A uniform ladder of length L and mass M leans against a frictionless vertical wall. The ladder makes an angle theta with ground and the coefficient of the static friction between the ground and the foot of the ladder is mu. What maximum height a man of mass m can climb up the ladder so that the ladder will not skid?

A uniform ladder of length L and mass M leans against a frictionless v

1.	A uniform ladder of length L and mass M leans against a frictionless vertical wall. The ladder makes an angle theta with ground and the coefficient of the static friction between the ground and the foot of the ladder is mu. What maximum height a man of mass m can climb up the ladder so that the ladder will not skid?
Answer» SOLUTION :The value of limiting friction betweenthe ground and the LADDER is `mu`n Fig. Let the man can climb a maximum height x up the ladder. The point P in figure denotes the position of the man. Here frictional force between the ladder and ground. f = `mu`n where n = contact or NORMAL force on ladder by ground = (Mg +mg) and the contact force on ladder by the wall is n. . At equilibrium NET force on the ladder is zero, i.e., n.- f=0 or, n. = `mu` (M+m) g Also the moment about the point B is zero n.L sin`theta -Mg"" (L)/(2)"cos"theta-mg.xcos theta =0` or, mxcos`theta = mu (M+m) L sin theta - (ML)/(2) cos theta` `:. x = (mu(M+m)Lsintheta-(ML)/(2)costheta)/(mcostheta)`