JNC Conditions Imply Mathematical Modeling Questions


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1. How far is the moon?
The moon’s orbit around the earth is nearly circular, with a period of 28 days.
Determine how far the moon is from (the center) of the earth using only this
and the following information, which was available to Newton.
a. The gravitational acceleration of the moon caused by the gravitational pull
of the earth is
G MEarth/r?,
where MEarth is the mass of the earth (but the values of G or MEarth were
not known separately).
b. The orbit of the moon (or any satellite, for that matter) is a balance
between the gravitational acceleration and the centrifugal force v2/r. That
G MEarth /r² = 02 /r,
GMEarth/r2 = 02/1.
where v = r ai is the velocity in the angular direction.
c. The gravitational acceleration for an object near the earth’s surface (r = a
= 6,400 km) is known to be g = 980 cm/s2.
2. The mass of the sun
Determine the mass of the sun in units of the earth mass (i.e., find M =
Msun/MEarth) using only the information provided in exercise 1 and the
following information:
a. the period of earth’s orbit is 1 year.
b. the sun-earth distance is, on average, about 1.5 x 108 km.
3. Geosynchronous satellite
you want to put a satellite in a geosynchronous orbit (so that the satellite
will always appear to be above the same spot on earth), how high (measured
from the center of the earth) must it be placed? You are given GMg = a_g = 4
x 1020 cm’s-2
4. Weighing a planet
Newton’s law of gravitation and the requirement that the centrifugal
acceleration of a body revolving around a planet should be equal to the
gravitational pull of the planet suggest a way to determine the mass M of any
planet with a satellite or moon. For this exercise you need to know G = 0.67
x 10-7 cm3s-2 gm ?
Determine the mass of a planet when you know only that its moon
revolves around it with a nearly circular orbit with r = 4 x 105 km once
every30 days.
5. Weighing Jupiter
Find the mass of Jupiter, given that its moon Callisto has a mean orbital
radius of 1.88 x 106 km and an orbital period of 16 days and 16.54 hours. For
this exercise you need to know G.

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conditions imply

formula for radius

The Golden Ratio


The Hofmeister Rule

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