crop circles

[hugme]

driving through middle georgia you pass a lot of farms. there are interesting sprinkler system on these farms...



the sprinkles on these farms are interesting... they work like a compass (the one used to draw a circle, not find north) there is a stationary point and then a large section that rotates around... this is a mathematical playground!!! here is what I decided to think about...

the large arm that turns around the circle is split into sections.. each section has a wheel



please excuse my lack of drawing skills

so each wheel out makes a larger and larger circle... the arm moves at the same rate so as you go out the arm each wheel moves faster and faster...

the question I posed in my head while I was driving was.. what is the quickest way to be able to figure out how fast each wheel needs to move in miles per hour... this was a pretty simple one... all you need to do is assume that the arm takes 1 hour to turn around the circle... then all you need to figure out the speed is the distance around the circle in miles (it's going to be a fraction)... do this for each circle... now.. take the time you want it to take for the arm to make 1 revelution around the circle and multiply that times the inverse.... so t being time and C being the circumfrance of the circle you come up with:

      C
     ----   =  speed
       t

now... chances are that it's not the circumfrance that you know... it's the length of each section... so you need this formula by the radius of the circle, radius being 'r'... which is:

     2r(pie)
     --------   = speed
        t

each sprinkler section is going to be the same length.... we will call this length 'x'. now number the sections so that the wheel closest to the center is 1... this value we will call 'a'.... so now our fomula has changed and is very managable.. so now the radius of the circle is the length of the wheel times the wheel number it is from the center.... so our finished working formula is:

      2ax(pie)
     -----------  = speed of the wheel
           t

where the variable 'x' is the length of the section
'a' is the wheel number starting at center and working out
and 't' is the time you want it to take for the arm to get all the way around

this sounds like a pretty simple equation... but this was my entire thought process... I was coming up with it just for fun while driving a 25 foot truck down a 4 lane highway...

Comments

[madmexican.livejournal.com]

it is still amazing to me how smart you are to figure that out while drive....althought it is obvious you have never seen this particular type of mobile hydro-agricultural mechanisims work...they move so much more slowly than you guessed......most usually make two cycles a day, and sometimes not even that.