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Re: PSE Example 8.6 via Reduce

  • To: mathgroup at smc.vnet.net
  • Subject: [mg112882] Re: PSE Example 8.6 via Reduce
  • From: Eduardo Cavazos <wayo.cavazos at gmail.com>
  • Date: Tue, 5 Oct 2010 05:31:45 -0400 (EDT)
  • References: <i7mq41$ovh$1@smc.vnet.net> <i8c8t0$g8f$1@smc.vnet.net>

On Oct 4, 5:05 am, Eduardo  Cavazos <wayo.cava... at gmail.com> wrote:
> On Sep 26, 1:44 am, Eduardo  Cavazos <wayo.cava... at gmail.com> wrote:
>
>
>
> > Here's example 8.6 from PSE [1]:
>
> >    http://i.imgur.com/HJLqs.png
>
> > Here I'll be working with the "exercise" portion of the example; i.e.
> > the case where there is friction between point A and point B.
>
> > Use Reduce on the set of equations which describe the system between
> > point A and point B:
>
> >    http://i.imgur.com/PLWza.png
>
> > The result shows that vB is about 12.9.
>
> > Now use Reduce on the set of equations which describe the system
> > between point B and point C. Also, plug in our value of 'vB' manually:
>
> >    http://i.imgur.com/rujUz.png
>
> > We get the correct answer for dBC; i.e. about 40.3.
>
> > Above I solved the problem in "piecemeal" fashion; first I used Reduce
> > on the system from A to B and used the result from that to work with
> > the system from B to C, also using Reduce.
>
> > I don't have alot of experience with Reduce, but it seems like, in
> > theory, I should be able to throw the whole set of equations at Reduce
> > in one step, instead of in two steps. However, when I try to do so,
> > Mathematica seems to have trouble with it (at least on this system;
> > Pentium M with about 1.5GB RAM). So something along the lines of:
>
> >    http://i.imgur.com/2swfE.png
>
> > So my questions are... Is this too much for Reduce to handle? Would
> > you recommend the piecemeal method instead?
>
> In particular, the trouble is that the computation ran over 3 hours
> without completing. When I split up the problem, each piece takes a
> few seconds.

Here's the code for the non piecemeal approach:

Timing[Reduce[{
    Subscript[\[CapitalEpsilon], A] ==
     Subscript[K, A] + Subscript[U, A],
    Subscript[K, A] == 1/2 m*Subscript[v, A]^2,
    Subscript[U, A] == m*g*Subscript[y, A],
    Subscript[\[CapitalEpsilon], B] ==
     Subscript[K, B] + Subscript[U, B],
    Subscript[K, B] == 1/2 m*Subscript[v, B]^2,
    Subscript[U, B] == m*g*Subscript[y, B],
    Subscript[\[CapitalDelta]E, AB] ==
     Subscript[\[CapitalEpsilon], B] - Subscript[\[CapitalEpsilon],
      A], Subscript[\[CapitalDelta]E,
     AB] == -Subscript[\[CapitalEpsilon], frictionAB],
    Subscript[\[CapitalEpsilon], frictionAB] ==
     Subscript[f, kAB]*Subscript[d, AB],
    Subscript[f, kAB] == Subscript[\[Mu], k]*Subscript[n, AB],
    Subscript[n, AB] == m*g*Cos[\[Theta]],
    Sin[\[Theta]] == Subscript[y, A]/Subscript[d, AB],
    Subscript[d, AB] > 0,
    Subscript[\[CapitalEpsilon], frictionAB] > 0,
    Subscript[v, B] > 0,
    Subscript[\[CapitalEpsilon], B] ==
     Subscript[K, B] + Subscript[U, B],
    Subscript[K, B] == 1/2 m*Subscript[v, B]^2,
    Subscript[U, B] == m*g*Subscript[y, B],
    Subscript[\[CapitalEpsilon], C] ==
     Subscript[K, C] + Subscript[U, C],
    Subscript[K, C] == 1/2 m*Subscript[v, C]^2,
    Subscript[U, C] == m*g*Subscript[y, C],
    Subscript[\[CapitalDelta]E, BC] ==
     Subscript[\[CapitalEpsilon], C] - Subscript[\[CapitalEpsilon],
      B], Subscript[\[CapitalDelta]E,
     BC] == -Subscript[\[CapitalEpsilon], frictionBC],
    Subscript[\[CapitalEpsilon], frictionBC] ==
     Subscript[f, kBC]*Subscript[d, BC],
    Subscript[f, kBC] == Subscript[\[Mu], k]*Subscript[n, BC],
    Subscript[n, BC] == m*g,
    Subscript[d, BC] > 0,
    Subscript[\[CapitalEpsilon], frictionBC] > 0
    } /. {
    Subscript[y, A] -> 20, Subscript[v, A] -> 0,
    Subscript[y, B] -> 0,
    Subscript[y, C] -> 0, Subscript[v, C] -> 0,
    Subscript[\[Mu], k] -> 0.210, g -> 9.8, \[Theta] -> 20 Degree
    }, Reals]]

Ed


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