Procedure
Part 1 – Parametric Plot: 1. Find a manipulatable parametric plot of the double pendulum. Make sure the plot allows you to change the initial starting angles, find the coordinates, and alter the duration of the animation. (*Script in Materials page) 2. Change the plot to show at least 30 seconds of the pendulum’s path. 3. Input π/2 as the initial angle for both (the angle of mass one) θ1 and (the angle of mass two) θ2. 4. Observe and record the path of the pendulum. 5. Record the coordinate plane positions of mass 1 and mass 2 at 5, 10,20, and 30 seconds. 6. Run the experiment again and see if there have been any changes. 7. Repeat steps 2 to 5 but alter the initial angle by a ten thousandth (π/2 à π/2.0001), a thousandth (π/2.001), a hundredth (π/2.01), and a tenth (π/2.1). 8. Observe the paths taken and compare them. Part 2 - Determining the chaotic nature: 1. Give all the variables numeric values: · g = 9.8 · L1 = 0.7 · L2 = 0.7 · m1 = 1 · m2 = 1 · θ1 = π · θ2 = 0 · θ1d0 = 0 · θ2d0 = 3.0 2. Find a plot that shows the difference between the angles of mass one and mass two of two double pendulums with different initial starting conditions over time. To do this, the plot must take the difference between mass 1 of both pendulums, repeat the process with mass 2, and take the logarithms of these two differences and plot them as a function of time. (** Script in Materials page) 3. Change the velocity of the θ2d0 to 3.0 for the first double pendulum on the plot. For the second double pendulum, change it to 3.1. Now that the pendulums have different initial starting conditions, run the plot. 4. Repeat step 3 but change the initial velocity by a ten thousandth, a thousandth, and a hundredth. 5. Observe the plots and see if the differences increase, stay the same, or decrease. |
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