements. From the standard deviations of the

j individual sub-sets an estimate was made of the

' average standard deviation of the measured length

] intervals reported in the table. The standard deviations were approximately 0.14 /u,m for both the 30-cm

and the 1 -meter intervals. It is evident that the variations in length appearing in the table are partly

random and partly due to real changes. In order to obtain adjusted length values corresponding to the

free-fall measurements the data were treated as though

all differences were real and as though the changes

took place at a uniform rate during the course of the

! intervening free-fall operations. Accordingly, the adTable 2. Measured length intervals for fused silica tubes

Tube

no. Date Measured lengths It 22 °C

u L,

Meter Meter

Ill May 21, 1965 0.999 950 16 0.299 953 04 May 24' 950 32 953 62 May 26 949 22 953 53 May 28 QdO AA 953 67 Jun 2 949 66 953 92

IV Jun 4 1.000 018 29 0.299 993 42 Jun 7 017 60 993 34 Jun 9 017 58 993 37 Jun 11-12 ni Q 0^ Ulo zo 993 26 Jun 21 017 75 993 32 V Apr 13 1.000 021 39 0.300 012 10 Apr 16 022 48 012 48 Apr 29 021 55 012 12 May 3 022 13 012 04 May 5 023 14 012 18

VI May 7 1.000 050 76 0.300 018 34 May 10 050 20 018 4S May 12 049 70 017 98 May 14 049 62 022 14 May 17 049 66 021 50 May 19 049 51 021 49

Note; Between the observations on the dates of May 12 and May 14 an adjustment was made to the lens diaphragm of the viewing system.

justed length intervals were calculated by linear interpolation between adjacent measured length intervals.

The adjusted values were then corrected from 22 °C to the observed chamber temperature for each set of

free fall measurements. The adjusted and corrected

length interval values appear in table 3. The time intervals presented in this table are the

mean times determined from each set of free-fall observations after correction for both frequency and

gating errors as described in the section on time

measurements. The corresponding values of the

acceleration due to gravity are calculated from the

relationship

L2 Li^ _ h tj_

The values in the final column are the standard

deviations of the gravity values computed from the

distribution of the pairs of individual free-fall time

measurements about the two mean intervals for each

set of free-fall observations. The standard deviation

of an individual free-fall determination may be calculated by multiplying the standard deviation of the

mean by the square root of the number of observations in the set, given in parentheses.

The mean values of the gravity measurements for

each of the four tubes are presented in table 4. The

standard deviations for the means were calculated from

the distributions of the individual set means. The

final result was computed as the unweighted mean

of the four tubes. The mean of the 36 individual sets gives the same value.

I

I 13
Table 3. Free-fall time and length measurements

I

I

i

Tube

No. Date1965

Eastern standard

time

Chamber

temperature

Adjusted lengths corrected to chamber

temperature Mean time intervals from free-fall observations Mean

acceleration Standard deviation

of mean and number

of observations u ii I,

°C Meter Meter Second oeconu. cmjs- cm/52

Ill Lvi<iy 719/735 21.98 0.999 950 19 0.299 953 18 0.348 391 1 1 0 1 SS 733 04 0 onno ( 1 «\ 22 22 12 950 36 953 50 394 77 735 92 1007 .uui 1 1 10 25 718/735 22^12 950 12 953 62 395 19 736 39 J019 .0011 (16) 25 1417/1435 22.14 949 59 953 58 398 52 738 96 .1012 .0015 (15) 27 22.18 949 32 953 57 389 84 732 47 1025 .UUl U I ID, 27 1 SI 9/1 "1^3 22.07 949 50 953 67 394 05 735 52 0997 .uuio 1 10, 29 730/748 22.20 949 63 953 77 393 86 735 40 .0998 •uui nni 1 ^^f\\ 1 1 10, 29 1938/1957 22.20 949 73 953 90 394 47 736 06 1028 on in i\f\

IV Jun 5 710/734 22.19 1.000 018 24 0 9QQ QQ^ 44 59 0 1 SS 71

1

97 QRO lOOU 0u.uuuo onofl ( 1 (\\

1 10 5 1Q37/1QSS 22.18 017 89 993 39 361 20 714 09 1026 .uuuD ( 10 8 749/815 22^15 017 70 993 38 365 96 717 66 !ioii .0007 (16) 8 1450/1530 22.12 017 66 993 38 369 53 720 38 .1011 .0007 (16) 10 22 08 017 85 993 37 370 44 721 10 1022 .UUUD \lo 10 1402/1422 22.16 018 13 993 31 370 38 720 98 .1021 OOOS (17^ 19 73S/ROO 22.35 018 34 993 35 361 81 714 41 .1016 ooin (ifi 20 919/936 22.21 00 01 993 34 364 30 716 28 1002 .UUU/ (ID,

