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The motion of an object acted on by a time-dependent, position-dependent force?

  1. Sep 15, 2011 #1
    Is there a non-iterative way to solve for the motion of an object under the influence of a time-dependent, position-dependent force?

    Here is an example problem:


    A bead on a straight wire is subjected to a net force.
    There is no friction between the bead and the wire.
    The bead and the wire do not experience any gravitational forces.
    The bead has a mass of 1kg.

    The net force acts along the wire, and is position and time dependent. It can be represented as

    F = 9x2 + 5xt - 10t2

    The bead starts at rest at x = 0.

    Where is the bead after 30 seconds have elapsed?

    How would one go about solving a problem like this numerically?
     
  2. jcsd
  3. Sep 15, 2011 #2
    Hi,
    Sadly there's no way around it :(...(due to the presence of squared and multiplied elements on the right hand side).
    This question however, is very reasonably stable using a simple, forward-Euler method. In other words,
    [tex]
    \Large
    x'(t) = \frac{(x_{n+1}-x_n)}{\delta t}
    x''(t) = \frac{x'_{n+1}-x'_{n}}{\delta t}
    [/tex]
    And substituting accordingly leads to very accurate and tolerable results.
    Have you tried using Mathematica, or Matlab? This can also be effectively modelled in C/C++ or any other programming language.
    I hope that helps,
    Daniel
     
  4. Sep 15, 2011 #3

    Ray Vickson

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    Science Advisor
    Homework Helper

    Sadly, the forward Euler method is often unstable and should almost never be used for solving a DE over s long interval. That's why so many alternative methods (such as Runge-Kutta, etc.) have been developed.

    RGV
     
  5. Sep 16, 2011 #4
    Oh, I definitely agree... but for short intervals, as is required in the query, as well as just for introductory means, it should work well enough...
    Of course, moving on to Runge-Kutta, or some implicit methods should elucidate any issues with accuracy, stability, so forth...
    Daniel
     
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