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Ladder against a frictionless wall

  1. Apr 23, 2015 #1
    A uniform ladder of length 9 m leans against a frictionless vertical wall making an angle of 47° with the ground. The coefficient of static friction between the ladder and the ground is 0.41.

    If your mass is 74 kg and the ladder's mass is 33 kg, how far up the ladder can you climb before it begins to slip?


    Given:
    L = 9 m
    M = 74 kg
    m = 33 kg
    μ = 0.41
    θ = 47°
    x = unknown


    Noted:
    FWL = Force from Wall to Ladder
    Ff = Force of Friction
    N = Normal Force


    I think I've solved it, but I just wanted to be sure. I was trying to get some practice in for my test tomorrow and found this question on these forums unanswered.

    Since nothing is moving, the system is in Equilibrium. So net torque = 0, or counter clockwise torques = clockwise torques. Equation for torque; t = Force x Moment Arm

    Mg(xL cosθ) + mg(0.5L cosθ) = FWL(L sinθ)

    The L cancels out from both sides, and the g cosθ is factored out of left side.
    g(xM + 0.5m) cosθ = FWL sinθ
    Divide both sides by sinθ
    g(xM + 0.5m) / tanθ = FWL
    Since system is at equilibrium FWL = Ff = μN = μ(M + m)g
    g(xM + 0.5m) / tanθ = μ(M + m)g
    Multiple both sides by tanθ, and g cancels from both sides
    xM + 0.5m = μ(M + m) tanθ
    xM = μ(M + m) tanθ - 0.5m
    x = [μ(M + m) tanθ - 0.5m] / M = 0.4127...

    Max length up ladder = xL ≈ 3.715 m
     
  2. jcsd
  3. Apr 23, 2015 #2

    ehild

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    Your solution is correct but the sentence above is not.

    It should be

    Since system is at equilibrium FWL = Ff ≤ μN = μ(M + m)g
     
  4. Apr 23, 2015 #3
    can you elaborate on that? I was simply plugging in the equation for Friction at that point. Why is Ff ≤ μN?
     
  5. Apr 23, 2015 #4

    BvU

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    Matter of careful wording: The system is also at equilibrium when xL = 0 but then ##F_f \ne \mu N##
     
  6. Apr 23, 2015 #5

    ehild

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    The force of static friction is a force that prevents sliding. It does not have a definite value, only a maximum: depending on the surfaces in contact, the maximum value is μsN, where μs is the coefficient of static friction. If the other forces exceed the static friction, the object starts to slide and is not in equilibrium any more. The friction becomes kinetic, and equal to μkN
     
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