A uniform ladder of length `L` and mass `m_(1)` rests against a frictionless wall. The ladder makes an angle `theta` with the horizontal. (a) Find the horizontal and vertical forces the ground exerts on the base of the ladder when a firefighter of mass `m_(2)` is a distance `x` from the bottom. (b) If the ladder is just on the verge of slipping when the firefighter is a distance d from the bottom, what is the coefficient of static friction between ladder and ground?

Explanation

`sumF_(x)=-n_(w)=0` …….i
`sumF_(Y)=n_(g)-m_(1)g-m_(2)g=0`……….ii
`sum tau_(A)=i-m_(1)g(L/2)costheta=m_(2)gxcostheta=n_(2)Lsintheta=0`

From the torque equation `n_(2)=[1/2m_(1)g+(x/L)m_(2)g]cottheta`
Then from eqn i
`f=n_(w)=[1/2m_(g)+(x/L)m_(2)g]cottheta`
from eqn ii
`mu=(f_(x=d))/(n_(g))=(((m_(1))/2+(m_(2)d)/L)cottheta)/(m_(1)+m_(2))`