A ladder is leaning against a vertical wall, and both ends of the ladder are at

Question

A ladder is leaning against a vertical wall, and both ends of the ladder are at the point of slipping. The coefficient of friction between the ladder and the horizontal surface is 1 = 0.205 and the coefficient of friction between the ladder and the wall is 2 = 0.253. Determine the maximum angle with the vertical the ladder can make without falling on the ground.

A ladder is leaning against a vertical wall, and both ends of the ladder are at the point of slipping. The coefficient of friction between the ladder and the horizontal surface is -0.205 and the coefficient of friction between the ladder and the wall is : 0.253. Determine the maximum angle with the vertical the ladder can make without faling on the ground. ladder can make without falling on the ground. Number

Explanation / Answer

Let the normal force of the wall on the ladder be N2 and the normal force of the ground on the ladder be N1.

Horizontal forces:

N2 = (u1)(N1) [1]

Vertical forces:

N1 + (u2)(N2) = mg [2]

Substitute [2] into [1]:

N2 = (u1)[mg - (u2)(N2)]

N2 = (u1)mg/[1 + (u1)(u2)] [3]

Torques about the point where the ladder meets the ground:

mg(L/2)sin(theta) = (N2)(L)cos(theta) + (u2)(N2)(L)sin(theta)

(1/2)mg = (N2)cot(theta) + (u2)(N2)

(1/2)mg = {(u1)mg/[1 + (u1)(u2)]}cot(theta) + (u2)(u1)mg/[1 + (u1)(u2)]

1/2 = {(u1)/[1 + (u1)(u2)]}cot(theta) + (u2)(u1)/[1 + (u1)(u2)]

[1 + (u1)(u2) - 2(u2)(u1)]/{2[1 + (u1)(u2)]} = {(u1)/[1 + (u1)(u2)]}cot(theta)

tan(theta) = {(u1) / [1 + (u1)(u2)]} {2[1 + (u1)(u2)]} / [1 - (u1)(u2)]

tan(theta) = 2(u1) / [1 - (u1)(u2)]

tan(theta) = 2(0.205) / [1 - (0.205)(0.253)]

theta = 77 degree

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