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Chapter 2
Two-Pan Equal-Arm Balances
2.1 Introduction
In fact, mass of a body is difficult to determine, but it can be compared easily
against a body of known mass. This process of comparing the masses of two bodies
is commonly known as weighing process. The weighing is carried out through a
balance. The balances can be broadly divided into two groups, namely (1) two-pan
balance and (2) single-pan balance. In this chapter, we will consider two-pan, equal-
arm balance.
2.2 Brief History
The equal-arm balance is one of the oldest measuring instruments; its invention may
date back to 5000 BC. It is one of the most accurate instruments yet is based on the
simplest principle of mechanics. In last 7000 years, it has gone a sea change from
a simple beam supported at its centre with two pans hanging from its ends through
a cord to the most sophisticated one with multiple bearings, inter-changeable pans
and remote control mechanism. Romans used the knife edges in eighteenth century.
This led to its faster development in Europe. By the end of nineteenth century, the
balance was modernised to such an extent that it became most precise mechanical
instrument. Metric convention of 1875 gave a boost to the manufacturing of standard
balances. International Bureau of Weights and Measures used interchangeable pan
balance of Messrs. Ruprecht, Austria (Fig. 2.1) [1]. In the first half of twentieth
century, National Physical Laboratory, Teddington, U.K. developed a balance,
which was the state of the art item. Many commercial firms, such as Messrs.
Stanton and Messrs. Oertling in U.K., Messrs. Sartorius and Messrs. Paul Bunge
in Germany and Messrs. Chyo Balance corporation in Japan, started manufacturing
standard balances having a precision better than one part per million. Later Messrs.
Chyo Balance Corporation, Japan, made commercially available balance with a
S.V. Gupta, Mass Metrology, Springer Series in Materials Science 155,
DOI 10.1007/978-3-642-23412-5 2, © Springer-Verlag Berlin Heidelberg 2012
19

20 2 Two-Pan Equal-Arm Balances
Fig. 2.1 Ruprechet 1 kg
balance
readability of 1 g in 1 kg. In around 1970, Messrs. Mettler Corporations started
manufacturing single-pan balances. However, electronic balances are overtaking
mechanical balances since 1970.
Ruprecht is one of the oldest balance manufacturing ﬁrms of Austria. The balance
shown in Fig. 2.1 belongs to BIPM, Sevres, France. It is a mechanically remote-
controlled balance. Remote controls are used for exchanging the two weights under
comparison and moving the rider from one notch to another. Indication of the beam
is read through a lamp and scale arrangement.
2.3 Other 1-kg Balances
2.3.1 Equal-Arm Interchangeable Pan Balance UK
Gould [2] designed and got fabricated a 1-kg balance at the National Physical
Laboratory, UK. It is an equal-arm three knife edge balance. It has the following
features
• Double equilateral triangular beam to get maximum rigidity
• Inter-changeable pans carrying the weights, so that transposition weighing can
be carried out
• Remote mechanical arrangements for releasing the beam and shifting rider from
one notch to any other notch

2.3 Equal-Arm Interchangeable Pan Balance UK 21
• A device to provide transposition of weights without breaking the contact of
central knife with its bearing plane. This ensures that the contact line of central
knife edge with its bearing plane remains unaltered
• Special crossed knife edge suspension arrangement, as shown in Fig. 2.2, for
hanging the two pans. The ﬁrst suspension A consists of the terminal bearing
plane and another knife edge. This knife edge is parallel to the terminal knife
edge. Second suspension B has a bearing plane and a knife edge at right angles
to ﬁrst knife edge to prevent any tilting. Third suspension C consists of a bearing
plane and a conical pivot at its lower end. The pan suspension P has a conical
hole and is suspended on the conical pivot of the third suspension C.
The balance is shown in Fig. 2.3. The mass value per division of the scale is 30 g.
Fig. 2.2 Suspension
assembly of a pan
Fig. 2.3 Equal-arm
interchangeable pans balance
(Gould, UK)

Fig. 2.4 NRLM balance with
two pans on each side
2.3.2 Equal Arm with Double Interchangeable Pan Balance
Chyo Balance Corporation of Japan manufactured and supplied a 1-kg balance to
National Research Laboratory of Metrology (NRLM), Japan. The NRLM balance
is also a three knife edge equal-arm balance but has two platforms on each side. The
two platforms are independently interchangeable. This allows eight comparisons
of two weights. In fact the balance can also take four weights at a time and can
compare any weight on one side to any of the two weights on the other side. The
effect of gravity due to two weights in different planes can also be evaluated in terms
of balance readings. The balance has a sensitivity reciprocal of 1 g per division.
A line diagram of the balance is shown in Fig. 2.4.
2.3.3 Substitution Balance NPL-India
A substitution balance [3] was designed and fabricated in mid-1970s. There were
two pans one above the other on each side of the balance. The pans have equally

2.3 Electronic Balance with Automation at NPL-India 23
Fig. 2.5 Substitution balance
NPL New Delhi
spaced vertical fins on which the weight can centrally rest. There are two loaders
one for each pan and having slits so that loader can move between the fins of the
pan freely. The loaders are actuated by a screw, which has right-handed threads on
its upper part and left-handed thread on the lower part. The loaders move up or down
by turning the screw. The screw is operated mechanically from a distance with the
help of bevelled 45 ˚ gears. When the screw is turned in one particular direction, one
loader moves up while the other loader moves down. On the right-hand side, the two
weights to be compared are placed on their respective loaders, which are in such a
position that none of the weights is on the pan. When the screw is turned clockwise
the weight on the lower loader rest on the fins of the lower pan on the right-hand side,
the loader is further moved down leaving sufficient space for the vertical movement
of the pan. In this position, weight on the upper loader will stay clear of the pan. The
lower weight is compared with the dummy weight on the left-hand pan. After taking
the necessary observations, the screw is turned anticlockwise and upper weight is
loaded while lower weight stay clear from the pan. Thus, two weights are compared
with each other. Due correction due to vertical distance between the two weights
is applied. For a better readability, the balance was provided with a lamp and scale
arrangement.
The balanced was able to give uncertainty of 100 g.
2.3.4 Electronic Balance with Automation at NPL-India
With the advent of electronic balances with better readability and automation,
a single-pan Mettler HK 1000 mass comparator with a dedicated computer was
introduced in NPL [4]. The mass comparator is electromagnetic force compensated,

