Ladders and rods against a wall
Answers are for parents and teachers.
1
A ladder of weight 400 N (5 m long, foot 3 m from the smooth wall, top 4 m up) carries a man of weight 200 N. The ground has μ = 2/5. How far along the ladder can he climb?
N = 600 N and F = S = 0.4 × 600 = 240 N. 0.75(200 + 200f) = 240, so f = 0.6 and s = 3 m.
2
A ladder of weight 120 N (same shape) stands on smooth ground, held by a rope at the foot that can take 315 N. A man of weight 480 N climbs it. How far along the ladder can he climb?
0.75(60 + 480f) = 315, so f = 0.75 and s = 3.75 m.
3
A rod of weight 120 N has tan θ = 4 and μ = 0.25. What is the greatest extra horizontal force pointing away from the wall that it can take?
S = 120/8 = 15 N. The most friction is 0.25 × 120 = 30 N, so X = 30 − 15 = 15 N.
4
A rod at θ = 30° to the ground rests against a smooth wall. Find the least μ.
tan 30° = 1/√3, so μ = √3/2 ≈ 0.866.
5
Can a rod rest against a smooth wall on smooth ground without a rope? Explain.
No: the wall's push S is horizontal and nothing balances it, so the sum of the horizontal forces is not zero.