Topics:

• Modeling continuous systems using Ordinary Differential Equations (ODE).
• Application: Pendulum.
  Pendulum Equation
  Pendulum history
• Introduction to R.
• Least-square line fitting to data points Reference
• R help files: Search R functions R reference card R help Index
• R files from lectures: beginnings.R simpleGraph.R

Assignment:

1. Spring-Mass System:
   
   • In Phaser ODE library use Linear Oscillation ODE for this problem. Set c=0, f=0, k=1, initial conditions x1=1, x2=1. Start m=1 and increase it to m=9 by increments of 1. For each value of m determine the (approximate) period of the oscillations in the XiVsTime view. Now, plot these periods as a function of m using R to determine how the period depends on mass.
   • Perform a similar experiment to determine the dependence of the period as function of k.
   • Resonance: Set c=0, k=1, m=2, x1=1, x2=1, f=0 and determine the period of the oscillations. Now, set f = 0.5 and determine an apprixomate value of the forcing period parameter a to set the system into resonance--unbounded oscillations. (Hint: Try to match the system period to the forcing function period)

2. Period of Pendulum:
   • In the pendulum equation in PHASER set m = 1, l = 1, and g = 1. What is the period of the pendulum motion starting with initial angle of 1.2 radians and zero velocity. To increase the period by 20% how much do you need to change the length of the pendulum?
   • If you take your grandfather's clock to the moon, does it slow down or speed up? By how much?
   • Assume that pendulum is at its stable equilibrium position. How much minimal initial velocity do you need to impart on the bob so that the pendulum goes over the top? Does this initial velocity depend on the length of the pendulum?

3. Energy conserved: The Energy of planar pendulum without friction is given by the function
   E(Θ , Θ') = (1/2)m(L^2)(Θ'^2) + mgL(1- cos(Θ)).
   This Energy function is supposed to be conserved along the solutions of the pendulum. Show this assertion by computing dE/dt using the chain rule, keeping in mind that Θ and Θ' depend on t, and the differential equations for pendulum. Observe that dE/dt = 0, so it must be constant function of t.

4. Free energy from EULER: Now, in the pendulum equation, take c= 0, m=1, l=1, g=1, the initial conditions Θ=0.5 and Θ'=0 and compute the solution for 20 time units using Euler's algorithm with step size of h=0.1. On the Time Panel of Phaser's Numerics Editor set "Skip iterations per plot" to 9 so that in the Xi Values view you will get the solution tabulated at integer values of time: you should have 21 rows of numbers. Now, with R, use these position and velocity numbers in the formula for the Energy of the pendulum above so that you will have 21 values of the Energy function. Plot these Energy values as a function of time. Are these Energy values constant? Increasing? Decreasing?

5. WINDOWS USERS:
   Please watch the cygwin_installation_demo and install cygwin on your laptop.
   MAC USERS:
   Do not install anything. Mac comes equipped with what you need.