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Man climbing a ladder

  1. Sep 3, 2016 #1
    1. The problem statement, all variables and given/known data
    Snap1.jpg

    A man of mass m climbs a ladder of mass M and length L. the ladder leans against a friction less wall and makes an angle α with a rough floor (coefficient μ).
    To which max height can he climb before the ladder slips.

    2. Relevant equations
    Friction: ##f=\mu N##

    3. The attempt at a solution
    The reaction in the base: ##R_x=(M+m)\mu g\sin(\alpha)##
    The perpendicular component of the lader and man's gravity makes torque round the base which is balanced by R, the reaction in the vertical wall:
    $$lmg\cos(\alpha)+\frac{L}{2}Mg\cos(\alpha)=RL\sin(\alpha)~~\rightarrow~~R=\frac{\left( lm+\frac{L}{2}M \right)g\cos(\alpha)}{L\sin(\alpha)}$$
    $$R=R_x:~~\frac{\left( lm+\frac{L}{2}M \right)g\cos(\alpha)}{L\sin(\alpha)}=(M+m)\mu g\sin(\alpha)$$
    The minimal height h:
    $$h=l\sin(\alpha)=\frac{2m\mu L\sin^2(\alpha)-ML\cos(\alpha)}{2m}\tan(\alpha)$$
    The result must be:
    $$h=L\frac{\mu(M+m)\sin(\alpha)-\frac{1}{2}M\cos(\alpha)}{m\cos(\alpha)}$$
     
  2. jcsd
  3. Sep 3, 2016 #2

    BvU

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    hi,
    can you explain the ##\sin\alpha## in your ##R_x## ?
     
  4. Sep 3, 2016 #3
    Thanks BvU, solved
     
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