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MAY xd 1971 NBS TECHNICAL NOTE 577

L<"°'ca

W*tis o«

Method of Calibrating Weights

for Piston Gages

U.S.

JVRTMENT

OF

)MMERCE

National

Bureau

of

ndards

S71

n
NATIONAL BUREAU OF STANDARDS

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\m bureau of stanmks

10 UNITED STATES DEPARTMENT OF COMMERCE

Maurice H. Stans, Secretary

>o , ^7 7

NATIONAL BUREAU OF STANDARDS • Lewis M. Branscomb, Director

Yi * NBS TECHNICAL NOTE 577

ISSUED MAY 1971

Nat. Bur. Stand. (U.S.), Tech. Note 577, 54 pages (May 1971)

CODEN: NBTNA

Method of Calibrating Weights

for Piston Gages

H. E. Aimer

Optical Physics Division

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C, 20402.

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Contents

Page

1. Introduction 1

2. Calibration Procedures 2 2.1 Preparation of Weights for Test 2

2.2 Comparison of Weights being Tested with Standards 3

2 .

3

Data Reduction 4 2.4 Uncertainty of Value of Weights 6

2.5 Checks for Blunders or Gross Errors 7

3. Weighing Methods 8

3.1 Direct Weighing 9 3.2 Direct-Reading Weighing 9 3.3 Substitution Weighing - One-Pan Constant Load Balances. . . 9

3.4 Substitution Weighing - Equal-Arm Balances 12

3.5 Transposition Weighing 13

4. Computation of Mass Value of Weight being Calibrated 18

4.1 Application of Corrections for Standard and Buoyant

Effect 18

4.2 Use of "Transfer Standard" 19

4.3 Added Weights 19

5. Air Density 24

References 25

Figures 26

Appendix 42

in

Method of Calibrating Weights for Piston Gages

H. E. Aimer

Generally weights for piston gages have odd

denominations that are often not readily calibrated by intercomparison methods. Therefore, these weights are frequently calibrated by direct comparison methods.

This paper presents direct comparison methods for

calibrating piston gage weights for use with both

equal-arm balances and single-pan balances. Methods of

estimating the uncertainty of the values obtained are given. Also included are methods of checking for

blunders or gross errors.

Keywords: Balance; buoyancy; calibration; standards;

substitution weighing; transposition weighing; true

mass; uncertainty; value.

1. Introduction

The method of calibrating pressure gage (piston gage) weights

described herein is adequate for most purposes. The method employs simple

weighing designs and includes using corrections for the standards and the

buoyant effect of the atmosphere. Aside from blunders, such as misrecording the balance indication or the weights used, these account for

the major sources of error in the mass value of the weights.

The comparisons may be made on any balance whose precision is adequate

for the requirements of the weights being tested. If the precision of the

balance is not known, the balance can be evaluated by tests such as described in ASTM's "Standard Methods of Testing Single-Arm Balances" [1]*.

Suitable mass standards, with established values or corrections, and

appropriate uncertainties are required. For some purposes, the

manufacturer's stated compliance with a class or other tolerance may be

sufficient. In other cases it may be convenient to calibrate the built-in

weights. Calibration procedures, such as are described in "Testing a Quick Weighing Balance" [2] or the ASTM's "Standard Methods of Testing

Single-Arm Balances" [1] may be used. The various dial settings may be

calibrated according to the method for calibrating dial-controlled weights

described on pages 742/709 of "Testing a Quick Weighing Balance" [2].

Calibration procedures for use with both equal-arm balances and one- pan constant- load balances are described. For one-pan constant-load

Figures in brackets indicate the literature references on page 25.

1
balances, both single and double substitution weighing methods using the

built-in weights as standards, as well as substitution methods using known

weights as standards, are described. For equal-arm balances, both the

substitution weighing and the transposition methods of weighing are

described. The computations are held to a minimum and will give the true

mass value, M in the equation m

Fe = %(1 - p^) + Mf(1 - p|)jkgL + YC,

Pa v . .. „ Pa,

equation (10) of "Reduction of Data for Piston Gage Pressure Measurements"

[3 ] as well as an appropriate estimate of the uncertainty of this value.

Each weight is compared directly with a standard or group of standards of

approximately the same mass as the weight being tested, by any of the

methods described in Section 3. If the standards and the weights being

tested are made of materials having approximately or nearly the same

density, the errors due to buoyant effects are minimized. Each comparison

will involve either duplicate measurements of the difference between the

weight under test and the standard, as in double substitution weighing, or

the same difference will be measured in each of the comparisons, as the

sensitivity difference in single transposition weighing. These differences

can be used to compute an estimate of the precision of measurement.

2. Calibration Procedures

The calibration of weights includes preparation of the weights for

test, their comparison with standards, and data reduction. Each of these

phases of the calibration is treated.

2.1 Preparation of Weights for Test

The weights are prepared for test as follows:

a. Unpack the weights carefully, being sure not to overlook any

of the small weights which may be a part of the set.

b. Clean the weights by wiping with a soft cloth of nonabrasive

material such as high quality cheesecloth. Solvents can be

used to remove any foreign material that is not readily

removed by wiping. Care must be exercised so that only

foreign material is removed. Be certain that the solvents

used will not injure the weights.

c. Place the weights and the standards in or near the balance

so that they will come to temperature equilibrium before the

weighings are made.

d. An information or summary sheet is prepared giving the

following information: (See Sample Summary Sheet, Figure 1)
1. Identify the weights to be tested by the owner,

set designation, and test number if one is used.

The identification of the instrument of which the

weights are a part should also be given.

2. The material of which the weights are made and

their density.

3. List all of the weights in such a manner that,

after calibration, the mass values found for them

may be entered next to the appropriate weight.

4. Any other information needed to make a complete

record of the test.

2.2 Comparison of Weights being Tested with Standards

Weighing Method - This suggested calibration procedure is based on the

double substitution method of weighing (see Section 3) because it can be

used with either one-pan constant-load balances or with equal-arm balances.

Other weighing methods (Section 3) may be used where desirable or where the

double substitution method is impracticable.

Standards - Generally the weights for piston gages have odd denominations not usually found in ordered sets of weights. Hence, the standard

for a given weight may consist of more than one known weight. The buoyant

effect of the atmosphere and associated errors can be minimized if, as far

as practicable, the material of the known weights used for standards is of

the same or nearly the same density as the material of which the weights

being tested are made.

