SOP No. 6

Recommended Standard Operations Procedure

for

Weighing by Transposition

1. Introduction

1.1. Purpose

This method eliminates possible. error in comparison of weights when

using an equal-arm balance, due to a small inequality in the length

of the arms. While this procedure is written to describe the

measurement of masses of weights for calibration purposes, it may be

used for general mass determinations, whenever accurate weighing is

required. (Note: This is a single - transposition procedure, hence

does not eliminate the effects of linear drift).

1.2. Prerequisites

1.2.1. Verify that standard weights are available with valid

certificates of calibration.

1.2.2. Verify that the balance to be used is in good working order.

This may be verified by maintenance of a valid control chart.

Otherwise, a series of trial measurements should be made to ascertain that the balance is functional and that its

sensitivity is adequate for th~ requirements of the

measurement.

1. 2.3. Verify the ability of the technician to make precision

weighings and that he/she has had training in this specific

procedure.

2. Methodology

2.1. Scope, Precision, Accuracy

The measurement principle is applicable to weighing with any

equal-arm balance. The range of weights that can be calibrated will

depend upon the capacity of the balance used. The precision of

intercomparison will depend upon the sensitivity of the balance, and

the care exercised in making the required weighings. The accuracy

will depend on the accuracy of calibration of the standard weights

and the precision of intercomparison.

2.2. Sumry

The masses of a standard and test weight are intercompared. The

loads are then interchanged and a second intercomparison is made. A

small calibrated weight is then added to one of the masses and a

SOP 6 - 1
third weighing is made to determine the sensitivity of the balance

under the load conditions.

2.3. Apparatus/Equipment

2.3.1. Equal arm balance of capacity and sensitivity sufficient for

the weights tested. The index scale of the balance is

conveniently numered 0 to 20 with 10 as the center division,

al though other numering systems such as 0 to 200 are

possible. The graduations should be n~bered so that

addition of a weight to the left arm will increase the

balance readings. A system in which the center division is 0

is not recommended since the negative readings that result

can cause observational and computational problems.

2.3.2. Standard weights with valid calibrations, traceable to NBS.

2.3.3. Small calibrated weights (usually deciml fractions) to be

used as tare weights.

. 2.3.4. Equipment capable of loading and unloading weights on balance

without daage to either (especially important in the case of

large weights).

2.4. Symbols

The following symols are used in this procedure.

S - standard weight

X - weight calibrated

T - counterweight

t - small calibrated. A subscript s or x is used to indicate the

larger weight with which it is associated.

sw - small calibrated weight used to evaluate the sensitivity of the

balance.

M - the mass of a specific weight. Subscripts s. x. t, sw. are used

to identify the weight.

AM - the apparent mass of a specific weight. Subscripts 8. x. t. sw

together with numerical ~ub-8ub8cripts as necessary are used to

identify the weight.

sop 6-2
2.5. Procedure

2.5.1. The following weighing sequences may be used:

Option A

Measurement No. Left Arm R.ight Arm Observation

1 X + txl S + ts 01

1

2 S + ts X + ~ 02

2

3 S + ts + sw X + tx2 03

2

Option B

Measurement No. Left Arm Right Arm Observation

1 S + ts X + txl 01

1

2 X + tx2 S + ts 02

2

3 X + tx2 + sw S + ts 03

2

2.5.2. Preliminary Procedures

Conduct preliminary measurements (without recording data) to

determine the values for ts and tx as described in 2.5.3.

This will serve to warm up the balance and to facilitate the

measurements. In a series of calibrations, this step may be

minimized or eliminated after the first series of

measurements. Depending on the corrections for S and X,

weights ts and tx may be unecessary in some cases.

2.5.3. Measurement Procedure (Option A)

All observations should be recorded on suitable data sheets

such as those in the Appendix. Record the laboratory amient

temperature. barometric pressure. and relative humidity.

2.5.3.1. Observation 1. Place the test weight(s) on the

left arm and the standard(s) on the right arm. Add

tx. and/or ts (small calibrated weights) as

necessary to obtain an approximate balance. Read

and record all data and the sum of the turning

points 01 (see GMP No.1).

