Mathematics & Science
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Applications of Differential EquationsThe Simple Pendulum(continued from last page...) Your little graphing session should have gone like this:
Plot[{linsol,actsol},{t,0,15}, Clearly the two solutions are very close to one another on this interval. In fact, they are almost indistinguishable, their differences only starting to show up as t moves further and further away from zero. What does this tell us about the validity of the linearization that we did earlier? It would seem to be telling us that the linearized problem approximates the actual problem extremely well. Why did the linearization do such a good job? Well, this problem had a fairly small initial velocity, i.e. v_{o}=1. In fact look at the y-values on the graph you just got--they're fairly small, somewhere on the order of 0.3. We said that the linear approximation would be best for small values of y, and the graph seems to be evidence of this. The Physical InterpretationWhile we're still looking at the picture, maybe we should have a quick chat about its physical interpretation. The y-values oscillate between -0.3 and 0.3 in a periodic fashion. Recall that the y-value in our mathematical model is actually the angle that the pendulum bar makes with the equilibrium position (straight down.) So the oscillation we see in the picture may be physically interpreted as the pendulum's position oscillating between 0.3 radians clockwise from equilibrium, and 0.3 radians counter-clockwise from equilibrium. (0.3 radians is about 17 degrees.) In the absence of friction, which was a requirement in our model, this interpretation of the solutions is perfectly reasonable. What if we use Larger Values of y?The displacement values would be larger if we used a larger initial velocity. Maybe we should investigate how the solutions compare if we increase the value of v_{o} to, say, 5. Go ahead and repeat all of the steps we've done so far in this laboratory, but just change the value of v_{o} in the first step to 5. Welcome back. Let's go look at the picture which you should have gotten... |
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ODE Laboratories: A Sabbatical Project by Christopher A. Barker ©2017 San Joaquin Delta College, 5151 Pacific Ave., Stockton, CA 95207, USA e-mail: cbarker@deltacollege.edu |