# Precise Measurement of Mass

## Gravitational Acceleration with Variations and Corrections

Gravitational acceleration can be measured by dropping an object in a vacuum chamber and measuring speed as a function of time as the object accelerates. This is the method made famous by Galileo. He is supposed to have dropped a large and a small object from the leaning tower of Pisa and found that they both hit the ground at the same time. Since then there has been a lot more work on the subject to define the gravitational acceleration.

1.0 Variations in the acceleration of gravity

Standard acceleration of gravity
g=32.17405 ft/sec2 or 9.80665 m/sec2

1.1 Standard Weight There is a common misconception that scales measure weight. In fact, most accurate scales measure mass. An object should “weigh” the same, no matter what scale it is weighed on, or where the scale is located. An ounce of gold must “weigh”one ounce in Miami or Boston. Otherwise you could buy the ounce of gold in Miami and sell it in Boston at a profit. If scales are used in commerce, an inspector will test them not for their accuracy in measuring force, but rather for their accuracy in measuring mass. In order to measure mass, some method must be used to compensate for the acceleration of gravity at the particular location in which a spring or load cell scale is used. This process is often called “calibrating the scale”. To calibrate a scale, a standard calibration mass is placed on the scale. The scale is then adjusted until it reads the appropriate standard weight. The standard weight is the weight the mass would have at standard gravity of 32.17405 ft/sec2 (9.80665 m/sec2).

A problem occurs when a load cell type scale is calibrated at one location and then moved to another location to weigh an object. For large scale capacities, it is often not possible to bring a calibration weight to the new site, either because this weight is not available, or because of the problems of shipping a calibration mass weighing many thousands of pounds. Therefore, it is necessary to correct for the change in the acceleration of gravity between the site where the scale was calibrated, and the site where the object is being measured. 1.2 Gravity Discussion For two spheres of uniform density, the force of attraction is proportional to the square of the distance between the center of the spheres, as shown in figure 1.

The earth is not a perfect sphere, nor is its density uniform, so that there is a variation in attraction due to the irregularities of the surface and the non-uniform density. An object on the surface is attracted to every small particle in the earth. Those particles that are close to the object exert the strongest influence, since the force is inversely proportional to the square of the distance. Although the average of the vector forces to each particle is approximately equal to a vector to the mass center of the earth, there will be some variation due to the concentration of dense materials in the earth’s crust at certain locations. There is a special science known as gravimetry which investigates variations in local gravity. The acceleration of gravity has been measured with an accuracy of better than 1 part in 10 million at many locations on earth.

If your object is located in a valley, there will be an attraction upward toward the peak of nearby mountains, and in fact there will also be a gravitational attraction sideways toward the mountain. These effects are of minor importance when measuring mass.

No one really knows what creates the force of gravity. Scientists have calculated that the attraction of gravity must act at least 10 billion times faster than the speed of light in order for the universe to be stable. There appears to be no limit to the extent of the attraction. Your body is being pulled outward from the earth by the attraction to the sun and the moon and every other object in the vast universe. In “outer space” (a location in space so distant from any massive bodies such as stars that gravitational influence is negligible) the force of gravity approaches zero. Gravity is also zero at the mass center of the earth, since the attraction is equal in all directions. Shuttle astronauts in earth orbit experience free fall, not lack of gravity. Centrifugal force due to their rotation about the earth exactly counteracts the attraction of gravity, so that they remain at a fixed altitude.

The force of attraction of an object to the earth is defined by the law of mutual attraction given in Figure 3.
where: G = universal gravitational constant, 6.672,59 E-11 nm2 /kg2,
Note: The currently accepted value of G given above has been obtained by experiment, and is therefore not known exactly.

r = radius from center of earth’s mass in meters = 6,378,100 m
Note: (this is a nominal value; actual radius varies)

m1 = current accepted mass of earth in kg = 5.9736 E24 kg
m2 = mass of object in kg

This force can also be defined by a variation of Newton’s second law of motion as given in Figure 4, where

g = acceleration of gravity in m/sec2

Combining the equation of mutual attraction with Newton’s second law yields an equation for the acceleration of gravity (figure 5) Example using nominal radius:

Gm1 = 3.985938 E14
g = Gm1 /r2 = 3.985938 E14/(6378100)2 = 9.7982 m/sec2

Note: This example is for the gravitational attraction only and doesn’t include effect of centrifugal force due to earth’s rotation. 1.3 Latitude Correction The most significant variable in determining the acceleration of gravity is the latitude. The value of g (shown in “gals” in fig. 6) is the smallest at the equator (due to the centrifugal force and the bulging of the earth).

The radius of the earth is approximately 22,000 meters more at the equator than at the north or south pole, due to bulge in earth which resulted from centrifugal forces while the earth was cooling. In addition, the centrifugal force due to the earth’s rotation counteracts the centripetal force due to the attraction of gravity.

The force is: Fcen = m2v2/R

where m2 = mass of object being weighed

R = distance to earth’s rotation axis

v = speed (zero at poles; 463 m/s at equator)

The traditional unit used by geologists for gravity is the “gal” (named after Galileo). 100 gals = 1m/sec2. Therefore, the standard acceleration of gravity is 980.665 gals or 980665 milligals.

The local value for g at sea level can be
calculated using the formula:
g = 9.80613 ( 1 – 0.0026325 cos 2L )
where L is the latitude in degrees.
g is in m/sec2

1.4 Altitude Correction For locations on the surface of the earth, the gravitational attraction is inversely proportional to the square of the distance to the mass center of the earth. Therefore, gravity decreases as you increase altitude.

At sea level g = Gm1 /r2 .
At an altitude H, gh = Gm1 /(r+h)2 = Gm1 /(r2 + 2rh+h2)

g/gh = (r2 + 2rh+h2) / r2 = 1 + 2h/r + h2/r2

h2/r2 is extremely small, so g/gh = 1 + 2h/r
gh = g/(1+2h/r)

In addition, the centrifugal force increases as the radius increases, resulting in a further decrease in the acceleration of gravity. This centrifugal effect depends on the latitude. The decrease in g due to rotation is zero at the poles and reaches a maximum of about 1.567 E-4 m/sec2 per km of altitude at the equator.

Formula to calculate reduction in measured weight
due to increase in altitude

gh = g ( 1 – 3.92 x 10-7 H )
where H = altitude in meters
g = gravity at sea level at the particular latitude
gh = at altitude H at the same latitude

Note: The above formula for altitude correction is based on the nominal radius of the earth and a latitude of 45 degrees. A more accurate calculation would be based on the actual radius and latitude at the specific location where the measurement was being made. However, the magnitude of the correction is small enough that this is not necessary.

The altitude correction is a relatively small number. If the altitude is increased by 3000 meters, then the measured weight decreases by 0.11%. However, high accuracy scales such as the Space Electronics Model YST series can detect changes in measured weight as small as that which results from an increase in elevation of only 40 meters, so the scale should be calibrated at the same altitude as the measurement.

1.5 Tidal Variations An object on the surface of the earth is attracted to every celestial body. Most of these masses are too far away to have any significance on weight, but the sun and the moon do have a significance. If you have a scale whose accuracy is 0.003 % or better, you will notice that the weight of an object varies as a function of the time of day. This effect is most pronounced during spring and fall when the sun and moon align. This produces the “neap tides” that often cause flooding of marinas at these critical dates. 