3. Basic Techniques and Modeling Strategies

• Inclusion of variable acceleration
  • a numerical integration scheme
  • use of functions

• Using energy as an indicator of accuracy

• Exercise 3.1: simple harmonic motion

  • initial conditions: x = 0, v = V0
  • harmonic acceleration: a = -K x,  with K = 4

  • (a) Take V0 = 1
    • start with dt = 0.05 and run the program until t = 2π (M_PI in cmath)
    • compare the analytic and numerical solutions as functions of time
    • plot the total mechanical energy E = 0.5 v² + 0.5 K x² as a function of time
    • reduce dt until "reasonable" agreement between the numerical and analytical results is obtained -- let's say, when the amplitude or the energy change by less than 1 percent

  • C++ and Python solutions to part (a)

  • (b) Now modify the termination condition to compute numerically the period of the motion, by finding the first time t > 0 when the particle crosses from x < 0 to x >= 0. Use linear interpolation to refine your estimate.
    • repeat your calculation of the period for V0 = 0.25, 0.5, 2, 4, 8.
    • plot the period as a function of V0 for V0 = 0.25, 0.5, 1, 2, 4, 8.

  • C++ solutions to part (b):  single V0,  internal loop over all V0,  script loop over all V0
  • Python solutions to part (b):  single V0,  internal loop over all V0,  script loop over all V0

• Exercise 3.2: not-so-simple periodic motion

  • repeat the previous exercise, but using the restoring force for a real pendulum:

         a = -K sin(x)

    with  K = 4  and the same initial conditions as above.

  • (a) what is the new expression for the energy?
    • what value of dt conserves energy to 1 percent?

  • (b) determine the period of the motion for V0 = 0.25, 0.5, 1, 2, 4, 8.
    • can you explain the behavior for V0 = 4, 8?
    • plot the period as a function of V0.

• Homework #2