Ladder Safety on a Slippery Workshop Floor
A maintenance worker must climb a uniform 4.0 m ladder of mass 14 kg to reach a ceiling light. The ladder leans against a smooth vertical wall at 68° to the horizontal. The floor is concrete and the coefficient of static friction between the ladder feet and the floor is 0.30. The worker has mass 75 kg and climbs slowly. His centre of mass may be taken as lying on the ladder at a distance x from the foot. The workshop supervisor is worried because the ladder is old and the floor has a slight oil film. Before allowing the climb, the supervisor wants to know the maximum height the worker can reach before the feet begin to slip, and whether the ladder is safe at the full 4.0 m length. The wall is assumed frictionless and the ladder is not tied at the top.
1. Calculate the maximum distance x along the ladder at which the worker can stand before slipping occurs.
Read the worked answer
Let (M=14\ \text{kg}), (m=75\ \text{kg}), (L=4.0\ \text{m}) and (\theta=68^\circ). The vertical forces give
[ N=(M+m)g=(89)(9.81)=873\ \text{N} ]
so the maximum static friction is
[ f_{\max}=\mu N=0.30(873)=262\ \text{N} ]
Taking moments about the foot of the ladder eliminates (N) and (f). The wall reaction (P) acts at the top:
[ P L\sin\theta = Mg\frac{L}{2}\cos\theta + mgx\cos\theta ]
At the slip limit (P=f_{\max}). Rearranging:
[ x=\frac{f_{\max}L\tan\theta/g-ML/2}{m} ]
[ x=\frac{(262)(4.0)(2.475)/9.81-(14)(2.0)}{75} ]
[ x=\frac{264-28}{75}=3.15\ \text{m} ]
So the worker can climb about 3.2 m along the ladder before the feet slip, assuming the wall is frictionless and the floor friction is exactly 0.30.
2. The worker says he will simply climb to the top because the ladder has never slipped before. Use your result to judge whether this is acceptable, and state two assumptions that could make the real situation less safe.
Read the worked answer
At the top, (x=4.0\ \text{m}), which is beyond the calculated limit of 3.15 m. The required wall reaction at the top is
[ P=\frac{g\cos\theta}{L\sin\theta}\left(\frac{ML}{2}+mL\right) ]
[ P\approx 325\ \text{N} ]
which exceeds the available friction of 262 N. The ladder would therefore be expected to slip before the worker reaches the top, so the claim that it has never slipped is not a safe basis for action. The worker should not climb above about 3 m without tying the ladder or improving friction.
Two assumptions that make the real situation less safe are: the wall may not be perfectly smooth, so the actual wall reaction direction is uncertain; and the oil film may reduce the effective coefficient of friction below 0.30, especially if the feet are wet or dirty. A third practical factor is that the worker may lean away from the ladder, increasing the moment.