Download Mass of the Earth RWLO

By Michael Dorneman
Student Directions:
Assoc.
Prof. Chem., MCCC
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 centripetal = 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 a 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,
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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?
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