THE EFFECT OF GRAVITATIONAL GRADIENT ON MASS
MEASUREMENTS
Speaker/Author: RT Mautjana
National Metrology Institute of South Africa (NMISA)
Private Bag X 34
Lynnwood Ridge
Pretoria
0040
South Africa
[email protected]
Phone: 012 841 4798 Fax: 012 841 2131
Abstract
The weight of an object is the force on the object due to gravitational acceleration (g). If the
local gravitational acceleration is known, the mass of an object can be calculated using
Newton’s 2nd Law, F=ma, where acceleration (a) is the gravitational acceleration (g), thus
F=mg. Mass measuring equipment measures force, and not mass as is commonly believed,
and then convert the force to mass, using Newton’s 2nd Law.
A mass comparator has a very high resolution, distinguished by extremely high repeatability
and detecting small changes of mass (weight). Comparators are typically used for mass
measurement analysis by National Measurement Institutes (NMI’s), using weighing
applications that requires absolute weighing at extremely high resolutions. The effect of
gravitational acceleration on an object, using the substitution method, cancels itself if the test
and reference has the same height, and are placed in the same position on the load-receptor.
However the cancelation is not perfect if the center of mass of the standard and test are
different with respect to the earth surface. The lack of cancellation is due to gravitational
gradient. Mass comparators are capable to of detecting these small differences mass (weight).
This paper illustrates the effect of small differences in the height mass standards on the mass
measurements.
1. Introduction
To determine mass measurement at high accuracy, it is necessary to know the gravitational
acceleration (g) at the location of the instrument. The gravitational acceleration can be
determined by measurement on site (using a gravity meter), calculation or interpolation of
measured values.
High accuracy mass determination is influenced by repeatability, linearity, sensitivity and
resolution of the balance/comparator. The substitution method reduces the influence of nonlinearity as it only uses a small section of the range for differential mass determination as the
mass of the reference standard is close to mass of the test standard mass. Comparators are
capable of detecting small differences between the test and reference standards, and
differences in center of mass of objects will influence measurement results if g is not
corrected for.
2. Background
The weight of an object is the force on the object due to gravitational acceleration (g). If the
local gravitational acceleration is known, the mass of an object can be calculated using
Newton’s 2nd Law, F=ma, where acceleration (a) is the gravitational acceleration (g), thus
F=mg. Mass measuring equipment measures force, and not mass as is commonly believed,
and then convert the force to mass, using Newton’s 2nd Law .
Local gravitational acceleration is defined as =
from
Newton‘s
law
of
gravitation
=
,
=
.
at the surface of the earth, as deduced
and
Newton
second
law (1)
Where [6]:
G - is the universal gravitational constant(6.673x 10-11N.m2.kg-2)
- mass of the earth(5.98 x 1024 kg)
- mass of a body
r - distance from the center of the earth to the surface (6378km)
There is a direct relationship between gravitational acceleration and the downwards force
(weight) experienced by objects on earth, as given by Newton second law equation given in
(1) above.
At height h above the surface gravitational acceleration (g) is given by
=(
.
)
(2)
The variation in gravity relative to the height of an object above the earth’s surface is
illustrated below in figure 1.
Figure 1.Gravitational acceleration relative to the height of an object
With reference to figure 1it is observed that the gravitational acceleration (g) decreases with
altitude or distance above the earth‘s surface.
The NMISA utilised the services of Counsel for Geoscience to measure gravitational
acceleration at three different heights to determine the corresponding gravitational gradient of
the laboratory as tabulated in Table 1 below, which shows the gravitational acceleration as
measured in February 2012.
Table 1. Gravitational acceleration at NMISA due to a change in height as measured by the
Counsel for Geosciences [1].
Height
0cm (Ground)
61.7cm
89.9cm
Measured g (m/s²)
9.7860986 ± 0.0000005
9.7860970 ± 0.0000005
9.7860962 ± 0.0000005
Figure 2 shows the variation in gravitational accelerations as determined from the measured
gravitational accelerations.
Figure 2. The gravitational acceleration gradient
A defined body has the same mass in any location; however it has a different weight in
different locations. The weight difference is caused by the variation in gravitational
acceleration of the earth, which decreases with an increase in height
When mass standards are compared in air they displace air which induces an air buoyancy
force. The actual weight of the body will equal the apparent weight plus the buoyant force.
Mass standards of the same mass, for example .1 kg, but of different densities, will have
different volumes and will displace different volumes of air, correction for the air buoyancy
effect is thus needed [2].
From the experimental data in figure 2 it is observed that the gravitational acceleration
gradient as determined in mass laboratory of NMISA as ( ) (s-²) = -3x10-6s-2.
3. Determination of Center of Mass and its effect on mass standards.
The mass laboratory of the National Measurement Institute of Switzerland (Metas) has made
several measurements for center of mass (CoM) for OIML class E1 mass standards and found
that center of mass is quite stable for each nominal value. They also discovered that the
quotient of CoM per height is similar for all nominal values, averaging from 0.454 to 0.455
[4].
Since NMISA does not have capabilities to measure CoM, only heights and diameters of the
available mass standards were measured and the quotient of CoM per height as measured by
Metas was utilised to determine CoM for the weights. Table 2 below shows the measured
heights and diameters for 100 g to 1 kg mass standards and their determined CoM.