Fig. 2.6 1 kg electronic mass
comparator
unequal arm and two knife edge balance. The coil is rigidly attached to the end of
the beam having a counterpoise weight. To the other end of the beam, the weighing
pan along with weights is attached. The coil moves in the annular gap of the magnet.
When a mass is placed on the pan, a position sensor detects and causes a current to
pass through the coil so that the beam comes to the original position. It has been
provided with a separate digital display and control unit. This unit has a built-in CL
data interface and digital display device. The measuring results are transferred via
the CL data interface to the control unit where the data is stabilised. The balance
is provided by a weight handler. It can take four weights at a time. Computer-
controlled motor places the weight on the pan in any desired sequence.
The balance has a capacity of 1,001.15g. The range of electronic compensation
is 1.5 mg. Readability is 1 g with a standard deviation of the same order. The four
weights on the weight handler can be compared in all possible combinations. All
environmental parameters are recorded and stored in the accompanying computer.
Buoyancy correction is applied to each and every individual weighing. The elec-
tronics mass comparator with its computer is shown in Fig. 2.6.
2.3.5 Hydrostatic Balance
A 1-kg mass comparator using principle of Nicholson’s hydrometer was constructed
at PTB Germany [5,6]. The force due to gravity acting on the weight placed in air is
compared against a constant buoyant force on the float immersed in the liquid. The
position of the floating system is kept unchanged by electromagnetic force compen-
sation. Alternatively the change in buoyant force is measured by measuring the posi-
tion of the float by laser interferometer. Readability of the comparator is 1 g and
standard deviation is 5 g. A line diagram of mass comparator is shown in Fig. 2.7.
On a granite plate 1, a bridge-shaped aluminium support 2 is located, consisting
of three columns, which are held together at the top by two mutually perpendicular
crossbeams. For the shake of clarity, only two front columns are shown. The support
carries the double-walled liquid container 3 with the buoyant body 4, consisting of
six hollow cylinders joined together to form a ring. The view from the top is shown
on the right of main figure. The buoyant body is connected to suspension frame 8 via

2.3 Electronic Balance with Automation at NPL-India 25
Fig. 2.7 Hydrostatic balance
three thin immersion rods 5, a three-armed holder 6 and a connecting rod 7 running
through a tube in the middle of the liquid container. The upper cross-piece of the
suspension frame contains the triple prism reflector 9 of the laser interferometer,
beneath which the beam splitter cube and reference reflector 10 are attached to a
stand by means of the bridge. The lower crosspiece of the suspension frame carries
1 kg weight 11. Up to six weights can be placed on the weight-changer 12 having a
motor-driven lift and turn-table. By an upward movement of the table, the weight is
removed from the suspension, the float being kept under constant load and centred
by means of an arrestment hook 13, preventing the buoyant body from emerging
from the liquid. The outer jacket of the double-walled liquid container is connected
to a water thermostat. The temperature remains constant within 5 mK. The inner
container holds 4.4 L of distilled, surfactant treated water. With the help of weights
with known mass differences from a few micrograms to one milligram, the position
of floats is calibrated by laser interferometer.

2.4 Installation of a Balance
2.4.1 Proper Environmental Conditions for Balance Room
It should be ensured that balance room is dust-free and rate of change in temperature
is a minimum. Practically, no temperature gradients should occur along the beam of
the balance. The balance should be kept on a reasonably vibration-free table.
2.4.2 Requirements for Location of the Balance
While choosing a location, the following points should be considered.
2.4.2.1 Vibrations
Balance should be located on a floor, which is reasonably free from vibrations. Prof.
Bessason et al. made a study of the vibration problem [7]. The following paragraphs
are based on their work.
To test the vibrations of the floor, take a flat bottom dish of diameter 15–20cm.
Pour mercury in it, place it on the floor to be tested and wait for 10 min. One can
see even small vibrations of the floor by observing the mercury surface. If a small
spot of light placed closed to the dish is thrown on the surface of the mercury and
its reflection is projected on a wall or ceiling, then movement of the spot will give
magnitude as well as the frequency of the vibrations. Low-frequency vibrations, in
the range of 1–5 and 5–100, are more injurious to the balance.
The vibration criterion, in terms of acceleration, velocity or displacement, can be
defined in several ways. Such as:
• Single peak value or a frequency-dependent peak value
• Single root mean square (RMS) value of energy
• One-third octave RMS spectrum
• Constant band width RMS spectrum
• Power spectral density
Depending on the tolerable measurement uncertainty, the criteria for vibration free
are again placed in three classes namely A, B, and C.
The vibration-free class is indicated in the Table 2.1 for different parameters and
desired uncertainty requirement.
Class A, B, C and D of frequency verses amplitude and acceleration are depicted
in Fig. 2.8.
Vibration criteria followed by various laboratories is shown in Table 2.2.

2.5 Evaluation of Metrological Data for a Balance 27
Table 2.1 Vibration free Classes for various classes
Parameter Range Desired
uncertainty
Vibration
criterion
Length
Primary length standard 1 m 10 11
A
Length measurement <2 m 10 6
A
General calibration 1 mm to 1 m 10 5
C
Mass
Calibration of level I standard 1 kg 10 8
A
Calibration of level II standards >1 kg 10 7
A
Calibration of legal standards 10 g to 10 kg 10 6
B
Calibration of working standards 10 kg to 20 kg 10 5
B
Calibration of weights in a working
Laboratory 10 g to 500 kg B
Density measurement 10 5
B
Pressure All range B
2.4.2.2 Sunlight
Sunlight should not fall directly on the balance especially through the side windows.
This would set up a variable temperature gradient along the beam.
2.4.2.3 Chemical Fumes
The location for the balance should be free from chemical fumes. However, in
a chemical laboratory where such fumes are unavoidable, there should be 24-h
running exhaust fan for proper ventilation and balance may be housed in a separate
chamber.
2.4.2.4 Air Draft
Direct air draft on the balance, such as by opening and closing of the door, should
be avoided in all cases.
2.5 Evaluation of Metrological Data for a Balance
2.5.1 Need for Evaluation
Mass of a given weight is determined by comparing it against a standard of known
mass using the balance as a comparator. The uncertainty in measurement of mass
will depend on the performance of the balance used. Therefore, it is imperative

Fig. 2.8 Classes of vibrations
that the balance is evaluated for its performance. The balance must be checked
periodically for its performance and any adjustment required due to wear and tear
should be carried out immediately.
Apart from visual examination following metrological data is important for an
equal-arm two-pan balance:
• Sensitivity or sensitivity reciprocal
• Time period or period of swing
• Stability/repeatability
• Accuracy of rider bar, if provided
• Equality of length of arms or arm ratio of the beam
• Repeatability (precision) of single or transposition weighing
• Repeatability (precision) of substitution weighing

