Solve Integrals and Work: 500 lb Bucket of Cement

[Jacobpm64]

Homework Statement

A worker on a scaffolding 75 ft above the ground needs to lift a 500 lb bucket of cement from the ground to a point 30 ft above the ground by pulling on a rope weighing 0.5 lb/ft. How much work is required?

Homework Equations

W = F * D

The Attempt at a Solution

I know that I would have to separate the problem into two parts. The part with the bucket is constant, and the part with the chain is not constant because certain parts of the chain move up different distances than other parts. I'm not sure how to set everything up though.

 

[Dick]

First express F as a function of distance above the ground then integrate it. Hint: F is linear and F(0 feet)=515 lbs, F(30 feet)=500 lbs.

 

[HallsofIvy]

Another way to do it: imagine a small section of rope of length, say $\Delta x$ at height x above the ground- small enough so the we can approximate the distance each point on that section has to be lifted by 30-x. It weighs [/itex]0.5\Delta x[/itex] pounds and must be lifted 30- x feet: the work done in lifting that section of rope is $0.5(30- x)\Delta x$ feet. Summing over all "small sections" for x= 0 to 30, gives a Riemann sum which can be converted into an integral.