Table 2.Center of Mass for NMISA Primary set (manufactured by Mettler Toledo)
Nominal
value
1 000
500
200
100
Height (mm)
80.6
63.8
47.0
38.2
Diameter
(mm)
47.9
37.9
27.9
21.9
Center of
mass (mm)
36.5
29.0
21.4
17.4
CoM/Height
0.453
0.454
0.455
0.455
The comparators were placed about a meter above the surface, and equation (2) was used to
determine applicable gravitational acceleration at each respective CoM where h was the
distance from the ground to the CoM of the weight and G, m and r had their usual meaning as
indicated in the previous section. From the calculated gravitational acceleration for each
CoM the influence due to the change in height on the mass was determined using equation (3)
below.
∆m =
.
. (∆h)
(3)
During determination of mass, if the test and reference mass standards have the same CoM
the actual value of gravitational acceleration cancels out since the standards are placed on the
same load-receptor of the comparator. However, such a cancellation is not perfect if the CoM
of the mass standards differ. The lack of cancellation is due to the presence of a gravitational
gradient and if it is left uncorrected for, it may produce some small errors as depicted in
Table 3 below.
Table 3.Effects due to change in the CoM of mass standards
CoM (mm)
g (m/s2)
∆m (µg)
36.4 OIML
28.75 SS
27.5 SS
19.5Pt-Ir
9.7860957273666
9.7860957546694
9.7860957508348
9.7860957792114
-8.430
-8.814
-5.978
500
28.9OIML
23.9
18.9
9.7860957503746
9.7860957657133
9.7860957810520
-3.663
-2.897
200
21.4 OIML
16.4
11.4
9.7860957733827
9.7860957887214
9.7860958040601
-1.006
-0.699
17.4OIML
12.4
7.4
9.7860957856536
9.7860958009923
9.7860958163310
-0.380
-0.227
Nominal (g)
1 000
100
Table 3 indicates that a reduction of about 7.7 mm in height from a 1 kg OIML shaped mass
standard, and the associated change in CoM, changes the mass by about 8.4 µg while a
reduction of 16.9 mm in height, and the associated change in CoM, change the mass by about
6 µg. Similarly a 5 mm height reduction from a 500g OIML shaped mass standard change
the mass by about 3.7 µg while reducing its height by 10 mm change the mass by about 2.9
µg. For smaller capacity mass standards the effect is much less significant since the
gravitational acceleration approaches a constant.
Table 4 indicates the effect of change due to CoM as a percentage, between the uncertainty of
measurement associated with OIML class E1 mass standards and the uncertainty of the
national measurement standard for mass, prototype No. 56 of the international kilogram.
Table 4. Change in mass as a percentage of uncertainty for OIML class E1 weights
Mass Standards (g)
1 000 (Pt-Ir)
1 000
500
200
100
U mg
± 0.010
± 0.170
± 0.084
± 0.034
± 0.017
∆m (%)
59.8
3.5 to 5.2
3.4 to 4.4
2.1 to 3.0
1.3 to 2.3
Table 4 shows that the changes in mass due to differences in the height of mass standards are
very small as their influence is less than 10% of the class E1 uncertainty. Although the effect
of small changes in height seemed very small, the effect could be significant especially in
disseminating from 1 kg national standard with a combined uncertainty of 5µg [5] to a 1 kg
stainless steel weight as the correction due to gravitational gradient can be more than the
uncertainty of the national standard.
4. Conclusion
Mass standards with the same center of mass, on the same balance load-receptor experience
the same gravitational acceleration. However, mass standards with different center of masses
experience different gravitational accelerations. The difference in the gravitational
acceleration for mass standards with different center of masses produces small errors of up to
about 8 µg on 1 kg mass standards. If this is left uncorrected, it may lead to incorrect results
for higher accuracy mass standards. The effect of gravitational gradient is thus significant for
higher capacity mass standards, with different center of masses, than for lower capacity mass
standards.
Acknowledgements
The author is grateful to Mr B van der Merwe, Mr H Liedberg and Mr V Ramnath of the
National Metrology Institute of South Africa for their inputs and constructive criticism which
resulted in the improvements to this manuscript.
5. References
1. R.H Stettler. Tie in of the three gravity base stations at the national metrology institute of
South Africa. Report No.2012-0087. Council for Geoscience, April 2012.
2. F. Mathis and C. MÜller-SchÖll. Mass Metrology Training Manual. Section 5. Air
buoyancy correction, Mettler Toledo, Switzerland, June 2010
3. National Physical Laboratory (NPL), http://www.npl.co.uk/reference/faqs/how-can-idetermine-my-local-values-of-gravitational-acceleration-and-altitude-(faq-pressure)
4. P. Fuchs, WG: Center of gravity. Federal Office of Metrology METAS, Switzerland,
2010
5. BIPM. Certificate for the 1kg mass prototype No 56 belonging to South Africa, No 88,
2005
6. M.R Spiegel. Theory and Problems of Theoretical Mechanics, 1982, McGraw-Hill Book
Company, SI (Metric) International edition.