Table 2.2 Vibration criteria adopted by various laboratories
Laboratory Vibration criterion Frequency range Amplitude Integration
time
JV Frequency dependent
peak Value
1–5 Hz v D 3:2 m=s 0
5–100 Hz a D 0:10 mm=s2
ISA Frequency dependent
Peak value
0:1–30 Hz d D 0:25 m 0
PSL Constant bandwidth,
0.125 Hz, RMS
spectrum envelope
5–50 Hz v D 0:8 m=s 8
NIST One-third Octave 1–20 Hz v D 0:15 m=s 1
RMS spectrum 20–100 Hz v D 3 m=s
BBN One-third Octave 4–8 Hz v D 24=f m=s Not specified
RMS spectrum 8–80 Hz v D 3 m=s
JV stands for Norwegian Metrology and Accreditation Service
ISA stands for Instrument Society of America
PSL stands for Primary Standards Laboratory at Sandia National
Laboratories in Albuquerque New Mexico
NIST stands for National Institute of Standards and Technology USA
BBN stands for BBN Systems of Technologies Cambridge
Symbols used are: v for velocity, d for displacement and a for acceleration
Frequency range, acceleration and amplitude tolerances for classes A, B, C and D are shown in
Fig. 2.8
2.5.2 Visual Examination
Before starting the actual test, one should examine the balances visually to ensure
that
• The knife edges are free from chipping or rough areas
• The bearing planes do not show any scratches or burs
• The clearances between knife and bearing planes are small, uniform in width and
are equal on both sides
Uniformity in width of clearances can be easily assessed by observing it along the
beam and against a strongly illuminated background.
The degree of parallelism at the entire bearing system can be assessed in early
stage of release of the beam by observing minutely backward or forward movement
of the tip of the pointer. In a balance fitted with a lamp and scale arrangement, up
and down movement of the spot of light on the scale is more pronounced and can
be used for the purpose.
• The arresting mechanism, pan stops and stirrups are working smoothly.
• The action of rider pick up bar is without any jerks and the rider is able to sit
firmly in the notches of the rider bar.

The results obtained will depend on the environment. Therefore, an area, which is
reasonably free from vibration and excessive air current, is selected. The changes in
temperature and relative humidity should also be minimal. The balance should be
placed on a sturdy stone table. The floor on which the balance is resting should be
rigid and preferably isolated from the floor. The balance should be on the ground
floor.
2.5.3 Sensitivity
Sensitivity is the rate of change in deflection. That is if change in deflection is Â
by a small addition of weight m in one of the pans. Then sensitivity S is given by:
S D Â=m (2.1)
In practice displacement of the indicating element is measured. Hence, sensitivity
is the ratio of displacement 1 of the indicating element between two positions of
equilibrium to the increase m of the load, which produces that displacement.
Thus, sensitivity S of a balance is given by
S D 1=m (2.2)
But in actual practice, from the utility point of view, it is the reciprocal of sensitivity
S.R., which is most often used. Sensitivity reciprocal S.R. is the change in load
required to produce a unit change in displacement of the indicating element. Thus,
S.R. is given by:
S:R: D m=1 (2.3)
As sensitivity may change with the load, sensitivity reciprocal S.R. is generally
determined at three loads namely at minimum, half, and full loads.
2.5.3.1 Sensitivity for Transpose Weighing
To determine sensitivity reciprocal, pans are appropriately loaded with load L, and
a small weight of mass m is then placed in the right-hand pan. The beam is balanced
so that the pointer moves almost between the extreme left of the scale and centre
of the scale. The rest point R1 is worked out by taking three left turning points
and two right turning points. The small weight is then transferred to the other pan
and the rest point R2 is similarly determined. Five such observations are taken;
difference between consecutive rest points is noted. Then mass of the small weight
divided by the mean of the differences gives the sensitivity reciprocal of the balance
at that load L. The S.R. is evaluated at the minimum, half and full capacity of the
balance.

Table 2.3 Observations for S.R.
S.no Load on
R.H.P
Load on
L.H.P
Observations Mean Rest point Difference in
rest points
1 L C m L 1.5 1.7 1.9 1.7 5.05
8.5 8.3 8.4 6.55
2 L L C m 7.4 7.6 7.8 7.6 11.6
15.7 15.5 15.6 6.60
3 L C m L 2.3 2.5 2.7 2.5 5.0
7.6 7.4 7.5 6.60
4 L L C m 6.4 6.6 6.8 6.6 11.6
16.7 16.5 16.6 6.55
5 L C m L 1.8 2.0 2.2 2.0 5.05
8.2 8.0 8.1 Mean difference=6.58
If L D 1 kg and m D 5 mg, then sensitivity reciprocal at 1 kg load is 5=6:58 D 0:7599 D 0:76 mg
In the case of small balances, with higher sensitivity, it is difﬁcult to get a proper
weight, as weight smaller than 1 mg is not available. In such a case, two weights
of equal denomination whose difference in mass values is known and is less than
1 mg are used for this purpose. The small weights are placed on each pan along with
the load at which the sensitivity reciprocal is to be determined. Weight with higher
mass value is placed on the right-hand pan. The rest point R1 is determined. Then
the small weights are interchanged and rest point R2 is again determined. The rest
of the procedure is same as described above.
Observations and their recording for measurement of sensitivity reciprocal are
given in Table 2.3.
2.5.3.2 Sensitivity for Substitution Weighing (Borda Method)
Load the balance appropriately; weights are adjusted in such a way that the pointer
moves between the extreme right and centre of the scale. Take observations for at
least three turning points, two of the extreme right and one sandwiched at the left.
Calculate the rest point. Let it be R1. Place a small appropriate weight of mass m
in the right-hand pan and note the observations and calculate the rest point. Let it
be R2. Weight must be such that the rest point is midway between extreme left and
centre of the scale. So that R2 R1 is largest and the relative measurement error is
least. Sensitivity reciprocal S.R. is given by
S:R: D m=.R2 R1/ (2.4)
2.5.3.3 Variation in Sensitivity
The sensitivity of a balance with coplanar three knife edges coplanar a D 0, can be
increased by

