50 m rope with 8 millimeters in diameter is dangling from an edge. density of rope =40 g/m. how much work to pull it up to edge?
// I've seen different variations of this problem, but I am unsure of how to setup as following along with the "Pumping liquid out of tank" classic problem, I don't see the need for p * g in the work formula here.
// The work formula I am referring to and am thinking I would use: Work = [Integral from b to a] (p*g *cross section area *lifting distance)dy
// I also notice that there is a diameter mentioned here, which I am not sure we would need.
Would someone please show me how to setup this integral properly?
Any guidance would be greatly appreciated.
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$\begingroup$The simplest method is to treat it as moving the total mass of the rope a height from the cg of the rope to the edge. That is, mgh where $m = 50\cdot 40/1000$ kg, $g = 9.81$ and $h = 25$ m.
If you want to go the calculus route, set it up as a series of infinitely thin discs being raised different heights:
Mass of each disc is $\pi\cdot .004^2\cdot \frac{.04}{\pi\cdot .004^2}dx = .04 \ dx$
Distance each disc travels is $x$ from $0$ to $50$.
$$W = \int_0^{50} mgx \ dx$$ $$W = \int_0^{50} .04\cdot 9.81x \ dx$$
$W = .02\cdot 9.81(x)^2$
$W = .02\cdot 9.81(50)^2$
$W = 490.5$ Joules
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