Expert: Paul Klarreich - 7/19/2006

Question
Consider 15 ft chain hanging from ceiling weighing 3lb./ft.

a.How much work is needed to take the bottom of the chain and raise it to the 15 ft level leaving the chain doubled and hanging vertically?


i got the integration of 22.5 dy from 0 to 7.5


am i on the right path is there a general eq.
b. how much work to raise to 12 ft level?  

Answer
Roger Asks in Category Calculus ...
 
Question:  Consider 15 ft chain hanging from ceiling weighing 3 lb./ft.

a.How much work is needed to take the bottom of the chain and raise it to the 15 ft level leaving the chain doubled and hanging vertically?

i got the integration of 22.5 dy from 0 to 7.5

am i on the right path is there a general eq.
b. how much work to raise to 12 ft level?
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[VIEW IN A FIXED FONT, LIKE COURIER.]
I think the approach could go this way:

At any time, part of the chain is supported by the ceiling hook (has to be attached to something) and part is being lifted.  We need to introduce a variable:

Let  h = the height of the bottom of the lifted part.  (Assuming the bottom of the chain originally reached the floor.)

Ceil
--+--
 |
 |
 |
 |
 |
 |
 | |<- equals 2h
 | |
 | |<- centroid of lifted part at height 3h/2
 | |
 0-0 <-- equals h
 .
 .
 .
 .
------
Floor

So at any time, the weight of the lifted part is equal to the length of the lifted part times your 3 lb/ft.

Now  dW = F ds, where s is the increase in height of the centroid.  And F is the weight, 3h.

And the value of h starts at zero and finishes at  7.5

So what happens when we lift by a distance of h, that is, the 'already lifted' part moves up to a height of h + dh?  The centroid of the lifted part actually increases in height by 2h.  Why is it 2h?  Because we take h away from the stationary part and put it below the moving part.  (It took me a while to realize this, and that delayed the answer somewhat.)

Ceil
--+--
 |
 |
 |
 |
 |
 |
 |
 |
 | |<- equals 2h
 | |
 | |<- centroid of lifted part at height 3h/2 + 2dh
 | |
 | | < another dh
 0-0 <-- equals h + dh  (one typing line is one dh.)
 .
 .
 .
 .
 .
------
Floor



So perhaps we want the:

{h=7.5
|       3h(2dh) = 6dh
}h=0

= 3h^2 from 0 to 7.5 =

          3(225)   675
3(7.5)^2 = ------ = ----
            4       4

..........................

Now I wonder if it could be done this way:

At the start, the bottom half of the chain weighs 7.5 * 3 = 22.5 lb.

Its centroid has h = 3.75 feet.

At the finish that bottom half has moved up so its centroid is now at

h = 7.5 + 3.75.

So it has increased in height by 7.5 feet.  The work done is F*d =
           15 225   3(225)   675
7.5(22.5) = -- --- = ------ = ---
            2 10      4       4

About your part b, I think you just do the same stuff, with the upper limit equal to 6 instead of 7.5.