V Apr 14 «1 9/ft9S 22.20 1.000 021 66 0.300 012 19 0.348 606 28 0.155 906 84 QHO 1000 0U.UUU 0007 i\f\\ t 1 1 D, 14 1 SI S/1 S97 22 16 021 90 012 27 610 50 910 06 1003 .UUU/ nnn7 (i^^i (ID 15 Rflfi/R9n ouu/o^u 22 14 022 16 012 37 613 76 912 45 .UUU/ 0007 t^(\ (ID 15 1 704/1 790 22 00 022 34 012 43 613 83 912 62 .UUUO (10 26 14,1 O/IAQQ 22.06 022 40 012 45 614 45 913 07 1006 000c; .UUUD I ID 27 7S6/fi30 22.12 022 21 012 36 620 33 917 60 1016 .UUUo OOOH 1 1 ft 110 27 1605/1621 22.08 021 95 012 28 622 82 919 S3 ]l019 .0008 (16) 28 745/810 22.17 021 78 012 19 622 82 919 53 .1020 .0006 (16) 30 739^759 22.15 021 80 012 13 626 86 922 61 1025 .UUl 00111 i^f\ ( 10 30 IdSft/l SI s 22 20 022 uiz 1 n A90 ZZ 1002 .UUlU 0010 i\fx (lO May 4 71 1/724 22.20 022 51 012 12 629 71 924 59 1013 ,uuuy OOOQ i\f\ (lo 4 14Sn/I SI A 22.22 023 03 012 18 629 21 924 29 .1039 .UUUJ OOOS (^f^ |1U

VI May 8 22.22 1.000 050 76 0.300 018 42 0.348 267 64 0.155 646 73 qon lOlQ u.uuuo \1U, g 22.24 050 49 018 49 271 03 649 31 ,1007 .UUUO onoa (91 \£. I 11 712/730 22.18 050 20 018 39 275 95 652 88 ]0974 .0009 (16)

1

1

049 93 018 13 279 74 655 93 .lUlO .UUUO (

1

15 733/802 22.14 049 74 022 01 310 18 680 52 .1007 .0009 (21) 15 1545/1610 22.17 049 74 021 69 304 69 676 30 .1020 .0016 (20) 18 757/814 22.15 049 72 021 53 313 35 682 81 .1016 .0008 (16) 18 1522/1541 22.18 049 67 021 52 310 71 680 64 .0986 .0011 (16)

Table 4. Summary of gravity values

Tube Number Mean acceleration Standard deviation No. of sets due to gravity of the mean

cm/s- cm/s^

Ill 8 980.1012 0,0004 IV 8 980.1015 .0003 V 12 980.1012 .0004 VI 8 980.1005 .0006

Unweighted mean

of four tubes 980.1011 0.0002

7. Sources of Bias

There are several potential sources of systematic

error in an experiment of this type. Although the

amounts of error contributed cannot always be evaluated accurately, limits can usually be set for them. It

is considered that the principal contributions to error

in the results of this measurement are the following:

longitudinal vibration of the falling tube, electrostatic

and magnetic forces, residual air in the vacuum chamber, lateral deflection of the faUing object, effects

from the optical windows of the chamber, time constants of the electronic circuits, and uncertainty in

the length standards and measurements.

7.1. Longitudinal Vibration

At the instant of release the fused silica tube rests

on the support cone which, upon release, moves!

downward with an acceleration about 10 percent in I excess of gravity. It is reasonable to expect that the

j release of the slight compression of the silica tube

| would cause it to vibrate longitudinally while falling, i As the displacement of the various parts of the tube

!

would cause an error in time indication, the magnitude '

of the error is worth examining. . The largest amplitudes would be those due to the

first mode of vibration in which there is a maximum

amplitude at the extremities of the tube and a node t at the center. For a fused silica tube or rod with

density 2210 kg/m^, modulus 7.2 X lO'ON/m^, and a

| length of one meter, the undamped frequency of the

first mode is about 2850 Hz, corresponding to a period

of 350 microseconds. The static compression of the

tube relative to the center of mass would be 0.113 /xm

and the maximum harmonic acceleration under these i conditions would be 3620 cm/s^. Since this accelera- i tion is greater than that of the support (approximately ; 1080 cm/s^), the surfaces cannot separate until the

j elastic deformation is reduced to a point where the ' sum of the harmonic acceleration and the acceleration

of the center of mass is less than 1080 cm/s"^.

