A 5 m long, 23.2 kg ladder rests up against a frictionless wall such that the angle between the ladder and the floor is 53∘ .

Question

If the coefficient of static friction between the ladder and the floor is 0.28, what is the furthest distance, d, a 89.3 kg person can climb up the ladder, in m?

Answer 1

Not giving a specific answer in case this is a graded problem.

However, the approach is pretty standard.

1. Set up a free body diagram with a ladder leaning against a wall with angle θ from the floor. There are five forces acting at different points on the ladder
2. F_Nx - the normal force of the wall acting horizontally away from the wall at the top of the ladder
3. F_W = Mg - the weight of the man on the ladder acting downward at some unknown distance d up the ladder
4. F_w = mg - the weight of the ladder acting downward at a point a distance L/2 up the ladder (length L)
5. F_Ny - the normal force of the floor acting vertically upward from the floor on the foot of the ladder
6. F_f = μ F_Ny - the force of friction acting horizontally toward the wall on the foot of the ladder to keep it from sliding out
7. Set up equations balancing the horizontal forces F_Nx and F_f, the vertical forces F_W, F_w, and F_Ny and finally the torques about a somewhat arbitrary pivot at the foot of the ladder. These torques are F_NxL sinθ clockwise, and F_Wdcosθ, and F_wLcosθ/2.counterclockwise.
8. Use substitution and algebra to solve for the unknown d in the last equation in terms of known values