Sensitivity Weight - The mass of the sensitivity weight used depends

in part on the on-scale range of the balance and in part on the mass differences between the weights being compared. The sensitivity weight should

be as large as practicable. Its mass may be from one-fifth to one-half the

on-scale range of the balance and it should be at least twice as large as the largest difference between the masses being compared. It may be

necessary to trim one of the weights by adding small known weights to it to

bring the difference between the masses to within these limits.

Comparison with Standard - Each weight of the set being calibrated is

compared by the double substitution method (or other suitable methods) with

"known" weights whose total mass is nominally equal to the mass of the

weight being tested.

In sets having two or more weights of the same denomination, it is

sometimes convenient to select one of the duplicate weights to be the

"standard" for calibrating the other weights of the same denomination.

This weight will be referred to as the "transfer standard". The "transfer

standard" is calibrated with the "known" weight as mentioned above.
Environmental and Other Data Required - The date and time the work was

done as well as the name of the observer should be entered on the observation sheet. The temperature, relative humidity, and barometric pressure at

the time the weighings are made are also recorded on the observation sheet,

so that buoyancy corrections can be computed and made, as required.

2.3 Data Reduction

Computation of Difference Between the Weight Being Calibrated and the

Standard - The difference between the weight being calibrated and the

standard is given by the difference in scale divisions between the balance

indications for the weight under test and the standard. The method of computing this difference and of converting it from scale divisions to mass

units is given for each method of weighing with the description of that

method in Section 3. But for the following discussion, it will be assumed

that the difference is in or has been converted to mass units.

The difference is expressed as follows:

(Wx

- S) = a where Wx

is the weight under test

S is the standard; it may consist of one or more

weights

"a" is the difference between W and S in mass units.

Application of Correction for Standard - The value of the standard may

be expressed either as one number or as a nominal value and a correction

,

as follows:

Mass Value of Std. = Norn. Value + Correction

In general, the "Mass Value" of the standard means its True Mass value and

its correction means its True Mass Correction unless otherwise indicated.

The computation of the value of the weight being tested, W , is further

described in Section 4.
When the value for S, whether expressed as one number or as a nominal

value and a correction, is in terms of true mass, as it should be, the com- puted value of Wx will be its true mass value within certain limits which

depend on the relative density of Wv and S (see Table 1) . If S and W„ have

the same or nearly the same density, the value found is for all practical

purposes the true mass value. If the correction for S is known only in

terms of apparent mass vs brass, as may be the case for some calibrated

sets of standards, true mass corrections must be computed. This

computation is as follows:

S TM - SAM + Pn (VS

" V

where S. M is the apparent mass vs brass value of the weight, that is

AM the nominal value + the apparent mass correction.

S_,M

is the true mass value of the weight

o is the density of normal air (1.2 mg/cm 3 at 20°C)

nV is the volume of an equivalent mass of normal brass at 20°C 1 B

V is the volume of the standard at 20 °C.

The true mass correction is:

Correction = S - Nominal Value

TM

Buoyancy Correction - Buoyancy corrections are needed to take into ac- count the difference in the buoyant effect of the air on weights of

materials having different densities [4], especially where the densities of

the materials are markedly different. It is a good practice to always

compute (see Section 4) , at least roughly, the magnitude of the correction

to establish the order of magnitude with reference to the desired accuracy.

If the correction is not significant it can be ignored.

If the volume of the weight is not known it may be computed from the mass and density relationship:

«wx Volwv = ET- x wx where Mu is the total mass of the weight, and wx

Dw

is its density at 20 °C.

Similarly the volume of the standard is:

Ms voi s - »j

where M

g

is the mass of the standard, and

Ds is its density at 20°C.

*By definition the density of normal brass is 8.4 g/cm 3 at 0°C; that is

approximately 8.3909 g/cm 3 at 20°C.

5
2.4 Uncertainty of Value of Weights

It is presumed that the weighings are being carried out by means of a measurement process whose parameters (precision, possible systematic

errors, etc.) are known and sufficient evidence is collected to insure that

the process is in a state of statistical control.* For each method of

weighing there will be available a standard deviation to be associated with

a single measurement of mass difference. This standard deviation will be

based on considerable history and would be used in preference to a standard

deviation based on the results of say one day's work. Such a value if

available provides the means for judging whether or not to accept that

day's measurements as being in control.

Uncertainty of Values Found for Weights - The uncertainty of the mass

value of the weights consists of two parts; the uncertainty due to random

errors of measurement and the systematic uncertainty due to the uncertainty

in the value of the standard. The limit of the uncertainty due to random

errors of measurement may be taken to be three times the standard

deviation, <j t where a is the standard deviation of the process.

Therefore

Uncertainty of value = 3o + uncertainty of standards.

Uncertainty figures or statements are valid only if the measurement process

was in a state of statistical control at the time the measurement to which

the uncertainty figure applies was made. Therefore, every uncertainty

figure associated with a mass value must be backed up by data adequate to

determine that the weighing process was in a state of statistical control

at the time the measurements used to determine that value were made.

*For a discussion of the procedures for estimating the parameters of the mass measurement process and for maintaining surveillance of the

measurement process, see NBS Technical Note 28 8, Measurement Philosophy of

the Pilot Program for Mass Measurement, by P. E. Pontius; and NBS Monograph

103, Realistic Uncertainties and the Mass Measurement Process, An

Illustrated Review, by P. E. Pontius and J. M. Cameron.
In many cases the allowable imprecision is so much greater than the

expected performance of the mass measurement process, based on previous

measurements, that the measurement error could not exceed the allowable

imprecision without some evident malfunctioning of the balance. In those cases it may be assumed that the results of the measurements are adequate if

the balance appears to be working properly; but a formal uncertainty

statement would not be made.

2.5 Checks for Blunders or Gross Errors

It is usually desirable to make some kind of check for at least gross

errors in the values being reported. This is especially true where the

standard consists of several weights because a weight may be overlooked

when recording those used.

Checks by Comparing Values - In sets having several weights of the same denomination, one form of check is to compare the values obtained for

weights having the same nominal value. If the value for one of these

duplicate weights is markedly different from the others, this may indicate

that the value is incorrect. For weights such as the "transfer standard"

used as the standard for other weights of the same denomination, of which

repeat weighings are made, the agreement of the results of the repeated

weighings constitutes an acceptable check on the value.

