Ladder slip: worker position and wall angle
A uniform ladder of length 6.0 m and mass 20 kg rests against a smooth vertical wall and on a rough horizontal floor. The foot of the ladder is 2.63 m from the wall, so the ladder makes an angle of 64° with the horizontal. The coefficient of static friction between the ladder foot and the floor is 0.32. A worker of mass 75 kg climbs slowly up the ladder to inspect a gutter. The wall is smooth enough to be treated as frictionless, and the floor is horizontal. The worker’s tools add negligible mass, but the worker’s position along the ladder matters because it changes the overturning moment. The ladder is lightweight compared with the worker but not negligible. The worker needs to reach a point near the top, and the site supervisor is worried that the ladder may slip before the worker gets there. The supervisor suggests moving the foot of the ladder closer to the wall, to 2.0 m from the wall, keeping the ladder length unchanged. You should model the worker as a point mass and use g = 9.8 m/s².
1. Find the greatest distance the worker can climb along the ladder before it slips, for the original 64° position.
Read the worked answer
Take moments about the ladder foot. Let R be the wall normal reaction and N the floor normal reaction. Vertically,
N = (20 + 75)g = 95g.
At limiting friction,
F = μN = 0.32 × 95g = 30.4g.
If the worker is distance x along the ladder, moments about the foot give
R × 6 sin64° = 20g × 3 cos64° + 75g × x cos64°.
Thus R = g cot64°(60 + 75x)/6. Setting R = 30.4g gives
60 + 75x = 30.4 × 6 / cot64° = 374,
so x = (374 − 60)/75 = 4.19 m. The worker can climb about 4.2 m along the ladder before slipping.
2. Investigate the supervisor’s suggestion of moving the foot to 2.0 m from the wall. Show whether the worker can then reach the top without slipping.
Read the worked answer
With the foot 2.0 m from the wall,
cosθ = 2.0/6.0 = 1/3, so θ = 70.5° and cotθ = 0.354.
At the top, x = 6.0, so
R = g × 0.354 × (60 + 450)/6 = 30.1g.
The maximum friction is still 30.4g, so R < Fmax and the ladder does not slip even at the top. The steeper angle reduces the horizontal moment arms enough to compensate for the worker being higher up the ladder.
3. State two modelling assumptions and explain whether each makes the no-slip prediction safer or less safe.
Read the worked answer
Two assumptions are that the worker is a point mass and that the wall is perfectly smooth. The point-mass assumption ignores the worker’s width and the fact that the worker’s centre of mass may not lie exactly on the ladder line; this can make the prediction less safe. The smooth-wall assumption removes any upward friction from the wall, which is conservative because it makes slipping more likely, so it is safer for design purposes. A further issue is that limiting friction is used; in practice μ may be lower if the floor is dusty or wet, so the calculated maximum x should not be treated as a safe working limit without a margin.