At the bottom edge of a smooth wall, an inclined plane is kept at an angle of `45^@`. A uniform ladder of length `l` and mass `M` rests on the inclined plane against the wall such that it is perpendicular to the incline.

a. If the plane is also smooth, which way will the ladder slide?
b. What is the minimum coefficient of friction necessary so that the ladder does not slip on the incline.

Answer

A, B, C

Explanation

(a)`N_(2)` and mg pass through `G.N_(1)` has has clockwise moment about `G` , so the ladder has a tendency to slip by rotating clockwise and the force of friction
`(f)` at `B` is then up the plane
(b) `sum _(A) = 0`
`:. Fl = mg((1)/(2) sin 45^(@)))` ...(i)
`sum F_(v) = 0`
`:. mg = N_(2) cos 45^(@) + f sin 45^(@)` ...(ii)
From Eqs.. (i) and (ii), `N_(2) = (3)/(2 sqrt2) mg`
and `f = (mg)/((2 sqrt2)`
or `mu_(min) = (f)/(N_(2))`
`= (1)/(3)`