Calculating Work to Lift Kite String from Ground: A Puzzle

[sinas]

Hi - Here's the entire problem to avoid confusion ><

"A kite is flying at a height of 500 ft and at a horizontal distance of 100 ft from the stringholder on the ground. The kite string weighs 1/16 oz/ft and is hanging in the shape of the parabola y=(x^2)/20 that joins the stringholder at (0,0) to the kite at (100,500). Calculate the work (in foot pounds) done in lifting the kite string from the ground to its present position."

I've been messing with arc length of the parabola because I think there might be a way to solve it with that, but I really don't know... Can someone give me a hint??

 

[vsage]

I would like to point out first and foremost that that question is flawed because hanging strings follow hyperbolic not parabolic functions. The potential energy gained by moving the kite from the ground to its current position is equal to the work done to move it there. Consider how much work it takes to move an infinitely small portion of that kite string up a height h where h varies according to the parabolic function h = (x^2)/20. Sum these together and you have the basics of an integral. Be careful not to integrate with respect to x but with respect to the arc-length though, or express the arclength in terms of x.

 

[sinas]

Ok I'm playing with that, but tell me something - if I were to calculate the work moving the entire string from 0 to the y-coordinate of the centroid of the curve, that wouldn't work, right? (no pun intended)

 

[vsage]

If I understand what you're trying to do right yes that would work. The centroid is the center of mass after all since the mass is distributed evenly along the string.