What is the maximum height a man can climb on a ladder before it slips?

[Karol]

Homework Statement

[/B]
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.

Homework Equations

Friction: ##f=\mu N##

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:
$$h=L\frac{\mu(M+m)\sin(\alpha)-\frac{1}{2}M\cos(\alpha)}{m\cos(\alpha)}$$

 

[BvU]

hi,
can you explain the ##\sin\alpha## in your ##R_x## ?

 

[Karol]

Thanks BvU, solved