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. R: Go to the web site http://www.r-project.org/ and download the appropriate version of R for your personal computer.

2. Spring-Mass System:
   
   • The Second-order differential equation describing the motion of a linear harmonic oscillator (mass-spring) is given by x''=(-k/m)x, where x is the displacement, k is the spring constant and m is the mass.
   • Convert this 2nd-order ODE to a pair of first order differential equations. Enter your pair of ODEs into Phaser Custom Equation Library.
   • Set k=1.6, initial conditions x1(0)=1.2, x2(0)=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 to determine how the period depends on mass.
   • Perform a similar experiment to determine the dependence of the period as function of k.
   • Set the values of k and m as you please; determine the period of oscillations for 10 initial conditions. How does the period depend on the initial conditions?

3. Period of Pendulum:
   • In the pendulum equation in PHASER set m = 1.2, l = 1.1, 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? (You need to look up the value of g for the moon.)
   • 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?

4. A series for period:
   • There is no formula to compute the period of a planar pendulum without friction. However, the following infinite series
     can be used to approximate the period to a desired accuracy by including a finite number of the terms of the series.
   • Now take g =1, l=1.2, the initial angular displacement of 60 degrees (no initial velocity) and using the first 3 terms of the series compute the approximate period.
   • Compare your answer with the one you get from Phaser simuation.