2.5.3.2. Observation 2. Place the standard weight(s) on the

left arm and the test weight(s) on the right arm

with appropriate small weights. as was done in

Ob8ervation 1. The tare weights carried with Sand

X may have to be chaged so that 02 is within one

division of 01 on a 0 to 20 division graduated

SOP 6- 3
scale. Read and record all data and the sum of the

turning points, 02.

2.5.3.3. Observation 3. Without disturbing weights already

in place from Observation 2, add a small calibrated

sensitivity weight (sw) to the left arm, sufficient

to displace the turning points 3 to 5 divisions on

a 20 division graduated scale. Record the sum of

the turning points, 03.

2.5.4. Measurement Procedure (Option B)

Conduct 3 measurements as in 2.5.3.1, 2.5.3.2, and 2.5.3.3

using the weighing schedule Option B.

3. Calculations

3.1. No Air Buoyancy Correction

. The following equations may be used to calculate the corrections

required for the test weight(s). The calibrated apparent mass values

for ts' tx, and sw must be used. Cs is the apparent mass correction

for the standard weight(s), S. The symbols Ns and Nx refer to the

nominal values of S and X, respectively.

Option A

AM +AM tsi tS2 -AM' -AM txl tx2 01 - 02 AMsw

Ox - Cs + + . + Ns - Nx 2 2 03 - 02

Option B

AM +AM tsi tS2 -AM -AM ~l ~2 02 - 01 AMsw

Ox - Cs + +. . + Ns - Nx 2 2 03 - 02

3.2. Air Buoyany Correction

3.2.1. Calculate the air density, PA' as described in Section 8 of

the Appendix to SOP No. 2 or obtain from Table 9.9.

3.2.2. Calculate the mass, !I, of the test weight X, using the mass

of the standard weight(s), the tare weights, and the sensitivity weights according to the optional sequence used. It

is assumed that the densities of the combination of tare

weights used do not differ significantly in the weighing

design.

Symols used: PA - air density

Mi - mass for weight i

Pi - density for weight i

SOP 6-4
KS(l

!I -

3.2.2.1. Optional Sequence A

_ PA~ 01 -

PSl 2

KSW(l - PA ) Kt + Kt 02 Psw sl s2 ( . + 1 03 - 02 2 2 (1 - ;~)Kt +Mt xl x2 _ PA )_

Pts

PA

1 -

Px

3.2.2.2. Optional Sequence B

Ksw 11 - P A ) Kt + Kt _ PA\l 02 - 01. \ Psw + sl s2

PsI' 2 03 - 02 2 (1 - ~)- Pts

Kt +Mt

xl 2 x2 (1

- ;~)

PA

1 -

Px

3.2.2.3. Calculate the mass correction, ex, as follows:

C:i-!l-Nx

when Nx is the nominal value for X

3.2.2.4. Calculate the apparent mass of X versus the desired

reference density of 8.0 g/cm3 or brass. It is

recommended that the apparent mass versus 8,0 g/cm3

be reported unless otherwise requested. The

density of X, Px' mut be in g/cm3.

3.2.2.4.1. Apparent mass versus 8.0 g/cm3

. 0012 (1 - )

Px

AK vs. 8.0 - !I 0.999850

KS(l

!I -

3.2.2.4.2. Apparent mass versus brass

AK vs. brass- !I

.0012 (1 - )

Px

0.999857

SOP 6-5
4. Assignent of Uncertainty

The limits of uncertainty, U, include estimates of the uncertainty of the

mass standards, Us' plus the uncertainty of measurement, Um, at the 99.73%

level of confidence. The latter is estimated by

ts

where s is the standard deviation of measurement and t is obtained from

Table 9.3.

Then U - :t (Us + ts)

4. l. Precision of Measurement lCown from Control Chart Performnce (SOP

No.9)

The value for s is obtained from the control chart data for

transposition measurements. Statistical control will need to be

verified by measurement of at least one check standai-d while the

. above measurements are in progress.

Use the value of t (corresponding to a probability level of 99.73%)

from Table 9.3 appropriate for the numer of degrees of freedom, ii,

on which the control limits of the control chart are based.

4.2. Precision Estimted from a Series of Measurements

Measure a stable test object at least 7 times, no two measurements of

which may be made on a single day. Calculate the mean and a standard

deviation in the conventional maer. The latter is the value of s

that is to be substituted in the equation given in Section 4. In this

case select the value. for t from Table 9.3 based on the number of

degrees of freedom involved in computing s.

