Hello All and Happidy New Year. I am setting here sipping on some Tangueray Rangpur and thought I would pose a rather fun calc applications problem.
"An amusement park ride has cars which move along a track (I hope I can get it across in my haphazard diagram). The cars are attached to arms that are 20 feet long and are connected to a common point on the axis of rotation(z-axis). The arms swing up and down along the track whose elevation is given by . Assuming all distances are measured in feet, find the length of the track".
I tried drawing the picture, but a I think I got it a little askew. Not too bad, though. It's height is 0 where it touches the x-axis and at the points over the y-axis it is 4 feet in elevation. The circumference of a circle with radius 20 is 125.66, so we know it'll be a wee bit more than that.
If anyone wants to try it, see what you come up with. I attempted it in both polar and spherical and arrived at the same answer using the arc length formulae. So, I am pretty sure I am correct.
I used @ for theta because that's easier in Paint.
Jan 5th 2008, 06:06 PM
It would appear no one seemed interested in tackling this problem. I hope my hackneyed diagram wasn't to blame:)
Anyway, if anyone would like to see what I came up with, let me know.
I thought it was a cool problem.
Jan 6th 2008, 12:24 AM
Well, I think cylindrical coordinates would be the best to work with, parametrizing by the angle . Then , and (using that our ride is at the end of an arm of length 20) . For cylindrical coordinates, , and so, in terms of , . Taking these derivatives and combining, I get that the length is: . I haven't tried to integrate it yet.
Jan 6th 2008, 12:27 AM
In fact, I'm not sure how to do that integral (if it can be done analytically; it looks like it might reduce to an elliptic integral).
Jan 6th 2008, 06:06 AM
Very good, Twisted(Bigsmile). That's the answer I got as well. I also tried it in spherical and got the same so that must be it. You're not sure how to do the integral?. I know exactly how to get it.......... Run it through technology.
I used my TI-92. It gave me 125.985. Since you done the polar in fine fashion, here's the spherical.
The spherical arc length formula is
We can find the derivatives and take the subs:
This gives the same exact result as the polar method.
Kind of a tough problem to set up. Don't even think about doing these integrals by hand. That's what computers are for.