Check by Weighing Groups - This kind of check cannot be used with

large weights whose mass is near the capacity of the largest balance

available. It is useful for weights which can be divided into groups not

exceeding the capacity of the available balances. The procedure is:

a. Divide the weights into convenient groups.

b. Compare each group with appropriate standards.

c. Compare the value found for the group with the sum of the values for the individual weights making up the group. They

should agree within the uncertainty of the measurements.

It is necessary to use some judgment in dividing the weights into

groups. Small weights should not be grouped with large weights.
3. Weighing Methods

Any one of the four weighing methods - direct , direct-reading , substitution , or transposition - may be used for calibrating piston gage

weights. The weighing method used depends in part on the balances and

weights available, in part on the requirements of the job at hand, and in

part on the preference of the person performing the test. Measurements

made by either the substitution or transposition weighing method require

more effort, but are in general more precise than those made by either the

direct or the direct-reading weighing methods. The direct and directreading weighing methods are generally used only where relatively imprecise

measurements can be tolerated and where only the minimum effort possible is

justified.

The following comparison of the direct weighing method with the single

transposition weighing method and direct-reading method with the single

substitution method gives an indication of the differences in precision.

Any weighing requires at least two observations. A direct weighing

requires a no-load reading and a load reading. If the standard deviation

of the mass measurement process, for one observation, is " O " (random

component of the uncertainty is 3a) , then the standard deviation of two

observations combined is a^2 (random component of the uncertainty of the

two observations combined is 3ov2) . In addition there must be added the

uncertainty due to the inequality of the arms. This may vary from a few

parts in a million to one or two parts in ten thousand, depending on the

condition and quality of the balance. In contrast to this is the

transposition method which also requires two observations, but since both

observations are measurements of the same thing, the difference between the

unknown weight and the standard (see Section 3.5, page 13), the standard

deviation of this difference is =- (the random component of the

30\V2 \ uncertainty of the difference is -j=-\ . In transposition weighing the

inequality of the arms drops out and does not become a part of the

uncertainty. Thus, if the same balance is used for both methods, the

uncertainty of the direct weighing method is twice that of the

transposition method plus the uncertainty due to the inequality of the

arms.
Both a direct-reading weighing and a single substitution weighing re- quire two observations; therefore the random portion of the uncertainty is

the same for both methods. In the direct-reading method the masses of the

built-in weights are considered to be equal their nominal mass values (if

corrections are applied it is no longer a direct-reading device) , and the

reading scale is assumed to have a one to one ratio with the indicated mass

units. In a substitution weighing the weight being calibrated is compared

with weights whose mass values are known and the reading scale is calibrated

as a part of the substitution weighing. The uncertainty of the directweighing method is greater than the uncertainty of the single substitution

weighing method by the uncertainty due to considering the nominal value of

the built-in weights to be their actual value plus the uncertainty due to

any reading scale error.

3.1 Direct Weighing

The equal-arm balance is the only type of balance that it is practical

to use for the direct weighing method. First read and record the no-load

,

also called the zero load, indication. Then place the weight being tested

on one of the pans, say the left pan; put sufficient known weights on the

other pan to just bring the balance to an equilibrium condition such that

the indication is the same as the no-load indication. The total mass of

the known weights is the mass of the weight being tested.

3.2 Direct-Reading Weighing

The direct-reading weighing method requires a balance that is capable

of directly indicating the weight of the load on the load receiving element

in mass units. Most one-pan balances meet this requirement. With no-load

on the pan, adjust the balance so that it reads zero; place the weight

being tested on the pan and by proper manipulation of the controls bring

the balance into a condition of equilibrium; read and record the

indication. Since this is a direct-reading balance, the indication is the

mass of the weight being tested, within the capability of the balance.

3.3 Substitution Weighing - One-Pan Constant Load Balances

Two substitution weighing modes for one-pan constant load balances are described. One uses known weights from an ordered set of weights as standards; the other mode uses the balance's built-in weights as standards. The

method of using the built-in weights for standards is the same in principle

as that using the usual known weights for standards though it may appear to
be different. With this method the built-in weights, removed from the

total load on the beam to bring the balance beam to a position of

equilibrium when a weight is placed on the pan, are the standards for that

weight. The dial settings indicate which weights have been removed from

the load.

To avoid negative balance indications or having to change dial

settings during a given weighing, it is convenient to place a small tare

weight on the balance pan and leave it on during the entire series of

weighings. A weight of about one-fifth the full scale may be used.

3.3.1 Single Substitution Weighing - Using Known Weights from an Ordered

Set as Standards (see Figure 2a for example)

.

Observation 1 - Place the weight being tested on the balance pan along

with the tare weight. Release the balance, read and record the

indication. This indication is designated I x .

Observation 2 - Remove the weight being tested, but not the tare

weight; and put enough standards on the pan to bring the balance into

approximately the same position of equilibrium as it had when the

weight being tested was on the pan, without changing the dial setting.

Read and record the indication. This indication is designated I

2

. Observation 3 - Place a sensitivity weight on the pan along with the

standards and the tare weight, if any. Read and record the

indication. This indication is designated I 3

.

is:

The difference between the weight under test, Wx , and the standards, S,

WX

- S - < J

1 " I 2>T^l7= a

where Ws is the mass of sensitivity weight and

"a" is the difference between ^and S in mass units.

3.3.2 Double Substitution Weighing - Using Known Weights from an Ordered

Set as Standards (see Figure 2b for example).- Indications I 1# I

2 ,and I

3 of

the double substitution weighing are obtained from observations 1, 2, and 3

of the double substitution weighing in the same manner as the corresponding

indications for single substitution weighing described above, Section

3.3.1. There is a fourth observation in double substitution weighing which

is described immediately below.

Observation 4 - Remove the standards from the pan, but leave the

sensitivity weight and tare weight on the pan. Put the weight being

tested on the pan with the sensitivity weight and tare weight. Read

and record the indication. This indication is designated 1^.

The difference between the weight under test, Wx , and the standards, S,

is:

10
wv- s «

I I" I

2 + X -- I

3 W

s

'x ^ 2 i7T17 = a

where Ws is the mass of the sensitivity weight and

"a" is the difference between \ and S in mass units.

3.3.3 Single Substitution Weighing One-pan Constant Load Balances Using

Built-in Weights for Standards. (See Figure 3a for example.)

