A painter wishes to know whether or not he can safely stand on a ladder. The lad
ID: 1366876 • Letter: A
Question
A painter wishes to know whether or not he can safely stand on a ladder. The ladder has a mass M1 = 8.0 kg which is uniformly distributed throughout its length L = 7.9 m. The ladder is propped up at an angle ? = 66?. The coefficient of static friction between the ground and the ladder is ? = 0.286, and the wall against which the ladder is resting is frictionless. Calculate the maximum mass of the painter for which the ladder will remain stable when he climbs a distance d = 5.2 m up the ladder.
If the painter does not stand on the ladder, what is the minimum angle theta for which the ladder will remain stable?
**please include as much explanation as possible!**
Explanation / Answer
The ladder is propped up at an angle = 66 degrees.
Hence, the angle at the top of the ladder and wall = top = 24 degrees.
Mass of the ladder = mL = 8 kgs.
The coefficient of static friction between the ground and the ladder = µs = 0.286
Total length of Ladder = L = 7.9 m
The length of ladder considered for the mass of the person who could stand stable L = 5.2m
This is a static equilibrium problem.
We need f = 0 and = 0
Force equations :
X – Component : FN1 – Ffr = 0
Y-Component : -mg - mLg + FN2 = 0.
Ffr = µs FN2
Torque equation (Around the bottom of the ladder )
FN1 L Sin – M g L Sin top – mL g (L/2) Sin top = 0
Hence,
Mass of the person M = mL x [(2.6 Sin top - µs Sin) / (µs Sin - Sin top)]
8 x [(2.6 Sin 24 – 0.286 x- 0.03) / (0.286 x -0.03 – Sin 24)]
8 x [(0.5 x-0.91 +0.009 ) / (-0.009 – 0.91]
8 x [(-2.366 + 0.009)/(-0.009 -0.91)]
8 x (2.227 / 0.919) = 19.38 kgs
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