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## Homework Statement

A 750N man raises a 200N bucket of water 30m up a helicoidal staircase. If we know that the staircase has a 10 meter radius and does exactly 4 complete turns in the 30m span, what was the total work done by the man against gravity?

Also, in the same scenario, what is the work required if the bucket had a small hole so that 50N leaked out at a constant rate while it was raised up the stairs?

## Homework Equations

[tex]\oint F dr[/tex]

## The Attempt at a Solution

Althought this can be solved with conservation of energy (750N+200N)*30m = 28,500J, I'd like to solve it using a line integral. I've solved the first part, but what's giving me a hard time is the 50N lost at a constant rate. Without the line integral, we can simply figure it out as [tex]\int \frac{50N}{30M}t[/tex] with t from 0 to 30, which represents the work with the variable force.

However, how would this be done with the line integral?

For the first part, I found the equation of the staircase:

x=10cos(t) dx=-10sin(t)

y=10sin(t) dy=10cos(t)

z=(30t)/8*Pi dz= 15/(4pi)

(As we want z=30 @ 8pi (4 complete turns))

I obtain: [tex]\oint 950dx + 950dy +950dz [/tex], t from 0 to 8pi

I can't seem to figure out how to with the variable mass.

Thanks for the help!

Marc