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Vectors minimum component

  1. Mar 10, 2016 #1
    1. The problem statement, all variables and given/known data
    The truck is to be towed using two ropes. If the
    resultant force is to be 950 N, directed along the positive x
    axis, determine the magnitudes of forces FA and FB acting
    on each rope and the angle q of FB so that the magnitude of
    FB is a minimum. FA acts at 20° from the x axis as shown.

    UgXVI2g.png

    2. Relevant equations


    3. The attempt at a solution
    I have no idea where to start because there is only 2 known values so i cant use sine law or cosine law. What's the first step ?
     
  2. jcsd
  3. Mar 11, 2016 #2

    ehild

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    Write the equations for the x and y components of the forces FA and FB. Solve for FB in terms of the angle q. At what q is FB minimum?
     
  4. Mar 11, 2016 #3
    FAx = FAcos(20)
    FAy = FAsin(20)


    FBx = FBcos(q)
    FBy = FBsin(q)

    can you help me with q ?
    i know q must make
    FBy=-FAy
    FBx + FAx = 950
    But i dont know how to find it.
     
  5. Mar 11, 2016 #4

    ehild

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    Plug in the expressions with the angles for the force components.
     
  6. Mar 11, 2016 #5
    FBx + FAx = 950
    FBcos(q) + FAcos(20) = 950
    FB cos(q) + 0.94FA = 950

    FBy=-FAy
    FB sin(q) = - 0.34 FA
     
  7. Mar 11, 2016 #6

    ehild

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    Now eliminate FA.
     
  8. Mar 11, 2016 #7
    Fb cos(q) + 0.94 Fa = 950
    Fb sin(q) + 0.34 Fa = 0
    --------------------------------------------- -

    [tex] \frac{Fbcos(q)}{0.94} - \frac{Fbsin(q)}{0.34}= \frac{950}{0.94}[/tex]
     
  9. Mar 11, 2016 #8

    ehild

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    Isolate FB.
     
  10. Mar 11, 2016 #9
    [tex]Fb ( \frac{cos(q)}{0.94} - \frac{sin(q)}{0.34})= \frac{950}{0.94}[/tex]

    [tex] Fb = \frac{950}{0.94} / (\frac{cos(q)}{0.94} - \frac{sin(q)}{0.34})[/tex]
     
  11. Mar 11, 2016 #10

    ehild

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    Fb is function of q and you need to find the minimum of that function. Have you studied Calculus?
     
  12. Mar 11, 2016 #11
    only calculus 1
     
  13. Mar 11, 2016 #12

    ehild

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    What is the derivative of a function at a minimum or maximum?
     
  14. Mar 11, 2016 #13
    zero
     
  15. Mar 11, 2016 #14

    ehild

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    Take the derivative of Fb with respect to q.
     
  16. Mar 11, 2016 #15
    derivative of cos and sin will result another sin and cos
    what should i do with it?
     
  17. Mar 11, 2016 #16

    ehild

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    That expression would be equal to zero. You will be able to solve that equation for q.
     
  18. Mar 11, 2016 #17
    The answer : :nb)
    CI5PCPg.png
    hmm,it looks like mission impossible . Do you know another method ehild ?
     
  19. Mar 11, 2016 #18

    ehild

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    It is a fraction equal to zero. What do you know about the numerator, if the fraction is zero?
     
  20. Mar 11, 2016 #19
    numerator must be zero which means q must be 0 for sine and 90 for cosine
     
  21. Mar 11, 2016 #20

    ehild

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    q is the same angle both for sine and cosine. What equation do you have for q?
     
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