A ladder is leaning against a vertical wall. The coefficient of friction between
ID: 2254266 • Letter: A
Question
A ladder is leaning against a vertical wall. The coefficient of friction between the ladder and the horizontal surface is ?1 = 0.125 and the coefficient of friction between the ladder and the wall is ?2 = 0.103. Determine the maximum angle with the vertical the ladder can make without falling on the ground.
alpha =______________degrees
Explanation / Answer
When the ladder is in equilibrium
Mg = N1 + u2 N2 ---------- (1)
Normal force at the top = frictional force at the bottom
N2 = u1 N1. ------------ (2)
Equation 1 is altered as
Mg = N1 + u1 u2 N1
Mg = N1 (1+ u1 u2) ---------------- (3)
If L is the length of the ladder and 'theta' is the angle of inclination from the vertical,
Taking moment about the bottom point of the ladder,
Mg*(L/2) sin(theta) = N2*L cos(theta) + u2 N2* L sin(theta)
Canceling L through out
Mg sin(theta) = 2 N2 (cos(theta) + u2 sin(theta))
Dividing by cos(theta)
Mg tan(theta) = 2 N2 (1+ u2 tan(theta))
From 3 & 2
N1*(1+ u2 u1) tan(theta) = 2 u1 N1 (1+ u2 tan(theta))
Canceling N1 through out
(1+ u2u1) tan(theta) = 2 u1 (1+ u2 tan(theta))
tan(theta) {1+ u2 u1-2 u1 u2} =2 u1
tan(theta) {1- u1 u2} =2 u1
tan(theta) = 2 u1/ {1- u1 u2}
tan(theta) = 2*0.125/{1-0.125*0.103}
theta = 14.212 degrees
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