(a) Making the beam longer
(b) Making the beam lighter
(c) Bringing centre of gravity closer to the central knife edge (centre of oscillation)
If all three knife edges are coplanar and there is no bending of the beam when
weights are placed in each pan, the sensitivity is independent of the value of weights
placed in each pan.
In actual practice, the beam bends slightly when loaded, so even when all the
three knife edges are coplanar, S (sensitivity) decreases with load.
To avoid the variation in sensitivity, terminal knife edges are kept slightly above
the central edge. With the knowledge of the elastic property of the beam, the gap
may be so chosen that the sensitivity remains practically the same.
Moreover, in practice the knife edges cannot be made perfectly sharp. So they are
slightly rounded off transversally to bear the load, otherwise they will get rounded
due to application of load in no time. In that case line of application of load will be
along the edge but passing through their respective centres of curvature, which will
be lower than the plane of the edges, so small positive value of gap is necessary.
Further due to wear and tear, the central knife gets rounded off thereby lowering
the horizontal plane passing through it, which decreases the value of gap, so it is
necessary to keep a finite positive value of gap initially.
The decrease in sensitivity is more in new balance especially if the knife edges
are not properly round off. This will show the increased sensitivity at the time of
initial verification or approval of models.
2.5.4 Period of Swing or Time Period
Period of swing or the time period is the time taken by the beam to complete one
oscillation. For a given condition of the balance and mass on the pans, the square of
time period is proportional to sensitivity. So any subsequent change in time period
is, therefore, a quick indication of a change in sensitivity. Similar to sensitivity,
time period also depends on the load. So it is determined at three loads namely at
minimum, half and full load.
The period is determined under the same three conditions of loading in which
sensitivity was determined. Load the balance with a load L and adjust so that pointer
moves around the centre of scale. Start measuring time, using a stopwatch, at the
instant when pointer stops and starts turning say from extreme left, time is taken
for n C 1 such turning points (n complete oscillations). Time period is the time
taken divided by n, the number of oscillations; five complete oscillations (six turning
points on one side) is the ideal number; however, three oscillations are also good
enough for less sensitive balances. Five such sets are taken. Mean of five such time
periods is taken and reported as the time period or period of swing of the balance at
that load.
Observations and their recording for measurement of time period are given in
Table 2.4.

Table 2.4 Observations for
time period
S.no. Number of
oscillations
Time (s)
1 5 156
2 5 157
3 5 155
4 5 156
5 5 156
Mean D 156 s
Timeperiod D 156=5 D 31:2 s
2.5.5 Accuracy of Rider Bar
A rider bar assembly is provided to make small changes in load without resorting to
the use of very small weights. The weights smaller than 1 mg are neither convenient
to use nor easy to make. Since changing the position of the rider affects the change in
effective loading to a particular pan, the accuracy of the rider scale is very important
[8]. The extreme notch on either side of the scale should lie in the vertical plane
passing through the corresponding terminal knife edge. This condition is necessary
as normally we ﬁnd out the mass of the rider by weighing, and we assume that load
equivalent to its mass is applied to the corresponding pan, when the rider is placed
at the extreme notch of the rider scale. In a 2-g micro balance if extreme notch of
the rider scale is shifted by 0.06 mm in a 120-mm beam, the error with a 5-mg rider
is 5 g. The notches should also be well formed and uniformly spaced so that rider
sits erect and applies correct proportional load to the speciﬁc pan. Quality of the
notches can be seen with a good magnifying glass. And equality in spacing between
the notches is measured with a long focal length microscope.
The matching of the extremes of the rider scale with corresponding terminal knife
edges is checked by either of the following methods:
2.5.5.1 Rider Exchange Method
The rider is kept on one end of the scale say left and a weight whose mass is
accurately known and nominally equal to its mass is placed in the right-hand pan.
Observations are taken and rest point R1 is calculated. Then rider is transferred to
the other extreme of the scale (right) and the weight at the same time is transferred
to the other pan. Observations are taken and rest point R2 is calculated. If the two
rest points are equal. Then effective mass of the rider is equal to that of the standard
weight. Otherwise effective mass of the rider is given as
Effective mass of the rider D mass of standard weight C S:R:.R2 R1/=2
Observations and their recording for effective mass of the rider are given in
Table 2.5.

Table 2.5 Observations for effective mass of rider
S.no Left hand Right hand Observations Mean Rest point
1 Rider on its scale 10 mg 4.6 4.8 4.0 4.8 9.1
13.5 13.3 13.4
2 10 mg Rider on its scale 4.5 4.7 4.9 4.7 11.15
17.7 17.5 17.6
13.8 13.6 13.7
4 10 mg Rider on its scale 3.6 3.8 4.0 3.8 11.2
18.7 18.5 18.6
12.6 12.4 12.5
If S.R. is 10g per division and mass of 5 mg weight is 5.005 mg, the effective mass of the rider is
given by:
Effective mass of rider D 5:005 C .9:13 11:18/ .0:01/=2
D 5:005 0:010 D 4:995 mg
Table 2.6 Calibration of rider bar
S.no Position of rider Load on other pan Observations Mean Rest point
14.5 14.3 14.4
2 Rider on pan 10 mg 3.5 3.7 3.9 3.7 10.15
16.7 16.5 16.6
15.8 15.6 15.7
4 Rider on pan 10 mg 3.6 3.8 4.0 3.8 10.2
16.7 16.5 16.6
12.6 12.4 12.5
Difference is 10:13 10:18 D 0:05, which is negligibly small. This indicates that extreme notch
of the rider scale lies in vertical plane passing through the corresponding terminal edge
2.5.5.2 Rider Transfer Method
The alternative method is to transfer the rider from the terminal notch to the pan
bellow. Balance is maintained by keeping an equivalent weight in the other pan.
The method is quite satisfactory for normal analytical balances say of 200 g capacity
with each notch equal to 0.1 mg. But the method is liable to error in a microbalance,
especially if the rider bar is offset vertically from the plane of knife edges. In such
cases, the centre of gravity will shift when rider moves from the rider bar to the pan
below it. This will cause change in sensitivity.
Observations and their recording for rider transfer method are given in Table 2.6

2.5.5.3 Positions of Centre of Rider Scale and Edge of the Fulcrum
A check may also be made if the centre of the rider scale lies vertically above the
central knife edge. Place rider at the central notch (zero of the rider scale) and take
observations and calculate the rest point. Then remove the rider without disturbing
anything, take observations and calculate the rest point. If the two rest points are
same within likely experimental error, then centre of the scale lies in the vertical
plane passing through the central edge.
As the differences in the rest points are likely to be small in all the aforesaid tests,
sufﬁciently large number of observations must be taken to eliminate the random
error.
2.5.6 Stability/ Repeatability
Stability is an old term, which essentially means repeatability of the balance.
The word “stability” comes from the conditions of the beam, which is stable if
its centre of gravity is below the fulcrum, and is unstable if above the point of
oscillation. If centre of gravity is much below the fulcrum and arm lengths are
small, then beam would come to rest quickly. It was taken in the sense that rest
points of the beam would be same or very close to each other. This property is
essentially the repeatability. Repeatability means that if a number of observations,
under certain conditions, are taken, then closeness of the observations is a measure
of repeatability. For the test for stability or repeatability, the balance case is kept
closed and the beam is released, rest point is calculated and the beam is arrested
again. The whole process is repeated say ten times without opening the balance
case or disturbing the weights on the pan. The standard deviation of the rest points
are calculated, which, in this case, is taken as a measure of stability.
Through the stability test, essentially the quality and adjustment of bearings and
whole release and arrestment mechanism are judged.
The test is carried out at minimum and full load, and any other load considered
being important. For example, if a balance is used for calibrating weights say 1 kg
and 500 g only, then this test should be carried out only at these two loads.
2.5.7 Repeatability of Weighing
Repeatability of a balance at a given load can be assessed by repeatedly comparing
similar weights and assessing their differences. Smaller is the dispersion in the
differences, better is the repeatability of weighing.
As there are three methods of comparing the weights, repeatability of weighing is
determined by using these three methods of comparison. Repeatability of direct and
transposition weighing can be determined by one experiment. In fact by interposing
observations for rest points at no load between successive observations when fully
loaded, the experiment is also used to ﬁnd out arms ratio of the beam.

