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(CUPS = Consortium
for Upper Level
published by John Wiley.)
Chapters in CUPS 'Classical Mechanics
- Motion Generator
- Anharmonic Oscillators
- Central Force Orbits
- Rotational Dynamics of Rigid Bodies
- Coupled Oscillations in One and
Suggested Classical Mechanics
Activities from the CUPS project
- Motion Generator has several numerical methods available. Carry out
Exercise 2.1 (Comparison of Numerical Methods) This involves analytical
solution to Motion subject to a 1D resistive force F = -c1*v It asks you
to repeat runs for different numerical methods and for different dt values.,This
enables you to determine the 'order of convergence' for each method
- Retrograde motion of planets has long been observed with fascination.
Under 'Systems', select Retrograde Motion. Run it and see how it beautifully
illustrates the reason for the retrograde behavior seen from Earth.
- In Maple, make a parametric plot of the x and y components of the line
joining Earth and Jupiter as they orbit the Sun. Take the period of Jupiter
to be 11.9 years and its distance to be 5.2 AU (1AU = Sun-Earth distance).
This should give you the same plot as seen from the 'Earth Window'.
- Carry out exercise 4.1.
This asks you to run the motion of small bodies at various points in the
Sun-Jupiter rotating frame, and observe the 'kinks' in the motion of the
small bodies. Carry out parts b)-d) where you offer a
reason for kinks, then change parameters to see if the changes affect the
- The body
at 60 degrees in the Sun-Jupiter 'lagrangian' system is theoretically stable
under small amplitude oscillations. Follow the suggestions in Exercise
4.3 for changing some of its initial (x,y) values; explore the range of
(x,y) values for which it is stable.
Rotational Dynamics of
- Set the system to 'Spinning Top'. Check under Initial Conditions that
the initial angular momentum is 20 units. Run the simulation and observe
the general behavior of a gyroscope - fairly smooth precession of the
rapidly spinning body under the given torque (usually due to its own weight).
- Now predict what will happen when you leave the torque the same but
decrease the initial angular momentum to nearly zero. Write down your prediction.
Now set the initial angular momentum value to 0.1 units, and run
the simulation. Record what you saw in the simulation.
- Compare the behaviors with large and small angular momenta. In simple
terms, why are they so different?
- Carry out exercise 7.3. This asks you to set up the dynamical matrix
for CO2, a linear molecule, with C in the middle, an O at each end, and
springs of constant K between the C and each O. The masses are known, and
the idea is to find the spring constant which reproduces the two frequencies
of 4.010 * 10^13 Hz, and 7.051*10^13 Hz. You are warned to choose the units
K with some care!
- Set up the same system in Maple. You will have 3 equations of motion
(F=ma), one for each mass (the variables might be x1, x2, and x3). With
masses and spring constants given, Maple will solve the system of 3 equations
and arrive at an analytical solution for the three modes. Before you
carry this out, think whether we will get 3 different frequencies from
the Maple simulation, whereas in Exercise 7.3 there are only two.
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