14
Assuming the limiting case in which the acceleration of the center of mass prior to separation is assumed

to be zero, the separation would occur at about 71

/AS after the release is initiated, approximately 81

percent of the first quarter period. At this point the

common velocity of the two parts is determined by the

velocity of the uniformly accelerated support, about

0.0765 cm/s. The resultant undamped ampHtude of

vibration of the fused silica tube corresponding to this speed and displacement would be 0.045 fjum. The

maximum effect of such an amplitude would result

in an error less than 0.0003 cm/s- in an individual

measurement. Such an error represents the upper

limit possible. The calculation has not included motion

of the center of mass of the silica tube which would

necessarily occur with a consequent reduction in the

harmonic velocity at the time of separation. Furthermore, the silver conductive coating on the tube would

add considerably to the damping of the vibration.

It is considered that errors of the above magnitude

would not occur consistently in a long series of meas- urements and can thus be considered neghgible.

7.2. Electrostatic Forces

It has been the experience of most investigators

in free-fall gravity experiments that electrostatic

forces constitute a major source of difficulty. The

separation of any two surfaces, particularly in a vacuum, usually leaves them in a charged condition.

It is not necessary for the surfaces to slide in order

to produce the charges. If one or both surfaces are poor conductors, the charges may remain highly

localized and the electrostatic forces arising from the

interaction of the intense fields may be quite substantial. The introduction of radioactive materials into

the chamber offers little help as the gas pressure is too low to afford many ions and the time too short for

them to dissipate the charges effectively.

In some early experiments with the apparatus

described here, a fused sihca rod was provided with

a polytetrafluoroethylene tip. The charge generated

by the separation of the tip from its seat was sufficient

to draw the falling rod several millimeters out of its path during the 0.5-s fall. The electrostatic charge problem can usually be

solved by making the entire falling object electrically conductive. In the case of the fused silica tubes

employed in this experiment, the objective was at- tained by coating the transparent tubes with silver

conductive paint of the type used for printed circuit

assemblies. The end-to-end resistance of the tubes

ranged from 4 to 10 Cl. The tubes were coated on the

outsides leaving uncoated only small areas in line

with the slit apertures. Since it was possible to vary

the separation between the tube and the support cone during the fall by changing the force applied

by the initial accelerating spring, it could be ascer- tained that any remaining electrostatic attraction was

below the limit of detection.

7.3. Magnetic Forces

Magnetic forces are not usually a substantial factor

in free-fall gravity measurements provided the falling

object can be constructed of nonferromagnetic materials. In the experiment described here, three of

the tubes employed contained small amounts, less

than 300 mg each, of steel in the form of thin plates

for the aperture edges. The other tube. No. Ill, contained no ferromagnetic materials. Since the magnetic

field of the earth was fairly uniform along the line of

fall of the silica tube, it is considered that the net

effect of magnetic attraction on the tube was negligible.

Effects caused by eddy currents would likewise be

small. An analysis by E. C. Lloyd ^ of the National

Bureau of Standards expresses the average ratio of

the force due to eddy current losses to the gravitational

force, FmlFg, for a conducting tube falling through a magnetic field discontinuity which is short compared

to the length of the tube.

Fg 1.27m(A/)

where A = a.rea enclosed by the tube, square meter,

f = time of fall, second,

(7= circumferential conductivity of the tube,

mho/meter,

/n = mass of the tube, kilogram,

A/ = length of the magnetic discontinuity,

meter,

A// = change in magnetic field strength, ampere/

meter.

For the experiment described here, A = 1.87 X 10"'*m^,

f = 0.35s, o"=100 mho/m, m = 0.1 kg. If one assumes

a value for A/ of 0.01 m and for AH of 12 A/m,

1.3 X 10-13.

7.4. Residual Air

The principal effects of air on a falhng object are the buoyant forces due to the density gradient in the

air and the viscous drag resulting from relative motion

of the object. The buoyant effect with fused sihca is less than 600 parts in one million at atmospheric pres- sure and, since it varies in direct proportion to the

pressure, can be considered negligible at pressures

below 16 N/m^. In the experiment the pressure at no

time exceeded 1.5 N/m^. The average pressure during

the 36 sets of free-fall measurements was 0.4 N/m-.