For repeatability of weighing for the method of substitution weighing, a separate
test is conducted.
Before starting the test for repeatability of single and transposition weighing,
observations are taken and rest point at no load is calculated. Two standard weights
of equal volume and each having a mass equal to the full capacity of the balance
are taken and are compared by transposition (double) weighing method. Rest points
in unloaded condition are calculated between two successive double (transposition)
weighing. Ten double weightings are performed with 11 rest points at no load. Last
observations should be at no load. The maximum departure from the mean of the
difference of the two weights or the standard deviation of these differences gives an
estimate of the overall repeatability of the balance.
Interposing observations of the rest points at no load between successive
transposition weighing brings out the fatigue characteristics of the beam. If the beam
is not properly designed, it may bend under a temporary heavy load and returns but
slowly to its former state. Consequently, the rest point may change and affect the
repeatability of weighing. To make the point clearer, a sample of observations and
calculations is given below and discussed. The test so carried out not only gives the
repeatability of single weighing, transposition weighing but also the stability and
equality of the length of the arms of the balance.
2.5.8 Equality of Arm Lengths
In an ideal balance, the length of the arms should be equal. But in practice there is
always some difference inbetween the two arm lengths of the balance. Normally,
the difference in arm lengths relative to mean arm length is calculated.
If lr, ll are the respective lengths of right and left arms, then relative difference in
arm lengths E is expressed as
E D .lr ll/=f.lr C ll/=2g (2.5)
To ﬁnd out the inequality, rest point of the balance at no load is determined. Two
pans are then loaded with weights of equal mass and volume. The mass of the
weights selected is equal to the total capacity of the balance. If the arms are unequal,
the rest point will change and a small weight will have to be placed on one of the
pans to restore the rest point to its original value. This small weight is the measure
of the inequality of arms. If M is the mass of each weight placed in the pans and m
is the mass of the small weight, then the arm above the pan in which m is placed is
shorter and E—the relative difference in arm length is given by
E D .lr ll/=f.lr C ll/=2g D m=M (2.6)
In actual practice as the two masses usually differ, the average of the rest points
obtained before and after changing the weights is taken that corresponds to truly
equal loads.

2.5.9 Calculation of Arm Ratio, Repeatability of Single
and Double Weighing
2.5.9.1 Procedure
Level the balance, which of course, is the ﬁrst thing to do, balance with poise nuts so
that the pointer of the balance swings equally on both sides of the central graduation
for freely swinging balance and rests in the centre of the scale for the damped
balances.
(a) Rest point at no load is determined.
(b) Two weights equal in mass, surface area and volume, mass of each is equal
to the full capacity of the balance, having distinction marks say W 1 and W 2
are placed; W 1 in left-hand pan and W 2 in right-hand pan. Rest point is
determined.
(c) Weights W 1 and W 2 are interchanged i.e. now W 1 is in right-hand pan and
W 2 in left-hand pan; the rest point is again determined.
The processes at (a), (b) and (c) are repeated ten times, the process at (a) is repeated
once more. The experiments begins and ends with weighing at no load i.e. there will
be 11 rest points at no load and 10 rest points each with two positions of weights
W 1 and W 2.
Let us denote ten rest points with the weight W 1 on left-hand pan by L1, L2; : : :,
L10 and R1, R2; : : :, R10 when the weight W 1 is on right-hand pan.
Take the mean of all L0
s and R0
s separately, let it be L and R then
L D .L1 C L2 C C L10/=10 (2.7)
and
R D .R1 C R2 C C R10/=10 (2.8)
Mean of L and R denoted by RL is given as
RL D .L C R/=2 (2.9)
If Sf is the sensitivity reciprocal at full load, then mf the mass equivalent of RL is
mf D RL Sf (2.10)
Take also mean of all O0
s and let it be O and is given as
O D .O1 C O2 C C O11/=11 (2.11)
If So is the sensitivity reciprocal at no load, mass equivalent of O is m2, such that
m2 D O So (2.12)

2.5.9.2 Arm Ratio
Then E the deviation of arm ratio from unity is given as
E D .mo mf/=W (2.13)
where W is the mean value of masses of W 1 and W 2.
If .mo mf/ is positive, then right arm in longer than the left arm.
If (mo mf/ is negative, then the left arm is longer than the right arm.
In the above calculations, it has been assumed that inequality of arm lengths is
such that the pointer moves within the scale on interchanging weights. If it is not
possible and weight of mass m is required to be placed say in the right-hand pan
when weight W 1 is in right-hand pan and is not required to be shifted when weights
W 1 and W 2 are interchanged. This means right arm is shorter in proportion to
m .mo mf/ and E—the deviation of arm ratio from unity is given by
E D Œm .mo mf/=W (2.14)
2.5.9.3 Repeatability of Single Weighing
Determine differences between consecutive rest points such as L1–O1, O2–R1,
L2–O2, O3–R2, and so on till O11–R10.
There will be 20 such differences. Standard deviation of these differences is
calculated and is multiplied by sensitivity ﬁgure, which gives the measure of the
repeatability of single weighing. In this case, it has been assumed that sensitivity
reciprocal with and without loads are equal if not then all O0
s are to be multiplied
by So and all L0
s and R0
s by Sf and then standard deviation of their differences
is calculated. For the relative repeatability, divide the above result by W . Both the
mass values should be in the same units. Fractional repeatability is expressed as k
parts in 10n
, where n is a positive whole number and k is in between 1 and 10.
2.5.9.4 Repeatability of Double Weighing
Determine L1–R1, L2–R2, and so on till L10–R10. There will be ten such
differences, calculate the mean and the standard deviation. One half of the value
of the standard deviation multiplied by the sensitivity reciprocal at full load gives
the precision of double weighing. For the relative repeatability, divide it by the load
value. Express the result as k parts in 10n
, where n is a positive whole number and
k is in between 1 and 10.
2.5.9.5 Overall Repeatability of the Balance While in Actual Use
Find the standard deviation of O1, O2, O11 and multiply it by the sensitivity
reciprocal at no load, let it be O. Similarly calculate separate standard deviations of

