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Gravity Constant Factors

by Ron Kurtus (revised 15 February 2016)

The acceleration due to gravity (g), seen in the gravity force equation F = mg, is considered a constant for objects relatively near the Earth's surface.

The gravity constant comes from the Universal Gravitation Equation at the Earth's surface. By substituting in values for the mass and radius of the Earth, you can calculate the value of the gravity constant at the Earth's surface.

The fact that the acceleration due to gravity is a constant facilitates the derivations of the gravity equations for falling objects, as well as those projected downward or upward. However, the value of g starts to vary at high altitudes.

Questions you may have include:

This lesson will answer those questions. Useful tool: Units Conversion



Derivation of gravity constant

The acceleration due to gravity constant comes from Newton's Universal Gravitation Equation, which shows the force of attraction between any two objects—typically astronomical objects:

F = GMm/R2

where

(See Universal Gravitation Equation for more information.)

One assumption made is that the mass of each object is concentrated at its center. Thus, if you considered a hypothetical point object of mass m that was at the surface of the Earth, the force between them would be:

F = GMEm/RE2

where

Since GME/RE2 is a constant, set:

g = GME/RE2

This is the gravity constant or acceleration due to gravity. Thus, the gravity equation is:

F = mg

Value of g

You can find the value of g by substituting the following items into the equation:

G = 6.674*10−11 m3/s2-kg

ME = 5.974*1024 kg

RE = 6.371*106 m

Note: Since the Earth is not a perfect sphere, the radius varies in different locations, including being greater at the equator and less at the poles. The accepted average or mean radius is 6371 km.

The result is:

g = (6.674*10−11 m3/s2-kg)(5.974*1024 kg)/(6.371*106 m)2

g = (6.674*10−11)(5.974*1024)/(40.590*1012) m/s2

g = 0.9823*101 m/s2

g = 9.823 m/s2

This value is close to the official value of g = 9.807 m/s2 or 32.174 ft/s2, defined by the international General Conference on Weights and Measures in 1901. Factors such as the rotation of the Earth and the effect of large masses of matter, such as mountains were taken into effect in their definition.

Although, the value of g varies from place to place around the world, we use the common values of:

g = 9.8 m/s2 or 32 ft/s2

On other planets

The same principles of gravity on Earth can apply to other astronomical bodies, when objects are relatively close to the planet or moon.

We typically consider "gravity" as concerning Earth. If you are talking about the force of gravity on another planet, you should say, "gravity on Mars" or such.

Acceleration due to gravity on the:

Variation with altitude

Although g is considered a constant, its value does vary with altitude or height from the ground. You can show the variation with height from the equation:

gh = GME/(RE + h)2

where

Height or altitude above Earth's surface

Height or altitude above Earth's surface

To facilitate calculations, it is easier to state h as a percentage or decimal fraction of RE.

For example, if h = 10% of RE or 0.1RE, then:

gh = GME/(1.1RE)2

gh = GME/1.21RE2

then

gh = 0.826GME/RE2 = 0.826g

Charting h and gh:

h %RE gh
67.31 m (220.8 ft) 0.001% 0.99998g = 9.8 m/s2
637.1 m (2207.8 ft) 0.01% 0.9998g = 9.8 m/s2
6.371 km (3.95 mi) 0.1% 0.998g = 9.78 m/s2
63.71 km (39.5 mi) 1% 0.980g = 9.6 m/s2
637.1 km (395 mi) 10% 0.826g = 8.09 m/s2

As you can see, the value of g starts to deviate from 9.8 m/s2 at about 6.4 km or 4 miles in altitude. At about 64 km or 40 mi, the change in g is sufficient to noticeably affect the results of gravity equations.

Effect on gravity derivations

The derivations of the equations for velocity, time and displacement for objects dropped, projected downward, or projected upward depend on g being a constant. Even a 1% or 2% variation in the value of g can affect the derivations.

(See Overview of Gravity Equation Derivations for more information.)

Summary

The acceleration due to gravity, g, is considered a constant and comes from the Universal Gravitation Equation, calculated at the Earth's surface. By substituting in values for the mass and radius of the Earth, you can find the value of g.

A constant acceleration due to gravity facilitates the derivations of the gravity equations. However, the value of g starts to vary at high altitudes.


Think clearly and logically


Resources and references

Ron Kurtus' Credentials

Websites

Gravity Resources

Acceleration Due to Gravity - TutorVista.com

The Value of g - Physics Classroom

Acceleration Due to Gravity - Haverford College

The Acceleration of Gravity - Physics Classroom

Books

Top-rated books on Simple Gravity Science

Top-rated books on Advanced Gravity Physics


Questions and comments

Do you have any questions, comments, or opinions on this subject? If so, send an email with your feedback. I will try to get back to you as soon as possible.


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