Observation 1 » Place the weight being tested on the balance pan with

the small tare weight, and set the dials at the appropriate settings

for that weight. That is, the balance indication is "on scale" when

it is in equilibrium. Release the balance, if in equilibrium, record

the dial settings. The weights represented by the dial settings are

the standards for the weighing. Read and record the balance

indication. This indication is designated Ij.

Observation 2 - Remove the weight being tested from the pan, leaving

the tare weight: set the dials at "0". Release the balance; read and record the indication. This is the reading with the standard on. This indication is designated I

2

. Observation 3 - Place a sensitivity weight on the pan. Read and

record the indication. This indication is designated I

3

. The difference between the weight under test, Wx

, and the standards, S,

(built-in weights indicated by dial settings in observation 1) is:

W

W„x

- S (I, - I,) — = a

1 2 I 3 - I 2 where W is the mass of the sensitivity weight and

"a" is the difference between W and S in mass units.

3.3.4 Double Substitution Weighing One-Pan Constant Load Balances Using

Built-in Weights for Standards. (See Figure 3b for example) . - Indications

I j , I

2

1 and I

3 of the double substitution weighing are obtained from

observations 1, 2, and 3 of the double substitution weighing in the same manner as the corresponding indications for the single substitution

weighing described above* Section 3.3.3. There is a fourth observation

in double substitution weighing which is described immediately below.

Observation 4 - Place the weight being tested on the pan along with

the sensitivity weight and the tare weight. Set the dials to setting

used in observation 1. (This in effect removes the standards from the

load on the balance.) Read and record the indication. This

indication is designated 1^.

The difference between the weight under test, W , and the standards, S,

(the standards are the built-in weights indicated by the dial settings in

observations 1 and 4 above) is: w _ 8m II" I 2 + I-" I3 W

s x 2 I

3 - I2

11
where W

g

is the mass of the sensitivity weight and

"a" is the difference between W and S in mass units.

3.4 Substitution Weighing - Equal-Arm Balances

When an equal-arm balance is used for substitution weighing the

weights being tested and the standards are interchanged on one of the pans

and a counterpoise weight placed on the other pan. The counterpoise weight

should have approximately the same mass as the weight being tested so that

the indicator will be "on scale" when the balance beam is in equilibrium.

For interchanging the weight being tested and the standard it is con- venient to select the pan on which increasing increments of load give

numerically increasing indications. When the reading-scale reads from left

to right, smallest numbers at the left end, adding weight to the left pan

increases the readings; conversely, when the reading-scale reads from right

to left adding weight to right pan increases the reading.

3.4.1 Single Substitution Weighing - Equal-arm Balance. (See Figure 4a for

example) . Assume that increased load on the left pan gives a numerically

larger reading.

Observation 1 - Place the weight being tested on the left pan and the

counterweight on the right pan. Release the balance, read and record

the indication. If the balance is undamped, turning points are read

and the rest point computed from them. l3l This indication is

designated I x

. Observation 2 - Remove the weight being tested and place enough known

weights, that is standards, on the pan to bring the balance beam to

approximately the same position of equilibrium as in observation 1.

Read and record the indication. This indication is designated I 2

. Observation 3 - Add the sensitivity weight to that pan which will

bring the indicator towards or past the center of the reading-scale.

This indication is designated I . The difference between the weight being tested, Wx , and the standards,

S, is: w

Wx

- S = (I,.- Ij, i

3 -

S

l2 | = a where W is the mass of the sensitivity weight and

"a" is the difference between Wx and S in mass units.

The absolute value of the sensitivity deflection, I 3 - I 2 is used, that

is without regard to sign, as indicated by the symbol |I

3 - I 2

|»

3.4.2 Double Substitution Weighing - Equal-arm Balance (see Figure 4b for

Indications i 1# I 2

, and I 3 of the double substitution weighing

are obtained from observations 1, 2, and 3 of the double substitution

weighing in the same manner as the corresponding indications for the single

substitution weighing described above, Section 3.4.1. The fourth observation of this double substitution weighing is described immediately below.

12
Observation 4 - Remove the standards from the left pan but leave the

sensitivity weight on pan. Put the weight being tested on the pan

with the sensitivity weight. Read and record the indication. This

indication is designated 1^.

The difference between the weight being tested, Wx

, and the standards,

S, is:

if- - s -

I i- J

2 + v: h

ws

2 |l 3 - I 2 | " a where W

g

is the mass of the sensitivity weight and

"a" is the difference between Wx and S in mass units.

The absolute value of the sensitivity deflection, I

3 - I 2

, is used.

3.5 Transposition Weighing

Transposition Weighing - Transposition weighing is done on two-pan

equal-arm balances. When this method of weighing is used counterpoise or tare weights are not required. The weight being tested is placed on one pan and the standard on the other pan; a reading is taken, then the weights

are transposed. Either single or double transposition weighing methods may

be used.

3.5.1 Single Transposition Weighing - Equal-arm Balance (see Figure 5a for

example)

.

Observation 1 - Place the weight being tested on one of the pans, say

the left pan, and standards on the other pan, the right pan. Adjust

the amount of known weight on the pan so that the indicator will be

"on scale" when the balance beam is in equilibrium. When the balance

beam is in equilibrium, read and record the indication. This

indication is designated Ij

.

Observation 2 - Remove the weights from their respective pans and

transpose them to the other pan. That is, the weight that was on the

left pan is put on the right pan and the weight on the right pan is

put on the left pan. Read and record indication. This indication is

designated I2

. Observation 3 - Add the sensitivity weight on the pan that will cause

the pointer to move towards or past the center of the reading scale.

Read and record the indication. This indication is designated I3

.

The difference between the weight being tested, W

x

, and the standard,

S» is: I x - i 2 wt W._ - S = -- —a x 2 |I 3 - I

2 | where W_ is the mass value of the sensitivity weight and

s"a" is the difference between W

x and S in mass units.

The absolute value of the sensitivity deflection, I

3 - I2

, that is

without regard to sign, is used. Whether the weight being tested is

heavier or lighter than the standard can be ascertained from the direction

of motion of the indicator or the rules described on pages 16 and L"B. 13
3.5.2 Double Transposition Weighing - Equal-arm Balance (see Figure 5b for

example). Indications I 1# I

2

, and I 3 of the double transposition weighing

are obtained from observations 1, 2, and 3 of the double transposition

weighing in the same manner as the corresponding indications for the single

transposition weighing described above, Section 3.5.1. The fourth observation of the double transposition weighing is described immediately below.