2.5 Requirement for Arm Ratio and Repeatability of Weighing 39
L1, L2; : : :, L10 and R1, R2; : : :, R10. Multiply each of them by Sf—the sensitivity
reciprocal at full load and let these are L and R respectively.
Then the mean standard deviation in unit of mass is given by:
D f.11 O2
C 10 L2
C 10 R2
/=28g1=2
(2.15)
is the measure of the overall repeatability of the balance.
2.5.10 Requirement for Arm Ratio and Repeatability
of Weighing
It is recommended that numerical value of overall repeatability should be less than
or equal to the value of the sensitivity reciprocal at full load.
[m .mo mf/] should be less than sensitivity reciprocal at full load. That is the
deviation from unity is not more than a fraction equal to sensitivity ﬁgure divided
by full load.
2.5.11 Test for Parallelism of Knife Edges
Ideally all the three knife edges of an equal-arm balance should be parallel to each
other in plan as well as in the elevation. Terminal knife edges must lie in one
horizontal plane. The horizontal plane containing the terminal knife edges may or
may not coincide with the horizontal plane containing the edge of the central knife.
Lack of parallelism of the knife edges as seen from top (in plan) is called as wind.
It is depicted in magniﬁed manner in Fig. 2.9.
If we see from the front of the balance, then the edges of the terminal knife edges
may not lie in one horizontal plane. The lack of lying terminal knife edges in one
horizontal plane is known as dip. This is shown in Fig. 2.10.
These can be measured by using surface plates and slip gauges and a dial
indicator gauge. Dip is eliminated by adjustment of the height of the one terminal
Fig. 2.9 Lack of parallelism
in plan (wind)

Fig. 2.10 Lack of
parallelism in vertical plane
(dip)
Fig. 2.11 Special bearing
with weight
knife edge to match with that of the other. The wind is brought to zero by adjusting
the side screws of the knife edge holder. The dip can also be tested by determining
the values of sensitivity at different loads. The dip value must be such that variation
in sensitivity with load is minimal.
The degree of parallelism in plan between the terminal knife edges is assessed by
weighing. Instead of regular bearing of the terminal edge, a bearing much smaller
say one-fifth of the length of the terminal knife edge carrying a weight equal to
one-fifth of the capacity of the balance is taken, and the rest points are calculated
in different positions of the special bearing one terminal knife edge. The bearing is
taken from one end of the terminal knife edge to its other end.
For bearing plane moving from front to the backside on the left terminal knife
edge, increasing values of rest points indicate that backside of terminal knife edge
is going away from the central knife edge. Opposite will be the result, while testing
the right terminal knife edge. The knife edge is adjusted with the screws of its holder
so that rest points all along the length of the knife edge are same.
Dip in the terminal knife edges is assessed by finding the sensitivity of the
balance in different positions of the bearing plane. If the hind portion of the knife
edge is higher, then sensitivity will decrease as the bearing plane is moved from
front to back.
The special bearing is shown in Fig. 2.11.
Variation in rest points will indicate the lack of parallelism of the terminal
edge with respect to central edge. Similarly calculating sensitivity of the balance
in different position of the bearing plane on the terminal knife edge will give an
assessment of the lack of uniformity of the dip. Weighing method is used to check
the equality of the arms and the correspondence with the rider bar.

2.6 Methods of Weighing 41
2.6 Methods of Weighing
There are two methods of calibration of weights. First most common is one to
one comparison i.e. comparing a weight against a standard of same denomination.
The second one is to take a group of weights and compare it against a standard of
suitable nominal value. In each method, there are three methods of weighting when
comparison is carried out with a two-pan equal-arm balance.
2.6.1 Direct Weighing
A body of unknown mass (weight under test) is placed in one pan of the balance
while the body of known mass, normally called standard weight, is placed in the
other pan. Observations are taken and rest point R1 is calculated when both the pans
of the balance are empty.
Then weight under test is placed in left-hand pan and standard mass is kept in
right-hand pan. Observations are taken and the rest point R2 is calculated.
Then mass of weight under test W is
W D S C .R2 R1/ S:R: (2.16)
S.R. stands for the sensitivity reciprocal of the balance i.e. mass value of one
division of its scale.
Equation (2.16) is valid on the following two assumptions viz.
1. Arm lengths are identically equal
2. a (the vertical separation between the horizontal planes touching the terminal
knife edges and through the central knife edge) is zero. Otherwise value of S.R.
will be different in two cases of weighing.
However, none of the two conditions is fulﬁlled in practice, especially when
accuracy demand is pretty high. Hence, direct weighing method should not be used
for any calibration work needing a relative standard uncertainty smaller than 10 4
.
The following two methods, therefore, are being dealt with in some detail.
2.6.2 Transposition Weighing
The weight under test W is placed in the left-hand pan and the standard weight S
in the right-hand pan. Observations are taken and the rest point R1 is calculated.
The two weights are then interchanged and the rest point R2 is similarly calculated.
Then W is given as
W D S C .R1 R2/.S:R:/=2 (2.17)

The only condition to be fulﬁlled by the balance is that arm lengths are equal enough
so that it is possible that the two rest points R1, R2 are within the scale without any
addition of extra mass on either side.
2.6.3 Substitution Weighing
In this method, only one mostly right-hand pan of the balance is used while on the
other (left) pan a constant but unknown mass (D) is placed so that observations are
possible when the weight under test W and standard weight S are placed in the
right-hand pan turn by turn. W is given as
W D S C .R2 R1/S:R: (2.18)
As S and W are not very much different and a is very small, either S or W may be
used for S.R.
In this case, there is no limitation on the balance regarding the inequality of the
arm lengths or mass of the pans.
2.7 Double Transposition and Substitution Weighing
For simultaneous determination of sensitivity reciprocal and obtaining difference in
mass of two weights with higher precision, double transposition method and double
substitution method is used.
2.7.1 Double Transposition Method
The procedure for simple double transposition weighing is as follows:
The weight under test of mass W is placed in the left-hand pan and the standard
weight of mass S in the right-hand pan. Observations are taken and the rest point
R1 is calculated. The two weights are then interchanged and the rest point R2 is
similarly calculated. This observation is repeated after allowing a time required
for normal interchanging of weights and the rest point is R3. The weights are
interchanged again bringing them to the original positions and rest point R4 is
calculated. Schematically it may be written as
Left-hand pan Right-hand pan Rest point
W S R1
S W R2
S W R3
W S R4

