A uniform ladder of length 2L and mass m leans against a wall in a vertical plane at an angle θ to the horizontal. The floor is rough, having a coefficient of static friction μ . A person of mass M st

Question

A uniform ladder of length 2L and mass m leans against a wall in a vertical plane at an angle $θ$ to the horizontal. The floor is rough, having a coefficient of static friction $μ$ . A person of mass M stands on the ladder at a distance D from its base (see figure). If the wall is frictionless, the maximum distance $(D_{max})$ up the ladder that the person can reach before the ladder slips is :

CT42 m5 diagram

TestHub · Accepted Answer

At limiting equilibrium, f = μ ( m + M ) g and the smooth wall exerts the horizontal reaction N w ​ = f . Taking moments about the base: N w ​ ( 2 L sin θ ) = m gL cos θ + M g D cos θ . Hence D ma x ​ = [ 2 μ ( 1 + M m ​ ) tan θ − M m ​ ] L .

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