Observation 4 - Remove the weights from their respective pans and

transpose them. Leave the sensitivity weight on the same pan that it

was in observation 3. This indication is designated 1^.

The difference between the weight being tested, Wx

, and the standard,

S, is:

i t - i 2 + i,- i 3 w

s w

* " s

5 ^^T -

a

where Ws is the mass value of the sensitivity weight and

"a" is the difference between Wx and S in mass units.

The absolute value of the sensitivity deflection, I

3 - I

2

, that is

without regard to sign, is used. Whether the weight being tested is

heavier or lighter than the standard can be ascertained from the direction

of motion of the indicator or the rule described below may be applied.

3.5.3 Determining the sign of the difference between two weights being

compared in a transposition weighing - To find the sign of the difference

"a", between two weights being compared by transposition weighing, in the

expression:

W - S = a Consider the following.

Example 1:

Obs. Load on Pans Balance

No. Left Right Indication

J o

1 W s Ii

2 s W 12

3 s+w

s W 13

Observation is the measurement of the no-load or zero load equilibrium

position. Observation 1 is the weighing with the weights W and S on the

pans as indicated. Observation 2 is the weighing with the weights transposed. Observation 3 is the sensitivity weighing. If we assume the

14
balance's zero load equilibrium position is constant and that the arms are equal, then:

W - S = I, - I

Q (D

is a measure of the difference between the weights W and S in scale divisions. Similarly

S - W = I 2 - I (2 ) is also a measure of the difference between the weights W and S in scale

divisions. We now have two measures of the difference between W and S.

Combining both measurements, we have:

W - S = Ii - Io (1)

S - w = I 2 - Io (2)

2W -2S = Ii-I -l2 + l0 (Sub. (2) from (1) (3)

Since the I 's cancel each other we can rewrite equation (3)

2W - 2S = I ! - 1 2 w - s =

Xl

I

lz

= a' (4) I I- J 2 The expression W - S = ——— states the difference between the weights in

scale divisions. Since I drops out when the two halves of the transposition weighing are combined, it is not necessary to find the zero load equilibrium position, Observation No. 0, The difference between W and S in mass units is stated in equation (5)

W - S = V J

Z WS = a (5)

& x

3 - 1^

Therefore, the complete transposition weighing requires only the three

weighings shown in the following example (Example 2)

Example 2:

Obs. Load on Pans Balance

No. Left Right Indication

1 W s Ii

2 S W I 2

3 s+w

s W *1

15
In Example 2, if we assume that the reading scale reads from left to right,

then since the sensitivity weight was added to the left pan we know that

w

the term m— in the expression,

I 3~ J 2 W - s =

I »- II

W

» - a

2 I

3 - I2 has a plus sign because I

3 is numerically greater than I

2

. Therefore the

sign for the difference "a" will depend on whether Ij or I 2 is greater; if

I j is greater, the sign is plus, and if I

2 is greater, the sign is minus.

From the sign for the difference "a", we can ascertain whether W or S is

the heavier weight. When the sign is plus, we have:

W - S = a or W=S + a and W is heavier than S.

When the sign is minus we have:

W - S = -a or W=S-a

and clearly S is heavier than W. Sometimes the sign for the difference is

not obvious and it may help to have rules for determining whether the sign

for the difference between the weights being compared is plus or minus.

There are two cases: Case I sensitivity weight added to load on left

pan; Case II sensitivity weight added to load on right pan. In each case

there are four possible combinations of plus and minus signs relating I 1 , la and 13 of the transposition weighing in the computation of the difference "a" between the weights being compared. These possibilities together

with the appropriate sign are outlined below.

I. Sensitivity weight added on left pan.

a. I j numerically greater than 1 2 , and 1

numerically greater than 1 2 , the sign for the

difference, "a" is plus (+)

b. I 1 numerically greater than I 2 , and 1 3 numerically

smaller than I 2

, the sign for the difference, "a",

is minus (-)

c. I j numerically smaller than I 2 , but 1 3 numerically

greater than I 2

, the sign for the difference, "a",

is minus (-) .

d. 1 1 numerically smaller than 1 2, and I 3 numerically

smaller than I 2

, the sign for the difference, "a",

is plus (+)

16
II. Sensitivity weight added on right pan.

a. I : numerically greater than I 2

, and I 3 numerically

greater than I 2 , the sign for the difference,

"a" , is minus (-)

.

b. ~L\ numerically greater than I2, but I3 numerically

smaller than I 2 , the sign for the difference "a"

,

is plus (+)

.

c. Ii numerically smaller than I2, but I 3 numerically

greater than I2, the sign for the difference, "a",

is plus (+)

.

d. Ii numerically smaller than I2, and 1 3 numerically

smaller than I2, the sign for the difference, "a",

is minus (-)

.

Stated symbolically

I. Sensitivity weight added on left pan

a. I x >I 2 and I 3 >I 2 sign for "a" plus (+)

b. I

t >I 2 and I

3 < I 2 sign for "a" minus (-)

c. I

x <I 2 and 1

3 >I 2 sign for "a" minus (-)

d. Ij <I 2 and I

3 <I 2 sign for "a" plus (+)

II. Sensitivity weight added on right pan

a. I

1 >I 2 and I 3 >I 2 sign for "a" minus (-)

b. Ij >I

2 and I

3 <I 2 sign for "a" plus (+)

c. I

x <I2 and I

3 >I 2 sign for "a" plus (+)

d. I

l <I 2 and I

3 <I 2 sign for "a" minus (-)

17
4. Computation of Mass Value of Weight being Calibrated

In the preceding section, weighing methods and the method of computing

the mass difference between the weight being tested were discussed and illustrated for each weighing method. The expression for this difference

had the form Wx

*> S == a. But the required result, the mass value

of the weight being calibrated, was not given. In the special case where

the nominal value of the standard, Sn

, may be considered its true mass

value and no buoyancy correction is needed, then substituting for standard,

S, its nominal value, Sn, in the expression:

Wx

- S = a we get wx

" sn - a and the mass value of Wx

- Sn+ a

The procedure is illustrated in figures 6 and 7.