2.7 Double Transposition and Substitution Weighing 43
Giving
W S D .1=4/ .R1 R2 C R4 R3/ S:R: (2.19)
There are many advantages of this method. The variance is reduced to half of that
in double weighing.
Without going into the mathematical proof here, it is sufficient to state that if
quantity Q is a linear combination of the measurement r1, r2, r3 and r4 and is
represented as:
Q D n1r1 C n2r2 C n3r3 C n4r4 (2.20)
Then variance in Q D s2
.n2
1 C n2
2 C n2
3 C n2
4/=.n1 C n2 C n3 C n4/2
(2.21)
where s2
is the variance of each measurement.
In this case, each n is equal to 1, if s is the standard deviation of each rest point,
then variance of (W S) is equal to .4=16/ s2
D .1=4/ s2
. In case of double
weighing, the variance is s2
=2.
Moreover, one may notice that position of weights in the first and fourth
steps is the same. Equality of R4 and R1 ensures the good repeatability. Due to
environmental conditions, quite often there is a constant drift. If the drift is linear
with respect to time, then the effect of the drift is eliminated. The effect will at least
be reduced if drift is not exactly linear. This will become clear from the following
calculations.
Let us assume the time taken for each comparison is constant. Let r1, r2, r3, r4
be the rest points with drift, and the drift is for each weighing, then
r1 D R1
r2 D R2 C
r3 D R3 C 2
r4 D R4 C 3
where R1, R2, R3 and R4 are the rest points without drift, i.e. R1 D R4. Then
.r1 r2 Cr4 r3/ D .R1 R2 CR4 C3 R3 2/ D .R1 R2 CR4 R3/
(2.22)
2.7.2 Double Transposition with Simultaneous Determination
of S.R.
The process of double transposition with simultaneous determination of S.R. is
carried out in five steps. The weight under test of mass W and a small weight of

mass m are placed in left-hand pan and standard weight of mass S in right-hand
pan and rest point R1 is determined. The weights are then interchanged and rest
point R2 is determined. For finding out the sensitivity reciprocal, in the weighing
process itself, the small weight of mass m is transferred from left pan to the right-
hand pan, and rest point R3 is determined. In the fourth step, standard weight and
the weight under test are interchanged and let the rest point be R4. In the final fifth
step, the small weight is transferred from the right-hand pan to the left-hand pan, and
rest point R5 is determined. First and second steps constitute one double weighing;
similarly third and fourth steps form another double weighing. Steps 2 and 3, and 4
and 5 constitute two double weighing for sensitivity reciprocal.
Left pan Right pan Rest point
W C m S R1
S C m W R2
S W C m R3
W S C m R4
W C m S R5
Giving
W S D m.R1 R2 C R4 R3/=.R2 R3 C R5 R4/ (2.23)
In this procedure, sensitivity reciprocal and difference in mass of two weights are
simultaneously determined. Also each determination is by double transposition.
Had we determined the sensitivity reciprocal and differences in mass separately,
we should have required eight comparisons instead of five. Hence the procedure is
labour saving.
Moreover, in this case also the position of weights in the first and fifth steps is
the same. Equality of R5 and R1 ensures the good repeatability. Here also it can
be shown that the effect of the drift is eliminated, if the drift is linear with respect
to time, and reduced if it is not exactly linear. This will become clear from the
following calculations.
Let us assume the time taken for each comparison is constant. Let r1, r2, r3 r4,
r5 are the new rest points, and if the drift is for each weighing, then r1 D R1;
r2 D R2 C ; r3 D R3 C 2, r4 D R4 C 3 and r5 D R5 C 4. R1, R2, R3, R4
and R5 are the rest points when there is no drift, i.e. R1 D R5.
Then
.r1 r2 C r4 r3/=.r2 r3 C r5 r4/
D .R1 R2 C R4 C 3 R3 2/=.R2 C R3 2 C R5
C4 R4 3/
D .R1 R2 C R4 R3/=.R2 R3 C R5 R4/ (2.24)

2.7 Double Transposition and Substitution Weighing 45
2.7.3 Double Substitution
The process of double substitution is carried out in four steps. Keep left-hand pan
loaded with the constant load. Weight under test of mass W , standard of mass S,
is placed successively in the right-hand pan and corresponding rest points R1 and
R2 are calculated. The observation of the second step is repeated after allowing the
time required for taking out and placing the weight in the right-hand pan and rest
point R3 is obtained. In the fourth step, weight under test is placed in the right-hand
pan instead of the standard weight. It is shown schematically below:
Constant load W R1
Constant load S R2
Constant load S R3
Constant load W R4
Giving
W S D .1=2/ .R2 R1 C R3 R4/ S:R: (2.25)
Here also, the positions of weights in the first and last step are same. So, linear drift
will be eliminated and greatly reduced if slightly nonlinear. The variance of W S
is also reduced to half of that of substitution weighing.
2.7.4 Double Substitution Weighing with Simultaneous
Determination of S.R.
The process of double substitution with simultaneous determination of S.R. is
carried out in five steps. Keep left-hand pan loaded with the constant load. The
weight under test of mass W , standard of mass S, standard S with a small mass m
and weight under test W with the small mass m are placed successively in the right-
hand pan and corresponding rest points R1, R2, R3 and R4 are calculated. In the fifth
step small mass m is removed from the right pan and rest point R5 is calculated. It
is shown below schematically:
Constant load W R1
Constant load S R2
Constant load S C m R3
Constant load W C m R4
Constant load W R5
Giving
W S D m.R2 R1 C R3 R4/=.R2 R3 C R5 R4/ (2.26)

In this case, the difference in mass of the weight under test and the standard as well
as the sensitivity reciprocal (S.R.) have been obtained two times, thus the shaving
of time has been achieved. Further the effect of linear drift is eliminated.
2.8 Maintenance of Standard Balances
It is recommended that for all measurements of mass, method of substitution
weighing should only be used. So all standard balances should, at least, be evaluated
for those parameters, which may affect the accuracy in determination of mass using
the method of substitution. So for routine purposes there is no need for arm ratio
test, etc. There are two parts of maintenance of standards
Physical: This includes maintenance of all components of the balance in good
working condition. Cleaning of different parts and levelling.
Metrological: For routine work, sensitivity reciprocal and repeatability of substi-
tution weighing in terms of standard deviations must be measured periodically and
a record of these two parameters must be maintained. Datewise records must be
maintained both in the table form and in graphical form. At least one of them must
be displayed along with the balance.
2.8.1 Category of Balances
In India, for legal metrology, there are three classes of balances namely reference,
secondary and working balances.
2.8.2 Reference Balances
These balances are kept and maintained by the Regional Reference Standards
Laboratories (RRSL). These laboratories are under the control of the Director
Legal Metrology, Central Government. Staff employed in these laboratories is well
qualiﬁed and mostly trained at NPL, New Delhi, and is supposed to know their job
well. So no speciﬁc write up is required for them.
2.8.3 Secondary Standard Balances
These balances are kept and maintained by Secondary Standard Laboratories of the
country’s State Governments and are in larger number.

