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Equilibrium of Forces Paradox

  1. Nov 19, 2015 #1
    Hi ! So a friend and I were solving some mechanics problems in class today. And we came across a pretty funny mathematical paradox. So basically we tackled the problem in different ways.....but we ended up with different equations...and none of us could prove the other wrong !!!!
    So here goes....
    The Problem:
    upload_2015-11-19_19-15-31.png
    The question....a very simple one really :
    "Express T in terms of P"
    1) His method...(or as he describes it "The Human Method")
    upload_2015-11-19_19-17-47.png
    He resolved vertical component of the force T, ie. Tcosθ
    Then he equated vertical components saying P = Tcosθ.... therefore T = Psecθ

    2)My way....
    Note: This was part of a much complex problem..... I'm not possessed to go through sooooo much trouble for such a small thing....and well...if i hadn't i wouldn't have found this....anyway
    upload_2015-11-19_19-29-18.png
    I resolved P instead,
    and got that T = Pcosθ...the exact opposite

    SO...the big question is ...
    T = Pcosθ vs. T = Psecθ

    Now ... i know practically speaking I might be wrong ...because according to me T < P .....however T would have to be greater ....since its offsetting the downward vertical force of P and has a leftward horizontal component.......but then again ...can't you make the same argument about resolving P ???????

    I know .. I've been blabbering alot...but it really seems mind boggling !!!!
    Mathematically both seem correct XD


    Update : There isn't any vertical acceleration !!!!

    -HvB99
     
    Last edited: Nov 19, 2015
  2. jcsd
  3. Nov 19, 2015 #2

    Dale

    Staff: Mentor

    You cannot express P in terms of T without some additional constraint information. For example, the condition that there is no vertical acceleration or something similar.
     
  4. Nov 19, 2015 #3
    there is no paradox. you cant solve this problem with this method because T and P is not in same line. one vector cant generate a plane.the first one you dont consider the vertical force of T and the second one you dont consider sinx of P. if you want to express vectors, you need 2 vectors to describe the other one.
     
  5. Nov 19, 2015 #4
    Yes....there isn't ...this was known...but i forgot to mention it...sorry!!!
     
  6. Nov 19, 2015 #5
    I'm sorry... but the other component....the one that isn't accounted for...as you said ...is the one causing a horizontal acceleration, since it is unbalanced( only that component ) .....

    I merely equated the components of the two forces that were in the same direction.
     
  7. Nov 19, 2015 #6
    How about the horizontal acceleration? What do you know about it?

    By the way, with just these two forces you cannot have and equilibrium.
    And it is good practice to state the problem completely before jumping to (or expecting) a "solution".
     
  8. Nov 19, 2015 #7

    Dale

    Staff: Mentor

    Given the additional information on the acceleration this is exactly correct. The vertical component of T is equal to P, and additionally T has a horizontal component that P does not.
     
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