4.1 Application of Correction for Standard and Buoyant Effect

In most cases, however, it is necessary to apply both a correction for

the standard and to make a correction for the difference in the buoyant effect of the atmosphere on the weight being calibrated and on the standard.

The expression Wx

- S = a gives the difference between the two weights in

mass units under the conditions existing when the comparison was made.

When the value of the standard is given as its nominal value, S , plus a correction, c, substitution for S its value, Sn+ c, in the expression:

Wx

- S = a (as computed)

we get Wx

- (S n

+c) = a (c is the correction for S

from previous calibration)

and W = S.+ c + a x n This procedure is illustrated in figure 8.

When the correction, c, for the standards, S, is in terms of true mass

the value found for Wx

is its true mass within certain limits which depend

on the relative densities of Wx and S. (See Table 1).

Where needed, the correction for the difference in the buoyant effect

may be computed as follows:

(Wx

- pVolw ) - (S - pVol s

) = a

(Wx

- S) - a - P(Vol s - Volw )

or (W„ - S) = a + P(Vol - Vol ) x

18
where P(Volw

- Vol s ) is the buoyancy correction term, often

x expressed as pAV

p

is the air density at the time of comparison

a is the indicated difference between the weights

Wx

is the weight being tested

S is the standard

The mass value of Wx when both the correction, c, for the standard, S,

and the buoyancy correction, pAV,are applied is computed as follows starting with the expression,

Wx

- S = a (as computed) wx

" ( sn+c ) = a (substituting for S its value S +c)

plus buoyancy correction » PAV

W

x

= S n

+ c + a + PAV

The value for Wx

is the true mass when the correction for S is in

terms of true mass, within the appropriate uncertainty. This procedure is

illustrated in figures 9 and 10.

4.2 Use of "Transfer Standard"

When one of the duplicate weights of a set having two or more weights

of the same denomination is used as a "transfer standard" (Section 2) , this

weight is calibrated, using one of the weighing methods described in

Section 3. The other weight or weights of the same denomination are then

compared with the "transfer standard" by a suitable weighing method. When

these comparisons have been completed, the "transfer standard" is

recalibrated. The mean of the two values found for the "transfer standard"

is the value used to establish the mass values of the weights for which it

served as a standard. This procedure is illustrated in figure 11.

4.3 Added Weights

When the mass difference between weights being compared is larger than

the on-scale range of the balance, it is necessary to add small weights to

one or the other of the weights to get an on-scale balance indication.

When an unknown weight is compared directly with a group of known weights,

this is not a problem because only enough known weights are placed in use

to bring the balance to the same equilibrium position as when the unknown

weight was on the pan. However, it is sometimes more convenient to use one

or more added small weights with the unknown weight so that a single large

standard may be used.

19
In substitution weighing, using a one-pan balance, this is a straightforward procedure. For example, assume that the mass of the weight being

calibrated is a little less than that of a convenient standard, as in example 1. (See Figure 12)

= I = I.

ile 1:

°i wx

+ w

A °2 S

°3 S + ws W

x

+ W

*

- I. w. - s

w^

(I i" I

2

) TpT7 = a = S - W + a Solving for Wx

waere

S

w.

is the weight being calibrated

is the standard

represents the added known weight

is the sensitivity weight

is the difference between the masses

If in the above example S is two pounds and W is, say, 1.95 lbs, it is

evident that this procedure is more convenient than if one had used standards equal to the mass of the weight being calibrated.

The use of an equal-arm balance permits greater flexibility because

the added weights can be placed on either pan. But this adds complexity to

the computations.

Consider the same kind of situation as in the example (Example 1) with

the one-pan balance where the standard is heavier than the weight being

calibrated and it is impractical to use a group standard, as would

be the case when a "transfer standard" is used. If the difference is known

prior to beginning the actual weighings of the calibration, then the

necessary small weights can be placed on the

calibrated as shown below. (See Figure 13a)

Example 2:

pan with the weight being

Observation

Load

Left

on Pans

Right Indication

°i WX+WA CW 1

l °2 S CW I

2

3 s +ws CW 13 w

x+wA CW

20
The difference is expressed as follows:

wx+ wA - s = V V hi h

w

X

s = a

* 2 |i 3+ Ij a

Wx = S - WA + a Solving for Wx CW is the counterweight and the other symbols have the same meaning

as above, for substitution weighing, with the one-pan balance.

But in many instances the magnitude of the difference between the

weights will not be known before beginning the calibrations and the first

half of the weighing will have been completed before it is apparent that

small weights need to be added. In that case the most economical procedure

is to place the added weight on the pan with the counterweight when the

standard is on the other pan, as in Example 3. (See Figure 13b).

Example 3:

Observation

Load

Left

on Pans

Right Indication

°1 WX CW I

.l °2 S cw+wA 9

°3 s+ws cw+wA *3

It is evident that the standard is heavier than Wx because additional

weights were placed in the right pan to bring the balance to an on-scale

equilibrium condition, when the standard was on the left pan. This is

really the same situation as in Example 2 because, if the counterweight

were increased by WA it would be necessary to add WA to the load on the

left pan in observation 1. The difference equation can be written in the

form: ws Wv

- (S - WA ) = (I,- I,)

^— = a

and

I

3 - I

2

W = S - WA + a Solving for Wx

.

This result is the same as that for Example 2.

The following argument will demonstrate that this is true:

Consider observation 1 as a measure of the difference between Wx and

CW,and observation 2 as a measurement of the difference between S and CW+WA

, then:

1 Wx

- CW = klj

2 S -(CW + WA

) - kl

2 °1~ °2 wx

- CW - s + CW + wA = Mlj- i

2

) 21
Since I

I and I

2 are in scale divisions a factor, k, is applied to convert

them into mass units. Generally k is equivalent to the term —

in the

foregoing examples. In this example 3 2 *-—i

i,- i

2 Wx

- S + WA = (I - I ) W

S = a wx

= s " WA + a Solving for W

x

.

Consider the situation where the weight under test is heavier than the

standard and the added weights were placed on the pan with the counterweight and the weight under test was placed on the other pan, as in Example

4. (See Figure 13c)

.

Example 4

Wx heavier than the standard, S, and the counterweight, CW.

Load on Pans

Observation Left Right Indication

°l wx

CW+W.

