Question

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.

Answer 1

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)`