Calc III Lab: Period versus Amplitude

Reset the parameters so that damping=0 and tau=0.

The period of an oscillation is the time that it takes for the position and velocity to return to their initial values. This corresponds to the time it takes for a trajectory in phase space to make a closed path. For our current values of the parameters, the amplitude of a trajectory is the largest value of x that the trajectory passes through. This will always occur when v=0.

Fix v=0 and let x=0.25. The amplitude for this trajectory will be 0.25 (Why?), but what is the period? Use the event stopping abilities of DsTool to find the period of the solution with initial condition (0.25,0). To do this

• Set v=0 as the event stopping criterion.
• Use the Forward and Continue buttons to begin a trajectory forward in time and continue that trajectory until it has traced out a closed curve in phase space.
• Read the value of time from the Final column of the Selected Point Panel. This number is the value of the period of oscillation for the given initial condition.

Question #3:

• Conduct the experiment above for x=0.25, 0.5, 1, 1.5, 2 and 2.5. On the graph provided, plot the period of each solution versus the amplitude of that solution.
• For the linear oscillator, you can explicitly check your graph. Explicitly compute the period of oscillation for damping=0 and tau=0 by solving for the general solution, x(t), of the linear oscillator. (Hint: recall that the period of sin(wt) is 2 Pi/w.) Does the period change as you change the initial condition?

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