A 1

l °2 S CW X

2 °3 s+ws CW 1

3

Using the same argument as for Example 3

Wx . wA - S - (l r I

2

) TY

-^

f-T

= a W

x

- S + WA + a

Transposition Weighing - In transposition weighing added weights may

be used during only part of the weighing or they may be used during the

entire weighing, as required. Again, consider the case where the standard

is heavier than the weights under test and the situation requires that the

added weights be employed during the weighing, as in Example 5. (See

Figure 14a)

.

Example 5:

Observat ion

Load

Left

on Pans

Right Ind ication

°i wx

+wA S 1

i °2 S wx+wA

J

2 °3 s +ws wx+wA *3

22
Compute as described for transposition weighing in Section 3.5.

*i" I2 ws Wx + W* - S = -i-—=- t X A 2 _ = a

2 |l

3 - I2

| Wx

S - WA + a Solving for Wx This result is the same as that obtained for substitution methods.

Now consider the same case as illustrated in Example 5 except that it

is necessary to employ the added weights during only one half of the transposition weighing, as in Example 6. (See Figure 14b)

.

Example 6:

Load on Pans

Observation Left Right Indication

°i WX+WA S

*l

°2 S wx

X

2 °3 s +ws wx *3

Treat each observation as a measurement of the difference between the

weights.

c-! wx

+ Wa - s » ki x

2 S - Wx

= kl 2

X ~ 2 2WX

+ WA - 2S = k{Ij- I

2

)

k is a factor to transform scale divisions into mass units.

2WX

- 2S - -WA + k(I - I)

Wx

- S - -jk + Mliiiii = a x 2 2

WA "x -

S - f

+ a

This result is similar to that obtained in the previous examples, the difference being that only half of the value of the added weights was applied.

Those of the above considerations that apply to substitution weighing

are equally applicable to both single and double substitution weighing

methods; and those that apply to transposition weighing are equally applicable to both single and double transposition weighing methods.

23
5. Air Density

In order to make the buoyancy correction computations (Section 4) it

is necessary to know the density of the atmosphere in which the weighings

were made. The density of the air may be computed in any of several ways

[4J[5J. Most of the methods given in the handbooks use tables which require

either dew point measurement or wet and dry bulb thermometer readings.

But, the approximate density of the air may be computed with sufficient ac- curacy (to about 10|Jg/cm 3

) directly from temperature, relative humidity and

barometric pressure by the formula given below [6] or by using the table

(see Appendix) derived from that formula. The equation is:

0.46554P - R.H. (0.00252T - 0.020582) rci

273.16 + T

where p = density of air in mg/cm3

P = barometric pressure in mm

R.H. = relative humidity in percent

T = temperature of air in degrees centigrade

The following example illustrates how the air density (p) is computed

using this formula.

Compute the density of the air when:

The temperature is 24. 8* C,

The relative humidity is 57%

The barometric pressure is 749.6 mm, Hg.

Substituting the temperature, relative humidity and barometric pressure for

the symbols in the equation above

0.46554 X 749.6 - 57(0.00252 X 24.8 - 0.020582)

P =

P =

273.16 + 24.8

348.9689 - 57(0.041914)

297.96

34 6.6599 _ , ,<-, „ ,„„,3 P = 297.96

ia63 mg/Cin

24
REFERENCES

[1] American Society for Testing Materials, 1968 Book of ASTM

Standards, Part 30 , Standard Methods of Testing SingleArm Balances, ASTM Designation E-319-68.

[2] National Bureau of Standards Handbook 77, Volume III,

Testing a Quick-Weighing Balance, pp 740/707 to 746/713.

[3] Cross, J. L., Reduction of Data for Piston Gage Pressure

Measurements, NBS Monograph 65.

[4] National Bureau of Standards Handbook 77, Volume III,

Circular 3, pp 671/53 to 683/65.

[5] Handbook of Chemistry and Physics, Chemical Rubber Publishing Company, Cleveland, Ohio.

[61 Bowman, Horace A. and Schoonover, Randall M. , Procedure

for High Precision Density Determinations by Hydrostatic

Weighing, NBS Journal of Research, Volume 71C (Engineering and Instrumentation) No. 3, 179-198 (1967).

25
FIGURE 1

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27
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28
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29
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FIGURE 6

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.

31
FIGURE 7

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the standards nor buoyancy corrections are required.

32
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33
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NATIONAL BUREAU OF IHNDAROI

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34
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35
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FIGURE 11

rORM NBS.345.0i

(S-I.8U

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OP STANDARDS

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36
Figure 11- Illustrating use of "transfer standard"

(a) First calibration of "transfer standard"

(Weight No. 132) using known weights;

(b) Calibration of other weights of the same denomination as the "transfer standard"

using the "transfer standard" as the

standard

.

(c) The repeat calibration of the "transfer

standard" after all the other weights

of the same denomination have been tested.

The mean of the two values (a) and (c)

found for the "transfer standard", Weight

No. 132, is the value of the weights

calibrated using the "transfer standard".

37
FTG1TRF, 12

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U.S. DEPARTMENT OF COMMERCE

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the computation of the mass value when added weights are used.

The application of the corrections for the added weight, the

standard, and the buoyancy correction.

38
o

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Illustrating computation of mass value when added weights are used,

double substitution weighing with equal-arm balance using "transfer

standard". (a) Weight under test lighter than standard, added

weights on pan with weight under test, (b) Weight under test lighter than standard, added weights on pan with counterweight when

standard is on the "load pan". (c) Weight under test heavier than

standard added weights on pan with counterweight when weight under

test is on the "load pan."

40
FIGURE 14

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Illustrating computation of mass value when added weights are used;

single transposition weighing using "transfer standard". Weight

under test lighter than standard. (a) Added weights on pan with

weight under test during entire weighing. (b) Added weights on pan

with weight under test during half of transposition.

41
APPENDIX

TABLE I

Magnitude of error introduced by considering

the true mass and apparent mass vs brass to be

the same for various ranges of densities.

Magnitude of Error Density Range

1 part in 10 5 7.85 g/cm 3 to 9.00 g/cm 3

2 parts in 10 5 7.35 g/cm3 to 9.75 g/cm3

3 parts in 10 5 6.94 g/cm3 to 10.6 g/cm3

4 parts in 10 5 6.56 g/cm3 to 11.62 g/cm3

5 parts in 10 5 6.23 g/cm 3 to 12.86 g/cm 3

1 part in 10 4 4.95 g/cm 3 to 27.63 g/cm3

3 parts in 10 1 * 2.7 g/cm 3

42
TABLE II

In Tables Ila and lib the temperature range is from 15° C to

35° C increments of 0.5°C. The relative humidity range is

from 10% to 90% in increments of 5%. The barometric

pressure range is from 575mm to 7 8 0mm in increments of 5mm.

