# Gravity on Earth versus on Moon

by Ron Kurtus

The force of gravity on the Earth is greater than the gravity on the Moon, because of the greater mass of the Earth. However, the Earth has a greater radius than the Moon, so that is a factor in the ratios of the forces of gravity.

Calculation of the force of gravity of each starts with Newton's Universal Gravitation Equation. Then values of the mass and the radius of each are used to determine the acceleration and force.

Questions you may have include:

• What is the gravitation equation?
• What is the value of gravity at the Earth's surface?
• What is gravity on the Moon's surface?
• How do the values of gravity vary with mass and radius?

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

## Gravitation equation

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

• F is the force of attraction, as measured in newtons (N) or kg-m/s2
• G is the Universal Gravitational Constant: 6.674*10−11 m3/s2-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)

If a much smaller object is near the surface of the Earth, the separation R becomes close to the radius of the Earth. Likewise for the separation on the Moon.

Since force equals mass times acceleration, the gravity equation becomes:

F = mg = GMm/R2

Thus:

g = GM/R2

This is the gravity constant or acceleration due to gravity.

## Acceleration due to gravity on Earth

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

gE = GME/RE2

where

• G = 6.674*10−11 m3/s2-kg
• ME = 5.974*1024 kg (mass of Earth)
• RE = 6.371*106 m (radius of Earth - distance from center to surface)
• RE2 = 40.590*1012 m2

Substituting in values:

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

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

∴ gE = 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:

gE = 9.8 m/s2 or 32 ft/s2

## Acceleration due to gravity on the Moon

You can find the value of gM on the Moon by substituting the following items into the equation:

gM = GMM/RM2

where

• G = 6.674*10−11 m3/s2-kg
• MM = 7.35*1022 kg (mass of the Moon)
• RM = 1.738*106 m (radius of the Moon)
• RM2 = 3.02*1012 m2

Substituting in values:

gM = (6.674*10−11 m3/s2-kg)(7.35*1022 kg)/(1.738*106 m)2

gM = (6.674*10−11)(7.35*1022)/(3.02*1012) m/s2

∴ gM = 1.62 m/s2

Thus, the acceleration due to gravity on the Moon is about 1.6 m/s2

## Comparison

Compare the mass and radius of the Earth and Moon to see the gravity ratio.

### Mass

Mass of Earth: ME = 5.974*1024 kg

Mass of Moon: MM = 7.35*1022 kg

ME = 81.3*MM

In other words, the mass of the Earth is about 81 times that of the Moon

Radius of Earth: RE = 6.371*106 m

Radius of Moon: RM = 1738*103 m

RE/RM = 6.371*106/1738*103 = 3.66

(RE/RM)2 = 13.4

RE2 = 13.4*RM2

In other words, the square of the Earth's radius is about 13.4 times that of the Moon

### Gravity

The acceleration due to gravity on the Moon is about 1/6 that of on the Earth:

gE = GME/RE2

gM = GMM/RM2

gE/gM = (GME/RE2)/(GMM/RM2)

gE/gM = (ME/MM)(RE2/RM2)

gE/gM = (81.3)(13.4) = 6

or

gM = gE/6

## 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.

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

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