Gravitational Forces Between the Earth and Moon

By Jesse T. Compton

Grade level:  High School Physics

Hamilton County Department of Education High School Course markers 9-12, 2002-2003;  4.H.1-  Apply the concepts of forces, motion, energy, electricity, and magnetism to the study of the earth and the universe.

Introduction / Task / Process / Evaluation / Conclusion / Credits / Answers

Introduction

 Every particle of matter in the universe attracts every other particle with a force that is directly proportional to the product of the masses of the particles and inversely proportional to the square of the distance between them.  Isaac Newton hypothesized that any object having mass always exerts an attractive gravitational force on all other massive objects.  The more massive the object, the stronger its gravitational pull.  The study of the motions of planets reveals another aspect of gravitational force.  The gravitational force (attraction) between two objects decreases in proportion to an increase of their square distance from one another.  Newton's equation used to calculate gravitational force is

F = (G x m1 x m2) / r²

where F is the gravitational force in Newton's (N), G is the gravitation constant in Nm²/Kg², m1 is the mass of object one in Kg, m2 is the mass of object 2 in Kg, and r² is the squared distance between the two centers of the objects in meters (m).

The Task

• Purpose of product -  The purpose of this product is for the student to apply the concept of gravitational force and use Newton's equation to calculate the gravitational force to something relevant (the gravitational force between the earth and the moon).
• Preceding and ensuing events -  Preceding this activity, the students will learn the difference between mass and weight, define and calculate mass from volume and density, and have an understanding of the unit of force called a newton.  Ensuing events  include labs, activities, and lessons on other concepts of forces, motion, and energy.
• Product improving learning -  This product will improve the mathematical skills of the student and expand his/her understanding of gravitational forces.  It will also give the student a chance to apply the concept of mass (which is important in physics) in a very relevant and practical manner.
• Product improvement or expansion -  This product could be expanded to include the calculation of gravitational forces of other planets and moons in our solar system.  The principles and calculations used in this activity can also be applied to the attraction between atoms and molecules in chemistry.

The Process

In this activity, you will be required to acquire proper data about the earth and its moon to be able to use Newton's equation and calculate the gravitational force between them.  You will need a calculator, paper, and a pencil.  According to Newton's equation for calculating gravitational force (F), you will need four pieces of information:

1. The gravitation constant (G) in Nm²/Kg².
2. The mass of object 1 (m1), in this case the mass of earth in kilograms (Kg).
3. The mass of object 2 (m2), in the case the mass of earth's moon in kilograms (Kg).
4. The average distance from the center of the earth to the center of the moon in meters (m).

Now that you have acquired the proper information for the equation, calculate the gravitational force between the earth and moon.

F= (G x m1 x m2) / r² =

The gravitational force between the earth and moon at the average distance between them may be expressed as F.  For the following changes in mass or distance or both, will the value of F remain constant, increase, or be reduced?  Calculate F for each of the following changes; it is understood that the factors not mentioned remain unchanged:

• The moon's mass is doubled.
• The earth's mass is doubled.
• Both the mass of the earth and that of the moon are doubled.
• The distance between the centers of the two are doubled.
• Both masses are doubled and the distance is halved.

Evaluation

Place your answers here.

The gravitational force between earth and its moon = F= (G x m1 x m2) / r²=

1. The gravitational constant (G) =                               .
2. The mass of the earth (m1) =                                 .
3. The mass of the moon (m2) =                              .
4. The average distance from the center of the earth to the center of the moon is                             .
5. If the moon's mass were doubled, F would =                                      .
6. If the earth's mass were doubled, F would =                                        .
7. If both the mass of the earth and that of the moon are doubled, F would =                                 .
8. If the distance between the centers of the two are doubled, F would =                                       .
9. If the masses of both earth and the moon are doubled, F would =                                           .

Conclusion

In this activity, the student is able to use mass in a practical application of gravitational force between the earth and the moon.  The student calculates gravitational force using Newton's equation for gravitational force between two objects.  The student also calculates gravitational forces using different masses of the earth and moon, and also different distances between the earth and moon.

Credits

• Booth, V. H. (1962). Physical Science, a study of matter and energy.  New York, NY: The Macmillan Company.
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• Chaisson, E., & McMillan, S. (2002).  Astronomy today.  Upper Saddler River, NJ: Prentice Hall.
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• Col, J. (2003). Enchanted learning.com. Retrieved March 31, 2003, from http://www.enchantedlearning.com/subjects/astronomy/planets/earth/Mass.shtml
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• Tal, K. (2003). The solar system. Retrieved March 31, 2003, from http://www.krysstal.com/solarsys_moon.html
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• Weisstein, E. W. (2003). Wolfram Research. Retrieved March 31, 2003, from http://scienceworld.wolfram.com/physics/GravitationalConstant.html
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• Bell, E. V. (2003). NSSDC photo gallery. Retrieved March 31, 2003, from http://nssdc.gsfc.nasa.gov/image/planetary/earth/gal_earth_moon.jpg

Answers

The gravitational force between earth and its moon = F= (G x m1 x m2) / r²= 1.99 x 10²⁰

1. The gravitational constant (G) = 6.67 x 10^-11  Nm² / Kg²                           .
2. The mass of the earth (m1) =  6.0 x 10²⁴  Kg                             .
3. The mass of the moon (m2) = 7.35 x 10²²  Kg                            .
4. The average distance from the center of the earth to the center of the moon is  384,400,000 m                            .
5. If the moon's mass were doubled, F would = 3.981 x 10²⁰  N                                   .
6. If the earth's mass were doubled, F would =   3.981 x 10²⁰ N                                    .
7. If both the mass of the earth and that of the moon are doubled, F would = 7.96 x 10²⁰ N                               .
8. If the distance between the centers of the two are doubled, F would =  4.98 x 10¹⁹ N                                    .