The viscous drag effects of air on a falling object

are more complex. At pressures above 100 N/m^ the

inertial effects of the gas tend to be the predominant

factor, while at lower pressures the predominant fac- tor tends to be momentum transfer from molecular

collisions with the object. Both effects overlap and

there are no sharp boundaries between them. Cook

[16] (Section 5.3) has discussed several cases and has

shown that, at low pressures for a rod of radius r and

density p, the ratio of the molecular forces to the

gravitational forces would be expected to be less than

Fm Col

jj

Fy 2prg

^ Informal communication to the author, January, 1962.

15
where a is the gas density, C is the average molecular

velocity, and U is the object speed. The gas pressures

present in this experiment were such that the Stokes

law forces would be expected to be of the same order

of magnitude or less.

If a = 5 X 10-12 kg/m3. C = 5 X 10^ m/s,

r = 7.5 X 10-'5 m, p = 2 X IQ-^ kg/m^,

and f/= 2 X 10''^ m/s, the ratio would be of the order of

^= 0.014 X 10-6.

From the estimate above it would be expected that

the total effect due to residual air would be less than

0.03 parts in a million. In the earlier course of the experiment a few determinations were made with atmospheric pressure in

the vacuum chamber. Since the air in the chamber is being accelerated downward at approximately the

acceleration due to gravity it might be expected that

the buoyant effect would be very small. For the limited

number of tests conducted, results at atmospheric

pressure did not differ significantly from those obtained at the normal pressures of 0.2 to 0.5 N/m^.

7.5. Deflection of the Falling Object

Any lateral impulse imparted to the falling tube

would cause it to rotate about a horizontal axis through

its center of mass. Since the reference slits lie both

above and below this point and in a plane about 8 mm

closer to the viewing microscope, the amount of error depends on the plane of rotation.

Rotations in a vertical plane perpendicular to the

optical axis were measured by photographing the fall- ing tube with a 25-^im wire stretched vertically across

the upper slit. The photographs were made through

the microscope of the optical sensing system, using

a short-duration flash synchronized with the falling

tube. The photographs showed random departures on both sides of the average line of fall varying up to a maximum of 0.2 mm.

Rotation in the plane containing the optical axis

could not be measured, but there was no evidence

from the sharpness of focus of the photographs that

the motion exceeded that in the other plane.

The motion observed in the photographs was probably due to a combination of both lateral translation

and rotation at the center of mass. Lateral translation

would theoretically contribute no error. A computation of the error due to pure rotation of

the maximum observed amount was made for motion

both in the plane of the optical axis and perpendicular

to it. The fact that the aperture plates were located off

the axis of the tube makes the computation slightly

different for the two planes. The computed magnitudes

of the errors were approximately the same for both

cases, however, about 8 X 10-'*g.

Errors due to rotation in the plane containing the

optical axis are positive or negative according to the

direction of rotation and thus are not likely to contribute systematic error. Rotation in a plane per-l I pendicular to the optical axis always results in a value \ of gravity which is too high, but it would be expected 1 that the average systematic error would not exceed

j 0.05 mGal. It did not appear feasible to apply a correc- I tion for this error as it could not be determined which I part of the 0.2-mm maximum displacement of the wire

image was due to rotation and which part was due to , lateral translation.

|

7.6. Ground Motion | No attempt was made to measure seismic ground

] motion during the observations. It is considered that

'

the greatest effect would arise from microseisms with ] periods on the order of 0.7 s. It was considered prefer-

^ able to operate the release without regard to the ] existing amplitude and phase of the ground motion i rather than to observe the seismic motion and time

the release for optimum conditions. In effect this

represents accepting a larger but purely random variability in preference to a smaller but less purely random

one.

I One source of ground movement which can operate | to produce systematic bias, however, is the shock | wave resulting from the release of the falling object.

In the present experiment, the mass of the moveable | carriage with the vacuum chamber was of the order

of 22 kg.

The transient disturbance introduced by the release i of the 22-kg carriage was studied with the aid of a ; Johnson-Matheson seismometer having a frequency | range of approximately one to five cycles per second,

j The measurements indicated that the pier to which | the position sensing station was attached experienced

a sharp upward impluse from the release of the car

| and returned to its original level within about 70 to I 80 ms. The amplitude of the motion was on the order

of 0.15 to 0.2 iim. The motion appeared to be heavily i damped as there was no appreciable overshoot on the

return. Random motion of the pier due to other disturbances within the building at times produced

motions of the same order of magnitude.

