Math challenge for you guys (trig, but may be possible with calculus, too)

[notfred]

Ok, so here's the problem: You have two gears of differing sizes; thier radii are called r1 and r2. You can assume that r1 is the smaller of the two radii.

These gears are circular, and thier center points are seperated by a distance L.

Given r1, r2, and L find the shortest length of chain that will fit around the the two gears.

Imagine a bike chain - you know the size of both the front and rear gear and the distance between them. You want to find the length of chain to fit such a bicycle.


Ok, here's my answer. If you want to try to find an answer yourself, don't read this until later. All my measurements are done in radians.

? = arcsin((r2 - r1)/L)
Length of chain = 2(L cos ?) + r1(p - 2?) + r2(p + 2?)

I have an example set of numbers to use, if you'd like to compare some answers. If you get an answer different from mine, please plug in these values and see if you get the same answer I did.

r1: 1.353
r2: 2.706
L: 17

When using these values, I found a required chain length of 46.859.
Please let me know if you get the same or a different answer.

Thanks

 

[Enfer Singe]

14.923p

 

[ifoundthetao]

hm.. my eqtn was diff, .. but let me draw a picture of it so we are on the same page.. but my eqtn is pi*r2+pi*r1+2sqrt(L^2+(r2-r1)^2)


Here is my drawing of it

 

[RaynorWolfcastle]

I don't feel like solving it, since it looks like it could be a mess to do but conceptually it's simple enough. You'd have to find two tangent lines to both circles (use derivatives). From there you can find the four intercept points (two on each circle). Once you have this, just measure the length between the the two intercept points on each circle and measure the length of arc that the chain "wraps" around each circle.

Should be straightforward enough to find a solution in terms of R1, R2, and L if you're willing to work through the math.

 

[notfred]

Any answers? I know you guys have solved harder matth problems before

 

[ifoundthetao]

does my stuff check out?

i got the exact same thing with my eqtn so you're all good

46.859 but technically, with those answers it should be 47, because of the sig digits.. the length should be 17.000 .. so i dont know w

 

[QuitBanningMe]

Bike? I google it and subtract the 1(derailer)
http://www.parktool.com/images/repair_help/rigorous.jpg
but I'm stupid

 

[ifoundthetao]

that was fun ... do you have any more? i havent got to any application trig in a while.. got any identities that need verification?

 

[notfred]

Originally posted by: ifoundthetao
does my stuff check out?

i got the exact same thing with my eqtn so you're all good

46.859 but technically, with those answers it should be 47, because of the sig digits.. the length should be 17.000 .. so i dont know w

assume this is a pure math question and not a physics question. The 17 is exact