Note: Repetitive measurements made on the same day evaluate only the

short-term standard deviation.

5. Report

5.1. Report resul ts as described in SOP No. l, Preparation of

CalibrationlTest Reports

SOP 6-6
Appendix

Transposition Data Sheet

for Equal-Arm Balance (Option A)

Test No.: Sheet No.: Date:

Item Identification: Balance:

Standard Identification: Observer:

Temperature: Pressure: Re1. Hum.:

,

Cs(Cs)- :t AMsw(Msw)-

Ps- Px- Paw -

Time Balance standard deviation -

Measure - Weights Turning Points Sum

ment

No. Left Arm Right Arm Low High

X + tx S + ts i

1 1 01-

tx - ts - i 1

S + ts X + tx

2

2 2 02-

ts - tx - 2 2

3 S + ts + sw X + ~ 03-

2 2

Time

AMt (Mt )- :t AMt (Mt )- :! sl si Xl xi

AMt (Mt )- :t AMt (Mt )- :!

s2 s2 x2 x2

Pt - Pt -

s x

SOP 6-7
Appendix

Transposition Data Sheet

for Equal-Arm Balance (Option B)

Test No. : Sheet No.: Date:

Item Identification: Balance:

Standard Identification: Observer:

Temperature: Pressure: ReL. Hum.:

,

Cs(Cs)- :! AMsw(Msw)-

Ps- Px- Psw -

Time Balance standard deviation -

Measure - Weights Tuming Points Sum

ment

No. Left Arm Right Arm Low High

S + ts X + tx

1 1 1

°l- ts - tx - 1 1

X + tx S + ts

2 2 2

02-

tx - t. -

2 2

3 X+tx+sw S + t. 03- 2 2

Time

AMtsl si (Mt )- :t AMt (Mt )- :! xi xl

AMt (Mt s2 s2)- :t AMt (Mt )- :! x2 x2

Pt - Pt -

s x

SOP 6-8
Appendix

Transposition Data Sheet

for Equal-Arm Balance (Option A)

Test No.: 135 Sheet No.: 1 Date: 8/26/86

Item Identification: 1000 lb No. 4211 Balance: Russell

Standard Identification: #11 and #22 Observer: HO

Temperature: 23. 5°C Pressure: 745.8 ro Hi ReL. Hum.: 50%

,

Cs(Cs)- 0.0047 1b :! 0.0030 lb AMsw(Msw)- 0.01 lb

3 3 3

Ps- 8.0 g/cm Px- 7.0 i/cm Psw - 7.8 it/cm

Time 1: 15 'D Balance standard deviation - 0.0018 lb

Measure - Weights Turning Points Sum

ment

No. Left Arm Right Arm Low High

X + tx S + ts 4.0

~

1

1 . 1 16.3 01- 20.35

tx - 3.6l7lb ts - 3.625lb 4,1

1 1

1'1.1

S + ts X + tx 5.6

2

2 2 14.1 02- 19.75

ts - tx - 3.6l2lb 5.7

2 2 14.0

H.Q

8.4

3 S + ts + sw X + tx 15,9. 03- 24.3

2 2 8.4

15.9

Time 1:40 'P

AMt (Ht: )- 3.625 :! .000075lb AMt (lIt )- 3.617 :! negliiib1e sl sl Xl Xl

AMt (Ut: )- 3.625 :! .000075lb AMt (lIt )- 3.612 :! neitligible s2 s2 x2 x2

Pt -

3 3

8.0 rz/.cm Pt - 7.8 rz/cm

S x

SOP 6-9
ANt + ANt- ANt - ANt 81 82 xi x2 01 - 02 . ( ~-~+ 2 + 2

3.625 + 3.625 . 3.617 - 3,612

Ox - .0047 + 2

Cz - .0047 + .0105 + .000659

Cz - 0.015859 lb

Unertainty - ta + U8

AMsw )

03 - 02

+

20.35 - 19.75

2

Degrees of freedom in s - 20

Uncertainty - 3.422 (0.0018) + (.0030 + .000075)

Uncertainty - 0.0092346 1b

~ - 0.0159 ~ 0.0092 1b

SOP 6.10

0,01

.

24.3 - 19.75