Chapter 5: Applications of Integration
Section 5.8: Work
With a rope weighing 12 lb/ft, roofing material weighing 50 lbs is hoisted 45 ft to the top of a building. Find the total amount of work done.
Since the force of gravity on the roofing material is constant, the work done raising just this material to the top of the building is the product of force and the distance through which the force moves. Hence, this work is 50 lbs × 45 ft = 2250 ft-lbs.
The force exerted by gravity on the rope decreases during the lift because the amount of rope hanging from the roof varies from 45 to 0 ft. So, with y measured positive from the ground, an infinitesimal segment of the rope of length dy weighs 1/2 dy lbs. This segment must be lifted through a distance 45−y ft. Hence, the work done in lifting just the rope is given by the integral
∫0451245−y ⅆy=110254 ft-lbs
Work done in hoisting just the building material to the roof:
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Work done in hoisting just the rope to the top of the building:
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Sum to get total work done
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+ = 110254
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