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I'm stumped on this problem. I've solved for the distance the person can climb in terms of N(b), but I cant get N(b) in known terms.

A ladder of mass 2M and length 2L is kept on a rough horizontal floor and leaned against a rough vertical wall. The coeff. of static friction between both the wall and the ladder and the floor and the ladder is u. A man of mass M starts climbing the ladder. Find the maximum distance the man could climb (x) before the ladder starts sliding.

Point (a) is defined as the point at which the ladder contacts the floor, and (b) is defined as the point at which the ladder contacts the wall. theta (known) is given as the angle between the top of the ladder and the wall.

N = Normal Force

F = Friction

What I have so far is:

x dir: N(b)-F(a) = 0

N(b) = F(a)

y dir: F(b) + N(a) - 2Mg - Mg = 0

F(b) + N(a) = 3Mg

Torque about A:

N(b)*2L(cos(theta)) + F(b)*2L(sin(theta)) - 2Mg*L(sin(theta)) + Mg*x(sin(theta) = 0

So solving for x in the torque equation is trivial, but I can't get anything else (N(a), N(b), F(a) or F(b)) in terms of M and g. Any insight is appreciated.

Edit: And because its about to slide:

F(a)=uN(a)

F(b)=uN(b)

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# Ladder on a wall problem

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