A uniform ladder of weight `W` and length `L` is resting between rough floor and smooth vertical wall. The ladder makes an angle `53^(@)` with horizontal. A man of weight `3W` climbs the ladder and stays at distance `L//3` as shown.
(a) Find the normal and friction force on the ladder at its base.
(b) Find the minimum coefficient of the static friction needed to prevent slipping at the base.

Explanation

The forces acting on the ladder are (i) weight of the man `3W`, (ii) weight of the ladder `W`, (iii) normal force at base `N_(1)`, (`iv`) friction at base `f_(s)` and (`v`) normal force at wall.
Since the rod is in equilibrium
`SigmaF_(x)=0implies f_(s)=N_(2)` ......(i)
`SigmaF_(y)=0impliesN_(1=3W+W=4W)` ........(ii)
`Sigma tau_(A)=oimpliesN_(2)xxLsin53^(@)`
`=3Wxx(L)/(3)cos53^(@)+Wxx(L)/(2)cos53^(@)`
`N_(2)xx4=3W+(3W)/(2)=(9W)/(2)`
`N_(2)=(9W)/(8)`
(a) Normal force at base `N_(1)=4W`
Friction at base `f_(s)=N_(2)=(9W)/(8)`
(b) `f_(s)le(f_(s))_(max)(=muN_(1))`
`(9W)/(8)lemuxx4W implies muge(9)/(32) implies mu_(min)=(9)/(32)`