Gravity of the Situation
Understanding, finding and correcting an essential value
by Philip Stein
Force measurements, particularly weighing, are among the most important and the most common made daily all over the world. Weighing is a measurement of the force (attraction) experienced between any mass and the Earth due to gravity. Nobody understands gravity as well as a clumsy man.
A little g is the notation used to represent a quantification of the local gravitational attraction between the Earth and another mass. Little g changes from place to place and time to time because the Earth and its circumstances change.
A big G denotes a quantification of the gravitational attraction of any mass to any other and is not dependent on time and place.
Some measurements and calibrations require knowledge of little g. Errors and uncertainties in little g fall right to the bottom line (a 1% error in g results in a 1% error in the force reported) and therefore exert an important influence on the correctness of measurement results.
Simple weighing, though, whether by double-pan balance or by substitution of known weights for unknown, is not subject to gravitational variation because the force of gravity is the same on the known and unknown masses.
When you calibrate a scale with a set of weights, you are actually using artifacts calibrated as masses. Moving a stainless steel kilogram does not change its mass, but its weight (force of attraction to the Earth) changes for different values of g. A 1kg mass compared in Sevres, France, to the grand master of all kilograms will have a 1kg mass anywhere. If you place it on a local scale and calibrate the scale to read 1kg of weight, then local g will have been taken into account.
The only time a little g correction would be needed is when a platform scale or balance was calibrated and then moved to a distant location without recalibration--a practice that is not recommended in any case.
The most obvious cases where little g must be corrected occur when the weight of a calibrated mass is compared to an opposing force not generated by the weight of another mass. Pressure from a deadweight piston gage, torque from a wrench or load cell, weight measured by springs, load cells or magnetic opposition, and current calibrations in a magnetic balance are some examples of such situations.
Finding the value of g within the United States
While the need to know and make corrections for little g arises very frequently, I have found that a lot of people who make measurements are not aware of this need (or how to make the corrections). Let's examine the units, terms and methods involved in finding the value of g.
The local acceleration of gravity is measured in gals, where 1 gal = 1 cm/sec2. Gravity at the surface of the earth is roughly 980 gals. One milligal is therefore about 1 ppm (part per million). A standard value of gravity has been set at 980.665 gals, and many times you will find instruments calibrated with a correction to this standard value.
Several different formulas can be used for finding the value of g for locations in the United States. Usually these formulas are based on latitude and sometimes altitude above sea level. These formulas are quite inaccurate, however, often being incorrect by 800 to 900 milligals, or about 0.1%. Obviously these formulas may be used if the stated uncertainty of a measurement is correspondingly coarse, but it's not a good idea.
Your local value of g can be calculated free of charge at the National Geodetic Service's (NGS) Web site. Find your latitude, longitude and altitude (a global positioning system receiver can help), and go to www.ngs.noaa.gov/TOOLS/Gravity/gravcon.html. (This URL is case sensitive, so be sure to type it exactly as it's shown or it won't work.)
As you can see by the map on the first Web page, a prediction based on latitude alone won't do very well except in Ohio, Indiana and Illinois, where the lines of equal gravity are straight and parallel, running almost exactly east to west.
How to calculate your local value of g
Press the 'Gravity Prediction' link, and enter your location data. Read the directions carefully at this point to make sure that you use the correct format for entry. Calculation takes a few minutes at most, and the result is returned along with an uncertainty. If you expect to be assessed at any time in the next 10 years, print out this page, and sign, date, frame and hang it on the laboratory wall.
Values of g from this Web site are calculated by interpolation among many measured points. This result is stated with a +/- value. My location in Pennington, NJ, for example, was 9.80151 +/- 2mgal.
Another option is to hire a survey firm or university to visit your location and measure the actual values. It's not possible to do better than about 0.5 mgal this way because even though the daily changes in g can be averaged by a long-term survey, a single calibration or measurement of a customer's instrument will not take these daily changes into account.
Your gravity correction might be as high as 0.2% of your measured value. This is large enough to make quite a difference to pressure, force and torque measurements, so it should be applied to the measurement data.
The expanded uncertainty of this correction is likely to be within 20 ppm anywhere in the United States. This value belongs in any uncertainty budget for pressure and the like. A quick discussion with NGS scientists indicated that this uncertainty is unlikely to be calculated according to current international practice and should probably be treated with a rectangular distribution.
Be careful to distinguish between the correction for g and the uncertainty of that correction. The correction is applied to the data, and the uncertainty of the correction is part of the overall uncertainty budget.
In cases where the correction is small, you might want to include the whole correction in the uncertainty and not apply it to the data. This will save you the trouble of correcting every measurement, at a cost of having to report a larger uncertainty.
Correcting measurements and calibrations
When the measurement or calibration for the local value of little g needs to be corrected, begin with the calibration certificate for your reference standard--the (traceable) instrument or artifact you will be using to make the measurement.
This certificate will tell you the value of gravity used during the calibration of that standard. This is most likely to be either the standard value, 980.665 gal, or the actual local gravity where the standard was calibrated. In either case, the calibration certificate should report the reference gravity if you need to know it.
Define a correction constant as a ratio consisting of your local gravity in the numerator and the reference gravity at the calibration source in the denominator. After determining each measurement value, multiply it by this correction constant to get a value appropriate for your location.
If you are calibrating deadweight pressure testers, torque sensors or force gages, report the corrected value along with the statement "corrected for local value of gravity." Don't report your local gravity. Your customers don't need it and might be confused if you provided this datum.
If you are calibrating masses that will be used on deadweight testers, torque calibrators or force gage calibrations, on the other hand, your customers will need to know your local gravity so they can define their own correction constant. You could also correct your measurements to standard gravity and report your results that way.
How often is recalibration needed?
Improvements made to measurement instruments have the most impact on changing these values; therefore, current values should be obtained from the NGS Web site.
The second largest changes are due to tides and the moon, which vary with a 12-hour period. The magnitude of this effect is about 200 microgals or about 0.2 ppm.
Other changes, such as atmospheric, ground water going up and down with rain and discharge, and tectonic plate movements, all have measurable effects on g, but they're too small to worry about--all on the order of 5 microgals. All of these uncertainties, plus those from interpolation by the Web site formula, are included in the reported uncertainty. None of these changes are of a large enough magnitude to be a cause for concern regarding drift and recalibration.
I would recommend that after getting a current value, applicants check back with the NGS every five to 10 years to make sure nothing has changed.
PHILIP STEIN is a metrology and quality consultant in private practice in Pennington, NJ. He holds a master's degree in measurement science from The George Washington University in Washington, DC, and is an ASQ Fellow.