Calc III Lab #5: Adding a Second Gravitational Mass
The assumption that our little particle is only being affected by the
gravitational force of a single object is a huge
simplifying assumption! Adding another gravitational body into our
scenario results in a wealth of new possible trajectories for
our particle. For example, the particle can
- Orbit around one body;
- Orbit around the other body;
- Orbit in a large path around both bodies;
- Move in a trajectory that takes it close to the first body, then
near the second, then near the first again, etc;
- Move in a highly irregular fashion that is difficult to analyze.
To study this new scenario, you must change the dynamical system from the
model labeled "One Gravitational Body" to the model labeled "Two
Gravitational Bodies." (Look under the "Model" menu on the command panel and follow the pull-right
menu for the category "Calc III Models.")
This model is the same
model as you saw at the end of Lab #4: There
is a planet of mass 1 (the earth) at position
(x,y)=(-1,0) and a smaller body (the moon) with mass
0.1 located at position (x,y)=(1,0). As we
cautioned in Lab #4, this model is only an
approximation to the earth-moon system; in reality the moon is only
about 1/100 the mass of the earth. (If you want to know where the
two "stars" are for this system, load the file
/u/calcIII/html/Dstool/data/two_stars.dstool)
Make sure that the position of our test particle is set to
(x,y)=(0,4).
Question #5:
Set the initial velocity of our particle to be
- (vx,vy)=(0,0)
- (vx,vy)=(0.5,0)
- (vx,vy)=(0.3,0)
Start a trajectory from each initial condition. What is happening
physically for each trajectory?
Question #6:
Choose other initial conditions and integrate some trajectories to get
a feel for some of the many possible orbits. Print a copy of some of these orbits and
attach the plot to your lab report.
Go To
Frederick J. Wicklin <fjw@geom.umn.edu>
Last modified: Mon Aug 15 07:48:04 2005
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