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Ladders and rods against a wall

الأسبوع 10 · اتزان سلم على مستويين متعامدين أحدهما خشن أو أملس، وساق على أرض خشنة وحائط أملس · 3 دروس
2

A ladder on smooth ground against a smooth wall held by a horizontal rope: the tension grows as someone climbs, and the greatest safe distance

Applied Mathematics Sec 3 Term 1, Unit 2, Lesson 2, Part 3
GoalFind the tension of the rope that holds a ladder on smooth ground, and how far a person may climb before it breaks.
New words
horizontal rope · a rope at the foot tied to the wall; it takes the place of frictionsmooth ground · the ground pushes only straight up
Explanation

On smooth ground there is no friction. A horizontal rope tied to the foot A and to the wall holds the ladder. The forces are the same as before, with the rope tension T in place of the friction F (the wall is on the right; up, right and anticlockwise are positive). Vertical forces: N = W + P. Horizontal forces: T = S. Moments about A give, for the 3-4-5 ladder (5 m long, foot 3 m from the wall), T = S = 0.75(W/2 + Pf). As the person climbs, f grows, so T grows. If the rope can bear at most Tmax, put T = Tmax and solve for f: the person cannot climb higher than that.

θNTSWPABQ
Worked examples
θNTS120 N400 NABQ
A ladder 5 m long and of weight 120 N has its foot 3 m from a smooth wall, on smooth ground, held by a horizontal rope at A. A man of weight 400 N climbs. The rope can take 240 N. How far can he climb? T = 0.75(60 + 400f) = 240 gives 60 + 400f = 320, so f = 0.65 and s = 3.25 m. Table of T for different f: f = 0 gives 45 N, 0.25 gives 120 N, 0.5 gives 195 N, 0.75 gives 270 N and 1 gives 345 N.
The same man is at the top of the ladder (f = 1). Find T and N. T = 0.75(60 + 400) = 345 N and N = 120 + 400 = 520 N.
Class activity

Hold a ruler leaning on a wall with your finger under its foot end resting on a slippery tray. Why does your finger need to pull towards the wall? Predict the direction before testing.

Practice
1
A ladder of weight 80 N (5 m long, foot 3 m from the wall) stands on smooth ground with a rope at the foot, and a man of weight 240 N stands at its middle. Find T.
2
A ladder of weight 100 N carries a man of weight 300 N (same shape, smooth ground, rope at the foot). The rope can take 150 N. How far can he climb?
3
A ladder of weight 80 N (same shape, smooth ground, rope at the foot) has a man of weight 240 N at its top. Find N, S and T.
4
Why does the tension rise as the man climbs?
5
A uniform ladder of weight 300 N has its top 6 m up the wall and its foot 3 m out, so tan θ = 2. The ground is smooth, and there is no man on it. Find T.