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Explanation of the Gravity Constant Factors by Ron Kurtus - Succeed in Understanding Physics. **Key words:** Universal Gravitation Equation, Earth, radius, mass, height, physical science, School for Champions. Copyright © Restrictions

# Gravity Constant Factors

by Ron Kurtus (revised 21 February 2011)

The value of **g** in the gravity force equation **F = mg** is the acceleration due to gravity. It 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:

- What is the derivation of the gravity constant?
- What is the value of the constant at the Earth's surface?
- How does the acceleration due to gravity vary with altitude?

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/R^{2}

where

**F**is the force of attraction, as measured in newtons (N) or kg-m/s^{2}**G**is the Universal Gravitational Constant: 6.674*10^{−11}m^{3}/s^{2}-kg**M**and**m**are the masses of the objects in kilograms (kg)**R**is the separation of the centers of the objects in meters (m)

(

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 = GM_{E}m/R_{E}^{2}

where

**F**is the force of attraction at the surface of the Earth^{}**G**is the Universal Gravitational Constant**M**is the mass of the Earth_{E}**m**is the mass of the object**R**is the separation between the center of the Earth and an object on its surface; it is also the radius of the Earth_{E}

Since **GM _{E}/R_{E}^{2}** is a constant, set:

g = GM_{E}/R_{E}^{2}

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}m^{3}/s^{2}-kg

M= 5.974*10_{E}^{24}kg

R= 6.371*10_{E}^{6}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}m^{3}/s^{2}-kg)(5.974*10^{24}kg)/(6.371*10^{6}m)^{2}

g= (6.674*10^{−11})(5.974*10^{24})/(40.590*10^{12}) m/s^{2}

g= 0.9823*10^{1}m/s^{2}

g= 9.823 m/s^{2}

This value is close to the official value of **g =** 9.807 m/s^{2} or 32.174 ft/s^{2}, 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/s^{2}or 32 ft/s^{2}

**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:

- Earth: 9.8 m/s
^{2} - Moon: 1.6 m/s
^{2} - Mars: 3.7 m/s
^{2} - Sun: 275 m/s
^{2}

## 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:

g_{h}= GM_{E}/(R_{E}+ h)^{2}

where

**g**is the acceleration due to gravity at height_{h}**h**-
**h**is the height above the Earth's surface or the altitude of the object

Height or altitude above Earth's surface

To facilitate calculations, it is easier to state **h** as a percentage or decimal fraction of **R _{E}**.

For example, if **h =** 10% of **R _{E}** or 0.1

**R**, then:

_{E}

g1.1_{h}= GM_{E}/(R_{E})^{2}

g1.21_{h}= GM_{E}/R_{E}^{2}

then

g0.826_{h}=GM= 0.826_{E}/R_{E}^{2}g

Charting **h** and **g _{h}**:

h |
%R_{E} |
g_{h} |

67.31 m (220.8 ft) | 0.001% | 0.99998g = 9.8 m/s^{2} |

637.1 m (2207.8 ft) | 0.01% | 0.9998g = 9.8 m/s^{2} |

6.371 km (3.95 mi) | 0.1% | 0.998g = 9.78 m/s^{2} |

63.71 km (39.5 mi) | 1% | 0.980g = 9.6 m/s^{2} |

637.1 km (395 mi) | 10% | 0.826g = 8.09 m/s^{2} |

As you can see, the value of **g** starts to deviate from 9.8 m/s^{2} 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

### Websites

**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**

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