Since the time measurements for the gravity determination were not initiated until about 118 ms after

release it is believed that the systematic effect of the

shock wave from the release of the carriage was

negligible.

7.7. Effects from Optical Windows

The vacuum chamber was equipped with three

pairs of optical windows to permit observations on

the fused silica tube inside. The windows, previously

described, were 4.5 mm thick. It was at first con- sidered that since any optical displacements which

they produced were included in the static length

measurements they were not Ukely to be the cause of

systematic error. This assumption proved to be far

from correct.

Initial measurements made with the apparatus

produced results which showed a long-term, quasiperiodic variability which could not be accounted for

16
by any reasonable assumption of either errors in the

measurements or secular changes in gravity.

During these measurements the vacuum chamber

was over-constrained in its attachment to the carriage,

the flat parallel ends of the chamber rested between

the two parallel cross members of the carriage with a hght friction fit and were secured against lateral move- ment by centering screws. The latch from which the

carriage was suspended was attached directly to the

upper cross-member of the carriage just above the

vacuum chamber. It was suspected that the chamber

was being loaded eccentrically by distortions of the

carriage frame resulting from release of the latch. If such eccentric loading should cause some elastic

bending of the vacuum chamber during the fall with

a resultant rotation of the optical windows, the effect

could be significant.

The displacement of the image of an object viewed

through a plane-parallel glass plate is sensitive to

tilt of the plate. The displacement for small angles of

tilt is on the order of

rf=w(l-i)

where T is the plate thickness, 6 is the angle of tilt and A' is the index of refraction. For a thickness of

4.5 mm and an index of refraction of 1.5 a rotation of one minute of arc will produce an apparent displacement of about 0.4 )U,m.

On the hypothesis that sufficient flexure of the chamber was taking place to produce window rotations of

this order of magnitude, and that the amount of flexure

would vary with slight differences of temperature between the brass chamber and aluminum alloy carriage, the vacuum chamber mount was redesigned.

The chamber was sHghtly shortened to produce a

positive clearance at the upper end in the carriage

and a single point support was provided at the lower

end on the central axis of the chamber.

The modification effectively eliminated the longterm variations in the observations. Later, when the

apparatus was moved from the older site in the East

Building of the Bureau to the new Gaithersburg site

where the observations reported here were made, the

carriage itself was modified. The connection between

the latch and the carriage was made through two independent yokes which were attached to the carriage

at points directly above the two vertical tubular structures. This arrangement eliminated the flexing of the

cross-members supporting the vacuum chamber.

7.8. Time Constants of the Circuits

The 10-stage photomultiplier tube used in the

measurements had a spectral sensitivity peak in the

400- to 500-nm range and was always used in the

dark-to-light transition condition. Connection to the

coupling amplifier circuit was less than 5 cm long and

unshielded to minimize capacitance loading of the

anode. The anode current was limited to 0.1 mA at

full illumination. The entire phototube circuit through

the coaxial cable connections to the timer panel was

capable of following illumination rise times at least

five times greater than those provided by the gravity

apparatus.

7.9. Uncertainty in the Length Standard

Of the two standards employed in the measurements, length and frequency, the length standard

was by far the more critical in terms of the difficulty

in obtaining the desired precision in its calibrations.

The standard, an H-section scale of invar was ruled

by the Societe Genovoise. The 1-m length was graduated into 1-mm intervals. The width of the engraved

lines was approximately 5 /u,m. Only the 0-, the 300-,

and the 1000-mm lines were used in the measurements.

The calibrations of the scale were limited to the in- tervals defined by those lines.

Table 1 shows the results of the calibrations of the

scale made by the Length Section of the National

Bureau of Standards. No determination was made of

the 300-mm interval in the 1961 calibration because

of the very small change observed in the 1000-mm

interval. The table shows that the standard deviations over the five-year period compare very favorably with

the estimated limits of error for the individual determinations. Since the scale was in active use over the

greater part of the five-year period, it was concluded

that the stability was excellent with no significant

drift. The mean corrections were selected as giving

the most reliable values for the lengths of the two

intervals.

Some estimate of the effect of the different length

determinations obtained for the invar scale can be

made by computing a series of gravity values in which

the successive length corrections are the only variable.

