Calculate the Mass of The Earth
By Michael Dorneman
Associate Professor of Chemistry
Mercer County Community College
Trenton, NJ.
Project Overview
Students will use real time data of a satellite’s motion and that satellite’s mass to
1) calculate the mass of the Earth
2) calculate the weight of a satellite in orbit around the Earth
3) calculate the weight of the Earth
(Note: The weight of the Earth is not the same as its mass.)
Student Learning Objectives
For this RWLO, the student will be able to:




Calculate mass, force, and weight
Describe the differences between mass and weight
Describe the differences and similarities between force and gravity
Apply Newton’s Second and Third Laws of Motion
Procedure
Time: Approximately 50 minutes
Materials: Online access, paper and pencils, calculators, printed Student Directions
and Student Worksheets, if desired
Prerequisites: The student should be able to use a calculator and scientific notation,
have an understanding of circular motion, of Newton’s Laws of Motion, and of Gravity.
Implementation: This RWLO can be used as an in class exercise if online access is
available, or as a homework assignment. This exercise is relevant to the topics of
Newton’s Law of Universal Gravitation, circular motion, and the difference between
weight and mass.
The instructor should thoroughly familiarize himself or herself with the J-Track 3D
windows that provide the satellite data, and the steps in acquiring the on-line dry mass
of satellites from the NSSDC Master Catalog prior to beginning this exercise.
The exercise has students fill out and complete a worksheet, the instructor will need to
print enough copies for his or her students, and familiarize themselves with the
calculations.
1. Prior to class, familiarize oneself with the J-Track 3D windows and NSSDC
Master Catalog.
NOTE: Students will have trouble with J-Track 3D and the NSSDC Master
Catalog. Emphasize that they must turn off their pop-up blocker for J-Track 3D.
Emphasize students may need to pick a different satellite if the one they choose
does not show up in the NSSDC Master Catalog. Using just the satellite name
without a number (in the Master Catalog) will often bring up the needed
information.
2. Print copies of the Student Worksheets and Student Directions.
3. Provide instruction in Key Concepts (gravity, acceleration, circular motion,
centripetal and centrifugal force, Newton’s Third Law) if necessary.
4. Hand out and discuss the student procedure. The instructor should demonstrate
the J-Track 3D data windows and teach online procedures if necessary. The
Instructor should clearly tell students that they will need to look at several
satellites. Data for all satellites is not in the NSSDC Master Catalog,
students will need to try several to find a satellite for which on – orbit dry
mass is given.
5. Hand out and discuss the Student Worksheet, and grading.
Content Material
By Michael Dorneman
Assoc. Prof. Chem., MCCC
Student Directions:
Overview:
There are over 8,000 artificial objects in orbit around the Earth, many of which are operational
satellites. Satellites fulfill a wide variety of missions, including recording weather, environmental,
and espionage data about the Earth, relaying communications (cell phones, TV, radio), locating
people and places (Global Positioning Satellites), and looking out into the Universe, among others.
For the satellite to stay in a circular orbit around the Earth instead of shooting out into space, it must
have centripetal force acting on it. One can calculate the centripetal force on the satellite using
Newton’s Second Law F = m a. To do this one must use the mass of the satellite and the
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acceleration due to the centripetal force:
acentripetal 
vSatellite
r
Where v is the velocity of the satellite. The “r” is the radius of the circular orbit. For your satellite
“r” is the distance to the center of the Earth. So “r” must be calculated by adding the radius of the
earth and the altitude of the satellite.
“r” = radius of the Earth + Altitude of Satellite
Now, remember that the force keeping the satellite in circular motion (in orbit) around the Earth, the
centripetal force, is (equals) the Force of Gravity. Newton’s Law of Universal Gravitation relates
the Force of Gravity on an object to the mass of the object, the distance from the object to the center
of the Earth, and the mass of the Earth. Newton’s Law of Universal Gravitation is:
FGravity 
G mSatellitemEarth
d2
With these formulas, and a bit of algebra, one can calculate the mass of the Earth.
So, in summary:
1. From the altitude of the satellite and the radius of the Earth, you can calculate the radius of
the orbit of the satellite, “r”
2. From the radius of the orbit of the satellite, “r”, and the velocity of the satellite, v, you can
calculate the centripetal acceleration, a centripetal
3. From the centripetal acceleration, a centripetal and the mass of the satellite you can calculate
the centripetal force, F centripetal, with Newton’s Second Law
4. The centripetal force, F centripetal , is (equals) the Force of Gravity between the satellite and
the Earth. From the Force of Gravity on the satellite, the mass of the satellite, and the
distance between the satellite and the center of the Earth “r”, you can calculate the mass of
the Earth.
Procedure:
Data Collection:
You must record all data and numbers you use on the worksheet, and clearly demonstrate the
calculations in the format you have used in lab, also on the worksheet. Fill in the worksheet as you
do this exercise. If you have a Pop-up blocker, set it to allow Pop-ups from NASA.
Go to http://science.nasa.gov/RealTime/ and read about J-Track 3D. About how many artificial
objects are in orbit around the Earth? How many of these are satellites? Then, on the menu on the
