Question

If the coefficient of static friction is 0.50, and the same ladder makes a 49° angle with respect to the horizontal, how far along the length of the ladder can a 64.0 kg person climb before the ladder begins to slip? The ladder is 8.6 m in length and has a mass of 21 kg.

Answer 1

Weight of Ladder = 21 * 9.8 = 205.8 N
Weight of the person = 64 *9.8 = 627.2 N
Length of Ladder, L = 8.6 m
Distant of person , d = ?
θ = 49 °
u = 0.5

At the bottom of the ladder, the vertical normal force on the ladder (upwards) is NV.
Resolving vertically:
NV = 205.8 + 627.2 = 833 N

At the top of the ladder the horizontal normal force on the ladder is NH.
NH = μ * NV
NH = 0.5 * 833
NH = 416.5 N

Taking moment about the bottom of the ladder:

416.5 * L * sin(49) = 205.8 * (L/2) * cos(49) + 627.2 * d* cos(49)
416.5 * 8.6 * sin(49) = 205.8 * (8.6/2) * cos(49) + 627.2 * d* cos(49)
Solving for d,
d = 5.16 m