Using the adopted mean corrections as a reference,

the caHbrations of 1960. 1962, 1964, and 1965 would

have resulted in gravity values differing from the refer- ence level by +0.5, +0.7, -1.0. and -0.2 mGal

respectively. The standard deviation of the mean

correction would be 0.4 mGal. This result illustrates one disadvantage in the use of an intermediary standard in a measurement requiring the greatest possible

accuracy.

7.10. Uncertainty in the Length Measurements

As previously indicated in Section 4.1, a determination of the two lengths, L\ and L2, of a silica tube consisted of ten measurements of each length with the

invar scale on one side of the carriage and ten measure- ments of each length with the scale on the other side.

The two sets were averaged to give the final value.

The individual sets showed good internal consistency even though a set required about one hour to complete. The standard deviation of a single length determination of either the 300-mm or the 1000-mm interval

was usually on the order of 0.6 or 0.7 /u.m; thus the

standard deviation of the mean of ten would be on the

order of 0.2 ^im. As indicated earlier, there was usually

some evidence of real change in the interval lengths over the complete series of free-fall experiments.

An interesting side effect in the length measurement

is the small error due to the relative shortening of the

17
invar scale and the silica tube when they are supported at rest in vertical positions. The length desired

is, of course, that of the silica tube in its relaxed condition in free fall. The length of the invar scale is determined in the length laboratory in a horizontal position

while supported at the Bessel (Airy) points. Neglecting

the sHght droop of the bar between the support

points, it can be considered in a relaxed condition

during this comparison. If now the scale and the tube

are supported on their lower ends in a vertical position,

both will be compressed slightly under their own

weight. The changes in lengths are in proportion to the

ratio of the densities of the materials to their respective

moduli of elasticity.

The decrease in length of a rod or tube standing on

its lower end is given by the relationship

M^f j

(Lo-y)dy

where AL is a change in length, p is the density, and

E is the modulus of elasticity. If the origin is taken as the bottom of the tube, at y = Lo the integral has the

value

AL = 0.5^E

and at y = 0.3Lo the integral is AL = 0.255 ^•

E

The fused silica used in the tubes had a density of

approximately 2210 kg/m'' with a modulus of 7.2x10'"

N/m-. Invar has a density of approximately 8000 kg/m''

and a modulus of 14.8x10'" N/m^. The computed

corrections for the tubes and invar length standard are

as follows:

30-cm 1-meter

interval interval

fim fjLm Invar -0.14 -0.26

Silica -0.08 -0.15

Difference -0.06 -0.11

Although these corrections are below the level of

significance in the measurements, the differences were systematic and were applied in the adjustment

of the lengths of the fused silica tubes.

Since the temperature of the silica tube was not

measured directly, the assumption was made that it tended to remain at some relatively stable level in

relation to the temperature of the vacuum chamber

which surrounded it. Some indication of the validity

of this assumption can be obtained from an auxiliary

set of measurements made on a brass tube of identical

size and construction as the fused silica tubes from

which the formal determination was made.

The brass tube was subjected to an identical set of

alternating length measurements and free-fall measurements as were the fused-silica tubes. The set of experi-

} ments with the brass tube followed immediately upon

j the conclusion of the determinations with the four silica i , tubes. The decision was made at the outset that re- 1

j gardless of what result was obtained with the brass

'

tube, the result would not be included in the value of \ gravity to be reported. ;i The length intervals for the brass tube were measj ured at vacuum chamber temperatures ranging from | 21.88 to 22.00 °C and corrected to 22 °C using a co-

| efficient of thermal expansion of 18.5 X 10^ per degree ! Celsius. The results are given in table 5.

| Table 5. Measured length intervals for brass tube

Date Measured len ;ths al 22 °C

Li

Meier Meter

July 2. 1965 0.999 733 06 0.299 962 06

July 7 734 23 962 30

July 12 734 50 962 18 July 14 734 34 961 94

July 16 734 16 962 02

Brass has a temperature coefficient such that a

j change of 0.1 °C would change the 1-m length by nearly

IfjLm. The table indicates that, aside from a single shift

between the first two measurements of the 1-m inter-

| val, the brass tube showed as good repeatability as the i fused silica tubes. Since the temperature coefficient

j of thermal expansion of brass is about 27 times that of i fused siHca, the evidence appears to substantiate the

