## Mathematics & Science
Learning Center |
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## Applications of Differential Equations## The Simple Pendulum
## The Initial Value ProblemOn the last page we carefully derived the differential equation governing
the motion of the simple pendulum. Once again we need to remind ourselves that
solving this equation by itself will lead to a As usual, for a second order initial value problem the initial conditions
will be in terms of Our new initial value problem might look something like this:
As a reminder: *y*= angle of displacement (a function of time,*t*, yet to be determined)*L*= length of the pendulum*g*= acceleration due to gravity*y*_{o}= initial angle of displacement*v*_{o}= initial angular velocity
Also, ## Linearizing the ProblemThis equation is non-linear and second order. It cannot be solved by standard
methods. The usual classroom approach is the So what would make a good linear approximation for sin( There are at least two common approaches that may be used to justify our next step:
Plot[{Sin[y],y},{y,-0.5,0.5}] Notice the braces enclosing the set of functions to be plotted. These are always necessary when more than a single function is to be plotted. It's now time to actually try the command. You can flip back here when needed in order to check the command. (Actually, you could even copy and
paste it into Mathematica.) Click on the button provided on the
left to switch. Don't forget to come back here when you're done!
Welcome back! Let's go discuss what you got... |
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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 |