Fasten a length of string to a hook. Securely fasten a mass to the other end of the string. Let θ represent the displacement of the mass from the vertical. Assume that θ is measured in radians, with positive displacements in the counterclockwise direction. Let ω represent the angular velocity of the mass, with positive angular velocity in the counterclockwise direction. Displace the mass 30° counter-clockwise (positive π/6 radians) from the vertical and release the mass from rest. Let the pendulum decay to a stable equilibrium point at θ = 0, ω = 0.
Important Note. You will not benefit as much as you should from this exercise if you don’t first complete parts a) and b) before attempting part c).
a) Without using any technology, sketch graphs of θ versus t and ω versus t approximating the motion of the pendulum.
b) Without using any technology, sketch graphs of ω versus θ. Place ω on the vertical axis, θ on the horizontal axis. Note: This is a lot harder than it looks. We suggest that you work with a partner or a group and compare solutions before moving on to part c).
c) Select Gallery→pendulum in the PPLANE6 Setup window. Adjust the damping parameter to D = 0.1, and set the display window so that −1 ≤ θ ≤ 1 and −1 ≤ ω ≤ 1. Select Options→Solution direction→Forward and use the Keyboard input window to start a solution trajectory with initial condition θ(0) = π/6 and ω(0) = 0. Compare this result with your hand-drawn solution in part b). Select Graph→Both and click your solution trajectory in the phase plane to produce plots of θ versus t and ω versus t. Compare these with your hand-drawn solutions in part a).
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