left side of the screen, Click on J-TRACK 3D (if asked, run the application from NASA).
After the small window fully loads and displays, each white dot in the picture represents the real
time location of a satellite. Clicking on a dot reveals that satellites designation or name, and the
circle or oval that represents its orbit. Click on a few satellites. Find one that is close to the Earth,
with a circular orbit. DO NOT use an Iridium satellite (their masses are not recorded). Now, find the
menu in the upper left hand corner of the J-Track 3D window, pick View, and then Satellite
Position. Record the designation or name of the satellite, its altitude, and velocity. Be certain to
include the units for those measurements.
Now go to the NSSDC Master Catalog Spacecraft Query Form: http://nssdc.gsfc.nasa.gov/nmc/scquery.html You must use a satellite for which the On-orbit dry mass is given. Type the name of
your satellite into the Query Form. Do not include the number. (There may be no data for a specific
satellite, but there will be data for many other members in the group.) Click on the Submit button. It
may take a few moments. Read about your satellite, find and record its mass (called On-orbit dry
mass). If information for your satellite is not available, or you get an error message, go back to the
J-Track 3D window and pick a different satellite. You may have to try several satellites.
You cannot do the exercise without the Name (and number), Mass (On-orbit dry mass), Velocity,
and Altitude of a satellite in orbit around the Earth.
You will need two other pieces of data, the radius of the Earth and a value for G, the Gravitational
Constant. The radius of the Earth is located at Earth Facts:
http://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html
Based on the direction of the orbit of your satellite, record and label the more appropriate radius of
the Earth. For the Gravitational constant G we will use 6.673 x 10–20 Km3 / Kg sec2
Calculations:
1. Calculate “r”for your satellite:
“r” = radius of the Earth + Altitude of Satellite
Units on your answer will be in kilometers. This is the value for d used in step 4.
2. Calculate the centripetal acceleration, a centripetal When you square v satellite use parentheses
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to include the units
acentripetal 
vSatellite
r 2
Units on your answer will be in kilometers/sec .
3. Calculate F centripetal, with Newton’s Second Law:
Fcentripetal = mSatellite acentripetal
Units on your answer will be in kilonewtons (KN). 1 KN = 224.8 lbs
4. Calculate the mass of the Earth from Newton’s Law of Universal Gravitation, after
rearrangement for m Earth:
F
d2
mEarth 
Gravity
G mSatellite
Remember, d is the distance between the center of the Earth and the satellite, calculated
earlier. Units on your answer will be in kilograms.
Answer the following questions:
What is the mass of the satellite? What is another name for the force of gravity between two
objects? What is the weight of the satellite (from the perspective of the Earth)? From the
perspective of the satellite, what is the weight of the Earth? Why? (Hint, Newton’s Third Law of
Motion)? What is the difference between Mass and Weight?
Name:_____________________________
Class:__________
Reference Number ________
Data:
Satellite Designation: ___________________________________________
Satellite Mass mSatellite
Satellite Altitude
=
Satellite Velocity, vSatellite
=
Gravitational Constant, G
Radius of the Earth
=
=
=
Calculations:
For each step, write the formula first, then replace the variables with measurements (include
units) then calculate and write your answer.
Calculation of “r” for your satellite:
(Note, this is also d for the Mass of the Earth calculation.)
Calculation of the centripetal acceleration, a centripetal Use parentheses to include the units when
you square v.
Calculation of the centripetal force, F centripetal:
Name:_____________________________
Class:__________
Reference Number ________
Calculation of the Mass of the Earth, m Earth: Be certain to square d before dividing.
What is the accepted value of the
Mass of the Earth? (See Earth Facts site above)
____________________________________
Try to explain why your calculated value for the Mass of the Earth is different from the accepted
value. Note, calculation error is not acceptable as an explanation.
What is the mass of your satellite?
_____________________________________
What is another name for the
force of gravity on an object?
_____________________________________
From the perspective of the Earth,
what is the weight of the satellite?
Use metric units (KN) for this answer.
_____________________________________
From the perspective of the satellite,
what is the weight of the Earth?
Hint, No new calculations need to be done. _____________________________________
Why? (Hint, one of Newton’s Laws of Motion)
What is the difference between Mass and Weight?
Assessment
The writer grades this activity out of five points, with deductions for the following:
No credit will be awarded if all calculations are not clearly demonstrated.
Up to one point will be deducted if any identifying information is missing or
incorrect.
Points will be deducted in ½ point increments for any of the following:
 incomplete or incorrect calculations
 missing units
 failure to write formulas before calculating
 failure to rewrite formulas with measurements replacing variables
 not calculating and writing answers at any stage
 incorrect answers
 if answers to narrative questions are incomplete, or incorrect, or badly written.
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Links to Course Competencies
This RWLO could be applied in the following courses: Physical Science
Concepts, other Physics Courses, Astronomy Courses, basic mathematics, et. al.
Specifically, this RWLO meets the following course objectives:

