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Find the tension in the rope and forces exerted by the floor

  1. Sep 17, 2009 #1
    I'm new here, so please bear with me.

    1. The problem statement, all variables and given/known data

    http://img15.imageshack.us/img15/9325/questionq.png [Broken]​
    [/URL]

    Verbatim from my textbook:
    As shown in Fig.7.40, the two sides of a step ladder BA and CA are 1.6 m long and
    hinged at A. A rope DE, 0.5 m is tied half way up. A weight 40 kg is suspended from
    a point F, 1.2 m from B along the ladder BA. Assuming the floor to be frictionless
    and neglecting the weight of the ladder, find the tension in the rope and forces
    exerted by the floor on the ladder. (Take g = 9.8 m/s2)

    2. Relevant equations

    Condition for equilibrium: Net force=0 and net Torque([tex]\tau[/tex])=0
    [tex]\tau[/tex]=rF sin[tex]\theta[/tex]

    3. The attempt at a solution
    http://img36.imageshack.us/img36/5815/questiono.png [Broken]​
    [/URL]

    Let the angle BAC be 2x, tension in string be T, hung weight of 40kg be W, and the reaction forces of the floor be R and S.

    Then, for the equilibrium of AB(net torque=0):
    0.4W*sinx + 0.8T*cosx = 4Rsinx which on simplification led to
    T = 2/3 *tanx(4R-W) ------(1)

    For the equilibrium of AC (Net torque=0)
    0.6*T*cosx = 1.6*S*sinx
    T = 8S/3 tanx --------(2)

    For vertical equilibrium of ladders, W = R + S -------(3)

    Equating 1 and 2,
    R - S = W/4 --------(4)

    Solving 3 and 4, I got R=245 and S=147 which are correct answers.

    The next question is to find the tension, which is what I'm having trouble with. Using either the law of sines or the law of cosines gives me x=24.6 degrees, and substituting that into eqns 1 or two gives me the wrong answer for the tension - the correct answer according to the book is T=96.7N.

    Any help is appreciated.
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Sep 17, 2009 #2
    AE and BD are 0.8 long, not 0.6, so your value for x is wrong.

    The values of R and S are independent both the length of the ladder and the length of
    the rope BTW. This is easy to see by considering the torque around F on the entire ladder
     
  4. Sep 17, 2009 #3
    That sucks, I don't usually do stupid things like that. Thank you very much.

    Also - is there a shorter way to do this problem?
     
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