Aladder is leaning against a vertical wall. there is negligible friction between the ladder and the wall, and the coefficient of static friction between the ladder and the ground is μs=0.145. the ladder is uniform, with its center of gravity located at its geometric center. at what maximum angle α, relative to the wall, can the ladder lean without slippin

α = 8.25°  

Explanation:

This is a balancing exercise, let's write Newton's equations

Translational equilibrium

X axis

      fr - Fx = 0

     Fx = fr

Axis y

     N - W = 0

     N = W = mg

The force (F) is for the support of the ladder on the wall

Rotational balance

We place our system and reference on the floor, at the point of support, the anti-clockwise turns are positive

       fx y -W x = 0

The friction force has as an expression

       fr = μ N

       fr y = mg x

      μ mg y = mg x

Distance is related by trigonometry

       sin α = x / L

       x = L sin α

       cos α = y / L

       y = L cos α

The angle α is measured from the wall

      μ L cos α = L sin α

      sin α / cos α = μ

      tan α = μ

      α = tan⁻¹ (μ)

      α = tan⁻¹  0.145

      α = 8.25°  

force tends to be low and long diustance masses are high