## 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.

## FAQs

### What is the most precise way to measure mass? ›

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.

**What is the scientific measurement for mass? ›**

The SI unit of mass is the **kilogram (kg)**. In science and technology, the weight of a body in a particular reference frame is defined as the force that gives the body an acceleration equal to the local acceleration of free fall in that reference frame.

**What is precise measurement in science? ›**

Precision and accuracy are two ways that scientists think about error. Accuracy refers to how close a measurement is to the true or accepted value. Precision refers to **how close measurements of the same item are to each other**. Precision is independent of accuracy.

**What is a precise measurement in physics? ›**

Precision is defined as '**the quality of being exact**' and refers to how close two or more measurements are to each other, regardless of whether those measurements are accurate or not. It is possible for precision measurements to not be accurate.

**What is precise mass? ›**

The precise mass of a compound is **its molecular weight calculated using the masses of the most abundant isotope of constituent elements**.

**What are 3 ways to measure mass? ›**

A number of tools exist for measuring mass in different environments. These include **balances and scales, measurement transducers, vibrating tube sensors, Newtonian mass measurement devices and the use of gravitational interaction between objects**.

**What are mass and weight measured in scientifically? ›**

**Mass is an intrinsic property of matter and is measured in kilograms**. The mass of a bird is a constant. A 15-gram bird is 15 grams, whether measured on the Earth, the Moon, or Mars. Weight is a measure ofthe force of gravity on a physical object and is measured in newtons.

**What is used for precision measurement? ›**

**Slide calipers** are an essential precision measuring tool providing a large measurement range, flexibility and accuracy up to 0.001" (25.4 μm). These tools typically measure outside, inside, depth and step dimensions.

**What is an example of a precise measurement? ›**

More Examples

**If the thermometer continuously registers the same temperature for several days**, the measurement is also precise. If you take the measurement of the mass of a body of 20 kg and you get 17.4,17,17.3 and 17.1, your weighing scale is precise but not very accurate.

**What are the 3 types of measurement in science? ›**

5–8) distinguished among three types of measurement: **fundamental, associative and derived**.

### What is the most precise measurement in physics? ›

A new assessment of **the electron** is the most precise measurement of a particle ever taken, at a precision of 1.3 parts in 10 trillion, and could tell us about new physics happening at the smallest scales.

**What are the three types of precision? ›**

Precision can assert itself in three different ways: Arithmetic precision - number of significant digits for a value. Stochastic precision - probability distribution of possible values. Granularity - grouping or level of aggregation of values.

**Why is precise measurement important in science? ›**

**In order to get the most reliable results in a scientific inquiry**, it is important to minimize bias and error, as well as to be precise and accurate in the collection of data. Both accuracy and precision have to do with how close a measurement is to its actual or true value.

**What is a precise scale? ›**

A precision scale is **a weighing instrument with high sensibility, designed, precisely, to measure the mass of solids and liquids with a high level of precision**. Generally, this type of scales has less capicity and are used to weigh smaller masses, when compared to the scales we use daily.

**What is accurate mass in chemistry? ›**

Accurate mass—**the experimentally determined mass of an ion measured to an appropriate degree of accuracy and precision used to determine, or limit the possibilities for, the elemental formula of the ion** [1].

**What instrument of mass measurement is the most accurate? ›**

**Mass comparators** are the most accurate mass measurement instruments available today. It is a special type of electronic scales designed to compare mass of two weights.

**What are 4 metric Ways units to measure mass? ›**

**What are the Metric Units of Mass?**

- definition.
- metric units.
- metric.
- units.
- mass.
- milligram.
- mg.
- gram.

**What are the 4 commonly used units to measure mass? ›**

The Metric System of Measurements uses the mass units: **gram (g), kilogram (kg) and tonne (t)**.

**How can you measure mass without gravity? ›**

In the absence of a significant gravitational field, the most direct way to measure mass is **through inertia**. The ZGMMD operation is based on Newtons second law of motion, which states that a force on an object is equal to the mass of that object multiplied by the acceleration.

**Is mass a scientific weight? ›**

In scientific contexts, **mass is the amount of "matter" in an object (though "matter" may be difficult to define), whereas weight is the force exerted on an object by gravity**.

### How do scientists measure weight? ›

Because we know the radius of the Earth, we can **use the Law of Universal Gravitation** to calculate the mass of the Earth in terms of the gravitational force on an object (its weight) at the Earth's surface, using the radius of the Earth as the distance.

