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.
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