Alternate Approach to Rope Problems
A weight hangs freely from a tall building by a rope of given weight and length. How much work is done
in pulling D feet of rope to the top of the building.
Let T be the combined weight of the rope and the attachment.
Let
 g be the weight density of just the rope.
D
Then the work is
 (T   g y ) dy
0
Explanation: If y feet of rope have been lifted then only T   g y lbs remain to be lifted.
We integrate this force function from 0 to D, the total amount of length lifted.
Example: A 100 lb weight hangs freely from a tall building by 30 feet of rope. The rope alone weighs 60
lbs. Find the work done in pulling 20 feet of rope to the top of the building.
Solution: T =160 lbs
 g = 60/30=2 lbs/ft D = 20.
20
Work =
 (160  2 y ) dy  2800 ft  lbs
0