(comments to: moloney@nextwork.rose-hulman.edu)

(CUPS = Consortium for Upper Level Physics Software, published by John Wiley.)

Chapters in CUPS 'Classical Mechanics Simulations'

- Motion Generator
- Anharmonic Oscillators
- Central Force Orbits
- Rotational Dynamics of Rigid Bodies
- Collisions
- Coupled Oscillations in One and Two Dimensions

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 kinks. - 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 Rigid Bodies

- 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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