j assumption that the temperature of the falling object | followed that of the vacuum chamber reasonably well,

j 7.11. Uncertainty in the Final Result

j From the foregoing discussion of systematic error it

\ is concluded that the principal known source of error | of a systematic nature lies in the uncertainty in the 1 value of the length standard. The uncertainty in the | time standard is too small to affect the final result | significantly. ] Effects such as seismic disturbances, lunar and solar | tides and random errors in the time and length measure- ments are included in the variability of the gravity '• values shown in tables 3 and 4. The uncertainty in the

final result is considered to be due principally to the . contributions of this variability and the uncertainty in

the evaluation of the invar length standard. ; From the standard deviations of 0.00021 cm/s^ for

| the length and time measurements and of 0.00039 cm/s^ ' for the length standard, it is estimated that the uncer- \ tainly in the final result is represented by a standard

| deviation of 0.00045 cm/s-. ^

8. Comparison with Other Gravity Values

The mean value of 980.1011 cm/s^ given in table 4

is that determined directly from the measured parameters of the experiment. Since there is a gravity

18
gradient of approximately 0.0003 cm/s^ni, it is necessary to know the efifective height at which the machine

determines the value. An expression has been derived

by Preston-Thomas et al. [10] by which the efifective

elevation above the optical axis at which gravity is determined can be computed. The efifective height,

//, can be expressed as:

.12 \t2-til

+ y (2«2 + 2fi + 3A<)j

where Ho = elevation of the center of gravity of the

silica tube above the optical axis at the

time of release,

At = time from the instant of release to the

instant of initiation of the time measurements,

t2 = time for the 1-meter interval,

ti = time for the 30-cm interval.

In the apparatus Ho had the value 56.8 cm, At was

0.1184 s and t2 and t2 were given typical values of

0.3486 and 0.1559 s respectively. The resulting value

of the efifective height above the optical axis is 14.2 cm. To establish a permanent reference point for the

absolute value at a location suitable for gravity meter

measurements, a location on the floor of the same room

was selected as shown in figure 9. The absolute measNORTH WALLOP BLDG.

3050-KGF

DEADWEIGHT

MACHINE

25,300 LBF DEAD WEIGHT MACHINE

NBS- I 2.204 METERS

ABOVE FLOOR

ROOM 129, BUILDING 202 NATIONAL BUREAU OF STANDARDS

(MARCH 1967)

DOWN

GRAVITY METER STATION NBS-2

ELEVATION 131.042 METERS

LAT 39° 7.'85 N LONG. 77° I3'.20 W

I

INSET SLAB

SCALE

(IN METERS)

urement site is designated NBS-1 and the reference

site is designated NBS-2.

The elevation of NBS-2 was estabhshed by a first order survey by the Levelling Branch of the Coast

and Geodetic Survey, Environmental Science Services

Administration, U.S. Department of Commerce.

Gravity meter connections linking NBS-1 and NBS2 to the national gravity base in the Department of

Commerce Building were made in November 1965

by the Gravity and Astronomy branch of the Coast

and Geodetic Survey. These connections also included

the station in the sub-subbasement of the East Building where Heyl's absolute measurement had been

made.

A summary of the gravity differences taken from the

report of the Coast and Geodetic Survey is given in

table 6. The differences are referred to the national

gravity base, specifically the top of the east elevated

pier in the gravity room of the Commerce Building.

The gravity meter observations were made with three

gravity meters in a double loop including two other

stations in the area. The standard error for the sta- tions NBS-1 and NBS-2 was 0.03 mGal and that for

the sub-subbasement of the East Building was 0.02

mGal.

Table 6. Gravity differences from national gravity base

Station description

Gravity value at the station minus gravity value at the national gravity base in the Department of Commerce Building

Support for optical axis of absolute gravity machine. Station NBS-1

mGal

-3.59

Permanent reference on floor of Room 129, Bldg. 202, Station NBS-2 -Z97

Floor of the sub-subbasement of the old NBS East Bldg. -19.46

Figure 9. Gravity reference station.

The optical axis of the absolute gravity apparatus

was 12.1 cm above the support plate which, in turn,

was 220.4 cm above the floor. The machine, therefore,

measured gravity at a point 232.5 + 14.2 cm above the

floor. An adjustment of + 0.0007 cm/s^ based on the observed gravity gradient of — 0.28 mGal/m of elevation

was appHed to the absolute result to correct it to station NBS — 2. The absolute value of gravity for

station NBS — 2, as determined by the experiment,

and the coordinates of NBS — 2 are:

^=980.1018 cm/s^

Lat = 39°07.85' north,

Long=77°13.20' west.

