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Ladder with 2 forces of friction and a person climbing (statics)

  1. Jul 26, 2013 #1
    1. The problem statement, all variables and given/known data
    A uniform ladder of mass m and length L rests against the wall as shown. The wall is frictionless. The coefficient of static friction between the floor and the ladder is μ. The ladder makes the angle θ with the wall. How far along the ladder can a person of mass m climb before the ladder begins to slide?

    2. Relevant equations
    f(friction)≤μN(normal force)
    t(torque)=r(moment arm)Fsinθ
    G=mg

    3. The attempt at a solution
    FBD is attached

    because the ladder is not moving
    t=-mgLsinθ/2+NfLsinθ-μNfLcosθ-mg(L-x)sinθ=0
    Fx=Nw-ff=0
    Fy=Nf+fw-2mg=0

    Solve t for Nf
    NfLsinθ-μNfLcosθ=mgLsinθ/2+mg(L-x)sinθ
    Nftanθ-μNf=mgtanθ/2+mg(L-x)tanθ/L
    Nf=mg(1/2+(L-x)/L)/(1-mu/tanθ)

    When θ is at its maximum without the ladder slipping, f=μN.
    ff=μmg(1/2+(L-x)/L)/(1-mu/tanθ)
    Solve Fxfor Nw
    Nw=μmg(1/2+(L-x)/L)/(1-mu/tanθ)
    Also fw=μ2mg(1/2+(L-x)/L)/(1-mu/tanθ)

    Solve Fy for x.
    mg(1/2+(L-x)/L)/(1-mu/tanθ)+μ2mg(1/2+(L-x)/L)/(1-mu/tanθ)-2mg=0
    3/2-x/L=2(1-μ/tanθ)/(1+μ2)
    x/L=3/2-2(1-μ/tanθ)/(1+μ2)
    x=L(3/2-2(1-μ/tanθ)/(1+μ2))
     

    Attached Files:

  2. jcsd
  3. Jul 26, 2013 #2

    ehild

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    The wall is frictionless, so [STRIKE]ff=0[/STRIKE]. fw=0
    Edit: It is fw that is zero
    ehild
     
    Last edited: Jul 26, 2013
  4. Jul 26, 2013 #3
    There is friction in the floor, so ff is not 0. However, fw should be 0 as the wall is frictionless.

    Dr Peter Vaughan
    BASIS Peoria Physics
     
  5. Jul 27, 2013 #4
    Okay, I feel really dumb. I mixed together this problem with the one below it when I was working. This is a lot easier.
    Solve for Nf using Fy.
    Nf=2mg
    Solve for x using t.
    -mg(L-x)sinθ-mgLsinθ/2+2mgLsinθ-2μmglcosθ=0
    -mgxsinθ+mgLsinθ/2-2μmglcosθ=0
    mgxsinθ=2μmglcosθ-mgLsinθ/2
    xtanθ=2μL-Ltanθ/2
    x=2μL/tanθ-L/2
    Please correct me if I'm wrong.
     
  6. Jul 27, 2013 #5

    ehild

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    Homework Helper
    Gold Member

    It is correct now.

    ehild
     
  7. Mar 31, 2015 #6
    Where are you getting an angle to plug in to your equations? The question just says that it touches the wall at angle theta.
     
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