# Week 6

### Assignment:

1. Small angle approximation:
• The Second-order differential equation given by x''=(-g/l)x is the small angle approximation to the pendulum equation by replacing sin(x) with x.
• Convert this 2nd-order ODE to a pair of first order differential equations. Enter your pair of ODEs into Phaser Custom Equation Library.
• Set g=l=1 and start the pendulum with 12 degrees displacement and release. Determine the period of the oscillations in the nonlinear pendulum equation (use Pendulum ODE in Phaser) and in the small angle approximation. What is the difference in the periods?
• Set g=l=1 and start the pendulum with 80 degrees displacement and release. Determine the period of the oscillations in the nonlinear pendulum equation and in the small angle approximation. What is the difference in the periods?

2. Lunar Pendulum:
• In the Pendulum ODE equation in PHASER set m = 1.1, l = 1.4, and g = 1. Note that x1 is the angular position and x2 is the angular velocity of the pendulum. What is the period of the pendulum motion starting with initial angle of 1.2 radians and zero velocity? To increase the period by 15% how much do you need to change the length of the pendulum? Use the Xi vs Time view to determine the period.
• 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 relative to earth.)

3. A series for period:
• There is no practical formula to compute the period of a planar pendulum without friction. However, the following infinite series
can be used to approximate the period (with zero initial velocity) to a desired accuracy by including a finite number of the terms of the series.
• Now take g =1, l=1.4, the initial angular displacement of 70 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.

4. Escape velocity: For this investigation use the Kepler ODE in the ODE Equation Library of Phaser. Suppose that a particle of mass = 1.1 is positioned at the coordinates (2.1, 0). For this problem, first you should load the equation defaults for Kepler ODE. Then set the appropriate initial conditions to follow the shapes of the orbits.
• Use the initial velocity components (0.0, 0.4) and admire the elliptical orbit. Double the mass to m=2.2. Notice that the orbit gets flattened. Find the value of the vertical velocity component (0.4) that gives the previous elliptical orbit with m = 1.1.
• Set m =1.1 and find the minimum vertical velocity so that the particle escapes to infinity; that is, its orbit ceases to be an ellipse. You might need to use a fairly long time and a big window size to get a good estimate of the vertical escape velocity.
• Next, double the mass of the particle. What is the new escape velocity?

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