Elevation = 131.042 m above mean sea level.

A comparison to the Potsdam system may be made

from the gravity meter connections to the national

gravity base in the Department of Commerce Building.

If a Potsdam value of 980.1180 cm/s- is assigned to that station, the corresponding value for NBS —

would be 980.1150 cm/s^. The NBS absolute resuh is 13.2 mGal less than this. This result is in general

agreement with the results of other recent absolute

gravity measurements.

19
9. Conclusion

A measurement of the absolute value of the acceleration due to gravity has been made and referenced to a permanent gravity meter station on the grounds of the

National Bureau of Standards near Gaithersburg,

Maryland. The value estabhshed for the reference

station is 980.1018 cm/s- with a standard deviation of

0.0005 cm/s". This result, transferred to the top of the

east elevated pier in the gravity room of the Department of Commerce Building. Washington, D.C.,

gives a value of 980.1048 cm/s- with the same standard

deviation.

The author is deeply indebted for the helpful advice

and assistance given by the staff of the National

Bureau of Standards during the course of the work.

He is particularly indebted to Mr. J. H. Troxell, who

made the length measurements during the final

determination.

1 0. References

[1] P. G. Roll, R. Krotkov. and R. H. Dicke. The equivalence of

inertial and passive gravitational mass, Ann. Phys., 26,

442-517 (1964).

[2] A. H. Cook. The absolute determination of the acceleration due

to gravity, Metrologia. 1, 84-114 (1965).

[3| F. Kiihnen and P. Furtwangler. Bestimniung der absoluten Gnisze der Schwerkraft zu Potsdam (Venifientlichung des

K. Preuszischen geodatischen Institutes. N. F. no. 27)

Berlin: P. Stankiewicz, 1906.

[4| P. R. Heyl and G. S. Cook. The value of gravity at Washington.

J. Res. NBS 17, 805-39 (1936)RP946.

[5] J. S. Clark, An absolute determination of the acceleration due

to gravity. Phil. Trans. Roy. Soc. London. Ser. A.. 238,

65-123(1939).

\j [6) H. L. Dryden, A reexamination of the Potsdam absolute deter- il mination of gravity, J. Res. NBS, 29, 303-14 (1942)RP1502. I

[7] H. Jeffreys, On the absolute measurement of gravity. Monthly i Notices Roy. Astron. Soc. Geophysical Suppls., 5, 398-408 \ (1949).

[8] P. N. Agalet~skii and K. N. Egorov, Rezul'taty absoliutnykh

opredelenii uskoreniia sily tiazhesti v punkte VNIIM (Leningrad). Izmeritel Tekhn., Vol. for 1956. no. 6. 29-34.

j [9] A. L Martsiniak, "Opredelenie absoliutnoi velichiny uskoreniia i sily tiazhesti po padeniiu zhezla v vakuume," Izmeritel I Tekhn. Vol. for 1956, no. 5, 11-15. | [10] H. Preston-Thomas and others, An absolute measurement of

j the acceleration due to gravity at Ottawa, Can. J. Phys. 38, I 824-52 (1960). i [11] A. Thulin, Une determination absolue de g au pavilion de ) Breteuil, par la methode de la chute d'une regie divisee, [ Ann. Geophys. 16, 105-27 (1960).

[12] J. E. Faller, An Absolute Interferometric Determination of the Acceleration of Gravity. Thesis, Princeton University,

1963.

[13] A. H. Cook, A new absolute determination of the acceleration due to gravity at the National Physical Laboratory, England.

Phil. Trans. Roy. Soc. London, Ser. A., 261, 211-252 (1967).

[14] C. Volet, L'intensite de la pesanteur determinee par I'obser- vation de la chute d'un corps, Comptes rendus de I'Academie

des sciences, 224, 1815-16 (1947).

[15] A. Sakuma, Etat actuel de la nouvelle determination absolue

de la pesanteur au Bureau international des poids et mesures.

International Association of Geodesy. Bulletin geodesique,

n.s., no. 69, 249-60 (1. sept. 1963).

[16] A. H. Cook, Recent developments in the absolute measurement

of gravity. International Association of Geodesy, Bulletin geodesique, n.s. no. 44, 34-59 (1. juin 1957).

(Paper 72C 1-264)

20

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