**How many ways can you measure mass? ›**

You can measure mass **using a balance**. A balance is different from a scale because it uses a known mass to measure the unknown mass where as a scale actually measures weight. Finding mass with a triple-beam balance or a double-pan balance is a form of measuring gravitational mass.

**Which is a precision instrument? ›**

Precision instruments are used to monitor surface and deep displacement, tilt (settlement) dynamics, crack relative tension, closure, subsidence, dislocation change, geoacoustics, stress and strain, and other physical parameters and environmental factors of the deformation.

**What are the 7 types of measurement? ›**

There are seven base units measurements namely **meter, kilogram, second, ampere, kelvin, mole, and candela**. The most common among these are kilograms, seconds, and meters.

**What are the 5 types of measurements? ›**

**types of measurements are:**

- Indirect method of measurement.
- Direct method of measurement.
- Fundamental method of measurement.
- Substitution method of measurement.
- Comparison method of measurement.

**What are the four scientific levels of measurement? ›**

Levels of Measurement | **Nominal, Ordinal, Interval and Ratio**.

**What is the most accurate scientific theory? ›**

**Einstein's general theory of relativity** is, alongside quantum electrodynamics, the most precisely tested physical theory.

**Is kg or g more precise? ›**

Of course 1 Kg is exactly 1000 g, Kilo in modern language means “1000”, so 1 Kg means “1000 g”. Then again, mathematically **1000g has more precision than 1Kg**.

**What is the difference between accurate and precise measurement? ›**

Accuracy and precision are both forms of measurement that define how close you are to hitting a target or achieving a goal. **Accuracy evaluates how close you are to the true value of the measurement, while precision shows how close the measured values are to one another**.

**What are the seven types of precision measuring instruments? ›**

These include **Multimeters, Voltage Testers, Videoscopes, Speedometer, Transducers, Capacitance Meters**, etc. Electronic equipment may be analog or digital as per the need.

### Why is precision so important in scientific research? ›

Precision in scientific investigations is important in order **to ensure we are getting the correct results**. Since we typically use models or samples to represent something much bigger, small errors may be magnified into large errors during the experiment.

**Is accuracy or precision more important to scientific measurement? ›**

**Precision does matter**. Accuracy is how close a measure value is to the true value. Accuracy matters too, but it's best when measurements are both precise and accurate.

**How do scientists get precise or specific size information about objects? ›**

Scientists use **sensitive measuring devices** to make measurements such as these. The measurements are usually made using SI units of measurement. Scientists often measure mass with a balance. In science, the volume of a liquid might be measured with a graduated cylinder.

**What does it mean to measure precision? ›**

Measurement precision is expressed numerically using measures of imprecision such as the standard deviation calculated from results obtained by carrying out replicate measurements on a suitable material under specified conditions (Figure 3).

**Why is precise measurement important in an experiment? ›**

**Precision represents the closeness between the different measured values of a quantity**. The measurements taken by scientists should be precise as well as accurate. Bad equipment and human error may lead to inaccurate results which are far from the true values.

**How does the precision of measurements affect the precision of scientific calculations? ›**

Precision of measured values refers to how close the agreement is between repeated measurements. The precision of a measuring tool is related to the size of its measurement increments. **The smaller the measurement increment, the more precise the tool**. Significant figures express the precision of a measuring tool.

**How do you find accurate and precise measurements? ›**

...

**How to measure accuracy and precision**

- Collect data. ...
- Determine the average value. ...
- Find the percent error. ...
- Record the absolute deviations. ...
- Calculate the average deviation.

**What is an example of precise and accurate? ›**

One example of accuracy is **the distance an arrow gets from the bullseye center**. Precision refers to how repeatable a measurement can be. A good example of precision is the distance between the second and first arrows, regardless of whether they are near the mark.

**How measurements can be precise but not accurate? ›**

Answer and Explanation: Measurement can be precise and not accurate **when there is a systematic error in the instrument itself**. In this case, the measurements obtained will have a closeness in the experimental values obtained but the actual result would not be closely related to the actual value.

**How do you make a measurement more precise? ›**

**Take Multiple Measurements**

The more samples you take for a given attribute, the more precise the representation of your measurement. In situations where sampling is destructive, or you can't take multiple measurements (e.g., growth rates in a culture), you can increase the number of replicates to compensate.