A worker places an aluminum ladder on a horizontal concrete slab against a vertical wooden wall at 30 degrees from the vertical. The ladder has length L=5m and m-30 kg. The ladder's CM is at a third of the length up. Worker Bob has mass M=90 kg and intends to climb up the ladder. Simultaneously, worker Charlie has mass M=90 kg and operates a rope through a single pulley to hoist a pail of mass m=30 kg. The axle of the pulley is anchored to the ladder at A=0.3 m along the ladder from the top point. Charlie is pulling the rope down with enough force to hoist the pail at uniform velocity. The mass of the pulley and rope are negligible. The friction in the pulley axle is negligible. The ladder is equipped with rubber booties and a rubber top. The kinetic friction coefficients are: rubber on dry concrete 0.9, rubber on dry wood 0.9, aluminum on wet concrete 0.2, aluminum on wet wood 0.2. 30 30° a) Calculate the safety limit in terms of maximal height off the ground that Bob can reach climbing up the ladder before the ladder starts sliding down. Assume dry environment and functional rubber coatings b) Repeated use has deteriorated the rubber coating of the ladder, exposing the aluminum under it. In addition, it rained the previous night and now both the wall and the concrete slab are wet. Calculate the safety limit in terms of maximal height off the ground that Bob can reach climbing up the ladder before the ladder starts sliding down. c) Alarmed by the unsafety of the worn-out ladder in wet conditions, Bob decides to "stabilize" it by suspending a second pail of mass m=30 kg off the ladder at a point at D=0.3m up the ladder length from the bottom. What is the new safety limit of the worn-out ladder for Bob in terms of maximal height off the ground in wet conditions?
A worker places an aluminum ladder on a horizontal concrete slab against a vertical wooden wall at 30 degrees from the vertical. The ladder has length L=5m and m-30 kg. The ladder's CM is at a third of the length up. Worker Bob has mass M=90 kg and intends to climb up the ladder. Simultaneously, worker Charlie has mass M=90 kg and operates a rope through a single pulley to hoist a pail of mass m=30 kg. The axle of the pulley is anchored to the ladder at A=0.3 m along the ladder from the top point. Charlie is pulling the rope down with enough force to hoist the pail at uniform velocity. The mass of the pulley and rope are negligible. The friction in the pulley axle is negligible. The ladder is equipped with rubber booties and a rubber top. The kinetic friction coefficients are: rubber on dry concrete 0.9, rubber on dry wood 0.9, aluminum on wet concrete 0.2, aluminum on wet wood 0.2. 30 30° a) Calculate the safety limit in terms of maximal height off the ground that Bob can reach climbing up the ladder before the ladder starts sliding down. Assume dry environment and functional rubber coatings b) Repeated use has deteriorated the rubber coating of the ladder, exposing the aluminum under it. In addition, it rained the previous night and now both the wall and the concrete slab are wet. Calculate the safety limit in terms of maximal height off the ground that Bob can reach climbing up the ladder before the ladder starts sliding down. c) Alarmed by the unsafety of the worn-out ladder in wet conditions, Bob decides to "stabilize" it by suspending a second pail of mass m=30 kg off the ladder at a point at D=0.3m up the ladder length from the bottom. What is the new safety limit of the worn-out ladder for Bob in terms of maximal height off the ground in wet conditions?
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Transcribed Image Text:Problem: Statics.
A worker places an aluminum ladder on a horizontal concrete slab against a vertical wooden wall at
30 degrees from the vertical. The ladder has length L=5m and m=30 kg. The ladder's CM is at a third
of the length up. Worker Bob has mass M-90 kg and intends to climb up the ladder. Simultaneously,
worker Charlie has mass M-90 kg and operates a rope through a single pulley to hoist a pail of mass
m=30 kg. The axle of the pulley is anchored to the ladder at A=0.3 m along the ladder from the top
point. Charlie is pulling the rope down with enough force to hoist the pail at uniform velocity. The mass
of the pulley and rope are negligible. The friction in the pulley axle is negligible. The ladder is equipped
with rubber booties and a rubber top. The kinetic friction coefficients are: rubber on dry concrete 0.9,
rubber on dry wood 0.9, aluminum on wet concrete 0.2, aluminum on wet wood 0.2.
30
30
a)
Calculate the safety limit in terms of maximal height off the ground that
Bob can reach climbing up the ladder before the ladder starts sliding down.
Assume dry environment and functional rubber coatings
b)
Repeated use has deteriorated the rubber coating of the ladder, exposing
the aluminum under it. In addition, it rained the previous night and now both the
wall and the concrete slab are wet. Calculate the safety limit in terms of maximal
height off the ground that Bob can reach climbing up the ladder before the ladder
starts sliding down.
c) Alarmed by the unsafety of the worn-out ladder in wet conditions, Bob
decides to "stabilize" it by suspending a second pail of mass m=30 kg off the ladder
at a point at D=0.3m up the ladder length from the bottom. What is the new safety
limit of the worn-out ladder for Bob in terms of maximal height off the ground in wet
conditions?
d)
Still disappointed with the ladder performance, Bob takes four old rubber
boots, cuts off the tops and slides the improvised rubber galoshes over the four
ends of the ladder. Assuming the same configuration as in part c but with rubber
booties, what is his new safety limit (with the extra pail, under wet conditions)? The
kinetic friction coefficient of rubber on wet concrete and rubber on wet wood is at
least 0.45
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