2.8 Maintenance of Standard Balances 47
Every secondary standard balance must be verified at least once in a year.
Reference standard weights must be used for this purpose. In between the two
verifications, the following parameters should be evaluated within the laboratory.
• Sensitivity figures at maximum, half and minimum capacity of the balance for
substituting weighing
• Variation in sensitivity figures if any
• Repeatability of substitution weighing at maximum capacity of the balance and
at the load equal to the lowest denomination of the weight which is going to be
verified on it
A datewise record of the values of all the previous sensitivity reciprocals and
repeatability should be kept.
Also after periodical testing, it should be ensured that
• Variation in sensitivity reciprocals is not more than 10% of the mean sensitivity
reciprocal.
• No value of the sensitivity reciprocal is more than the prescribed value. For ready
reference, these should be given in the record sheet for sensitivity reciprocal.
• The value of repeatability is smaller than that of sensitivity reciprocal at full load.
2.8.4 Working Standard Balances
Every working standard balance must be evaluated at least once in 6 months. The
secondary standard weights must be used for this purpose.
Parameters for which it should be evaluated are
• Sensitivity reciprocals at maximum, half and minimum capacity of the balance
for substituting weighing
• Variation in the values of sensitivity reciprocals
• Repeatability of substitution weighing at maximum capacity of the balance and
at the load equal to the lowest denomination of the weight which is going to be
verified on it
A datewise record of the values of the aforesaid parameters should be kept.
It should be ensured that
• Variation in sensitivity reciprocals is not more than 20% of the mean value of the
sensitivity reciprocal.
• No sensitivity figure is more than the prescribed value.
For ready reference, these figures should be indicted on a record card, which should
be kept inside the balance.
• Repeatability is numerically smaller than or equal to half the value of the
sensitivity reciprocal at full load.

References
1. G. Girard, The organs of the convention du metre the kilogram and special researches in mass
measurement. Lecture delivered in advanced course in Metrology, held at NPL, New Delhi,
(Ruprecht balance, 1985)
2. F.A. Gould, A knife-edge balance for weighing of the highest accuracy”. Proc. Phys. Soc. B.
42, 817 (1949)
3. S.V. Gupta, Mass standards-kilogram. NPL Tech. Bull. VII(3) (1975)
4. T. Lal, S.V. Gupta, A. Kumar, Automation in mass measurement. MAPAN-J Metro. Soc. India
7, 71–80 (1992)
5. M. Kochseik, R. Probst, Investigation of a hydrostatic weighing method for 1 kg mass
comparator Metrologia 19, 137–146 (1984)
6. M. Kochseik, R. Probst, R. Schwartz, Mass comparison according to a hydrostatic weighing,
with an uncertainty smaller than 5:10 9
, Proc. 10th Conf. IMEKO, TC 3 on Measurement of
force and Mass, (1984), 91–95
7. B. Bessason, C. Madshus, H.A. Froystein, H. Kolbjornsen, Vibration criteria for metrology
laboratories. Meas. Sci. Technol. 1009–1014 (1999).
8. P.F. Weatherhill, Calibration of the beam notches, 1030. J. Am. Chem. Soc. 52, 1938–1944
For further reading about two pan balances and related topics
9. A.H. Corwin, Micro-chemical balances. Industr. Eng. Chem. (Anal.) 1, 258 (1944)
10. V.Y. Kuzmin, Basic features in primary standard Equal-arm balances. Meas. Tech. (USA) 31,
1064–1068 (1988)
11. Encyclopaedia Britannica,Balance, 1059–1064 (1989)
12. M. Theisen, Etude sur le a balance. Trav. Bur. Int. Poids Measure 5, 8, (1986)
13. F.A. Gould, in Balances, A Dictionary of Applied Physics, vol. 3, ed. by R. Glazebrook
(Macmillan, London, 1923), p. 113
14. A.F. Hodsman, The effective radius of curvature of knife edge. J Sci Instrum. 29, 330 (1952)
15. M. Kochseik, R. Kruger, H. Kunzmann, Setup of a laser interferometer for measurement of the
beam oscillations of a balance. Bull. OIML. 70, 1–6, (1978)
16. G.F. Hodsman, A method of testing bearing materials for chemical balance. J. Sci. Instrum. 26,
341 (1949)
17. J.J. Manely, Observations on the anomalous behaviour of the balance, an account of devices
for increasing accuracy in weighing. Phil. Trans. Roy. Soc. London, A 210, 387, (1910)
18. F.B. Hugh-Jones, The modern balance and its development. J. Phys. E. Sci. Instrum. 15,
981–987, (1982)
19. E. Debler, K. Winter, Improvement of the weighing accuracy of a 50 kg beam balance. IMEKO,
(1986)
20. R. Schwartz, M. Mecke, M. Firlus, A 10 kg comparison balance with computer controlled
weight changing mechanism PTB, submitted as CCM- Doc/88-6 (1988)
21. P. Pinot, Comparator for mass standards – experimental study of correlation coefﬁcient of
measured parameters. Metrologia 28, 27–32, (1991)
22. A.V. Nazarenko, et al., Comparator for checking large weights. Meas. Tech. (USA) 30,
1164–1167, (1987)
23. R. Spurny, Standard balance with upper scale limits up to 10 kg. Meas. Tech. (USA) 29, 90–92,
(1986)
24. L.B. Zurich, S.G. Weissgias, Mettler Dictionary of Weighing Terms (Metttlers Instruments,
Switzerland, 1988), pp. 37–38
25. S.N. Afanasov, S.V. Biryuzov, et al., Estimating the metrological characteristics of a precision
balance. Meas. Tech. (USA) 29, 88–90, (1986)
26. H.V. Moyer, Theory of balance. J. Chem. Edu. 17, 540 (1940)
27. E.P. Osadchii, P.N. Timoshenke, Mathematical model of a precision balance. Meas. Tech.
(USA) 32, 402–404 (1989)
28. C. Xi, L. Yuan, A study of digital force balance measuring device. J. Sci. Instrum. (China) 11,
77–82 (1990)

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