The student will be able to quantitatively discuss circular motion,
centrifugal and centripetal forces in circular motion,

Define acceleration and perform simple calculations to find acceleration

State Newton’s three Laws of Motion and explain each,
use Newton’s Second (the equation F = ma) and Third Laws of Motion.

Define force and give its units

Write and use Newton’s Law of Universal Gravitation, use this in
conjunction with Newton’s Second Law of Motion,

Describe the differences and relationship between mass, weight, and
force.
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Supplementary Resources
A note to the Instructor:

All distance and velocity data from NASA in this exercise are in
Kilometers. The value for G is usually given with units of meters (6.673
meters cubed kilograms per second squared). To eliminate the extra
conversions, the students are given G with units of Km. Forces will be in
units of Kilonewtons. One kilonewton equals 1000 newtons or 224.8 lbs of
force.
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Recommendations
Recommendations for Integration:
It may be best to use this RWLO after discussing all disparate topics listed in the
Student Learning Objectives above.
Depending on the sophistication of the students and the course, the instructor
could leave out some or all of the student directions and, after a discussion of the
principals involved and where to get the data, allow students to work out some or
all of the mathematics on their own.
Back-up:
1. Go to the sites and copy the necessary data for students to use.For
several different satellites you will need to have the Satellites’

Designations

On-orbit Dry Masses (= mSatellite )

Velocities (= vSatellite )

Altitudes
2. Other data necessary for all students are the Polar Radius of the Earth
(= 6356.8 km) and the value for the Gravitational Constant given in the
student directions.
At http://science.nasa.gov/newhome/copyright.htm one finds this notice:
"The written materials and pictures within the Science @ NASA websites are
presented without copyright and are the property of the U. S. Government. While
no copyright is asserted for NASA written materials or pictures, you must respect
the copyrights on commercial materials which may, on occasion, be used here by
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records. To preserve the scientific integrity of the information, we ask that the
content not be altered . We also ask that NASA's Marshall Space Flight Center
and Science@NASA be credited. The NASA logo and the name Science@NASA
are the property of NASA. The NASA logo and the stories and images in the
Science@NASA websites may not be used to imply official endorsement of nonNASA activities. "
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