The barometric reduction factor to the barometric pressure

for atmospheric humidity is given in the body of Table Ila

for various temperatures and relative humidities. The air

density, in milligrams per cubic centimeter, for the stated

conditions is given in the body of Table lib to 0.01mg/cm3

. To get the air density in pounds per cubic foot, multiply

the air density in mg/cm 3 by 0.06242783; and to get the air

density in pounds per cubic inch multiply the air density in

rng/cm 3 by 0.000036127.

The following procedure is used to find the air density, p , with Tables Ila and lib.

Enter Table I la throuqh the temperature and relative humidity. In the column headed Air Temp °C

of Table Ila, find the temperature nearest the

thermometer reading, then go across the table to

the column under the relative humidity nearest the

relative humidity reading. The number in the body

of the table at the intersection of the relative

humidity column and the temperature line is the

reduction factor and is subtracted from the

barometer reading. The resulting figure is the

reduced barometer reading and is used to enter

Table lib. In the column headed Air Temp ° C of

Table lib, find the temperature nearest the

thermometer reading, then go across the table to

the column under the barometric pressure nearest

the reduced barometer reading. The figure in the

body of Table lib at the intersection of the barometric pressure column and the temperature line is

the density of the air. The following example

illustrates the procedure for using the tables.

Find the air density ( p

) when:

The temperature is 24.8°

C

The relative humidity is 57%

The barometric pressure is 749.6mmHg.

43
Step 1. In Table Ila the temperature nearest 24.3 °C is

25.0 *C, go along the line to the column under

relative humidity nearest 57%, which is the 55%

column. The number at the intersection of the

25 *C line and the 55% column is 5.0.

Step 2. Subtract 5.0 from the barometer reading, 74 9.6.

749.6mm - 5.0 = 744.6mm. This is the reduced

barometric pressure.

Step 3. In Table lib, find the 25.0°C line (the temperature in the table nearest 24.8 °C) across the

table to the column under barometer pressure

nearest 744.6mm, which is the 745mm column. The

number at the intersection of the 25° C line and

745mm column, 1.16, is the air density, p, in

milligrams per cubic centiineter, for the stated

conditions.

The air density found, 1.16 mg/cm , in lb/ft3 is

1.16 x 0.06242788 = 0.07241634 lb/ft3

in lb/in. 3 is

1.16 x 0.000036127 = 0.00041907 lb/in3

44
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47
TABLE III

CONVERSION TABLE

Some Useful Conversion Factors

To Convert

To

lb/ft3

lb/ft3

lb/in3

lb/in3

mg/cm3 mg/cm3 g/cm3 g/cm3 grams

grams

kilograms

kilograms

milligrams

ounces

ounces

pounds

pounds

pounds

cu in.

cu ft

cm3 cm3 cm

cm

feet

ounces

pounds

inches

From Multiply By

mg/cm3 X 0.062427886

g/cm3 X 62.427886

mg/cm3 X 0.000036127

g/cm3 X 0.0361272

lb/ft 3 X 16.018465

lb/in3 X 27679.9028

lb/ft3 X 0.016018465

lb/in3 X 27.679903

ounces X 28.349523125

pounds X 453.59237

ounces X 0.028349523

pounds X 0.45359237

pounds X 453592.37

grams X 0.03527396

kilograms X 35.27396

grams X 0.00220462

kilograms X 2.204623

milligrams X 0.0000022046

cm3 X 0.06102374

cm3 X 0.00003531467

cu in. X 16.387064

cu ft X 28316.846592

feet X 30.48

inches X 2.54

cm X 0.0328084

pounds X 16

ounces X 0.0625

cm X 0.3937008

48
TABLE IV

DENSITIES AND VOLUMES PER UNIT MASS

OF

SELECTED MATERIALS

Density at 20°C

Material sZ cm

Brass (Normal) 8.3909

Stainless Steel 8.0

Stain less Steel 7.92

Stainless Steel 7.89

Stainless Steel 7.84

Stainless Steel 7.8

Stainless Steel 7.76

Stainless Steel 7.75

Steel 7.83

Cast Iron (Gray) 7.0

Cast Iron (White) 7.6

Nickel Ch romium 8.34

Nickel Ch romium 8.41

Nickel Ch romium 8.5

Tantalum 16.6

Platinum 21.37

Platinum 21.5

Platinum- Iridium 21.54

Aluminum 2.7

lb/in 3

0.30314

0.2890

0.2861

0.2850

0.2832

0.2818

0.2803

0.2800

0.2829

0.2529

0.2746

0.3013

0.3038

0.3071

0.5997

0.7720

0. 7767

0. 7782

0.0975

Volume at 20°C

cm 3 in 3 Per gram Per lb

0.119177 3.29881

0.1250 3.4600

0.1263 3.4950

0.1267 3.5083

0.1276 3.5306

0. 1282 3.5487

0.1289 3.5670

0.1290 3.5716

0.1277 3.5351

0.1429 3.9543

0.1316 3.3948

0.1199 3.3190

0.1189 3.2913

0.1176 3.2565

0.0602 1.6675

0.0468 1.2 95 3

0.0465 1.2874

0.0464 1.2850

0.3704 10.2522

49

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1. PUBLICATION OR REPORT NO. Technical Note 577

2. Gov't Accession No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

Method of Calibrating Weights for Piston Gages

5. Publication Date May 1971

6. Performing Organization Code

7. AUTHOR(S)

H. E. Aimer

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16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant bibliography or literature survey, mention it here.) Generally weights for piston gages have odd denominations that are often not readily calibrated by intercomparison methods. Therefore

these weights are frequently calibrated by direct comparison

methods. This paper presents direct comparison methods for calibrating piston gage weights for use with both equal-arm balances

and single-pan balances. Methods of estimating the uncertainty of

the values obtained are given. Also included are methods of

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17. KEY WORDS (Alphabetical order, separated by semicolons)

balance; buoyancy; calibration; standards; substitution weighing; transposition

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