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 μ1 = 0.235 and the coefficient of friction between the ladder and the wall is μ2 = 0.153. Determine the maximum angle with the vertical the ladder can make without falling on the ground.

Answer 1

First consider the forces involved. W: weight N1: normal force from ground N2: normal force from wall F1: normal force from ground F2: normal force from wall The coefficients of friction tell us that F1 < μ1N1 and F2 < μ2N2. The ladder does not move, so the forces in each direction are balanced. N1 + F2 = W N2 = F1 The ladder does not rotate, so the torques are balanced around each end. At the bottom of the ladder the two forces which act at the bottom produce no torque, and likewise at the top. W * L/2 * sin(θ) = N2 * L * cos(θ) + F2 * L * sin(θ) W * L/2 * cos(θ) = N1 * L * cos(θ) + F1 * L * sin(θ) Put all of this together to solve for θ. You should end up with two inequalities, θ < k1 and θ < k2. The smaller restriction is your answer.