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How much work does the fish do?

  1. Mar 24, 2004 #1
    Let's say there's a creek.

    Down in the creek, there is a current.

    The vector field that describes this current is

    Cur[x y z] = [(ðy)^x]i + [(y^4)+(xyz)]j + (2z + e^z)k

    Nota bene:
    ð = pi

    The force is in Newtons.

    X, Y, and Z are the spatial dimensions in meters, whose origin is a piece of bait in this case.
    You see, there's also this guy who's fishin' in the creek. His bait's down there, situated on the origin.

    A fish sees it, and circles around it one complete time. The fish is unsure during this period, and maintains a distance of one meter.

    This motion takes exactly seconds for the fish.
    So, the motion can be described as a vector-valued function of t, time (sec.)

    Fis(t) = [cos(t)]i + [sin(t)]j + [0]k

    How much work is done by the fish?

    Ok... so I made this problem up... that's why it's so weird. :)

    I need help setting it up. I know I need to use a line integral.

    The upper limit, t, in seconds, will be , while the lower will obviously be 0.

    So, first off, I need to find the integrand, which is the dot product of Cur[Fis(t)] and Fis'(t).

    To begin

    Cur[Fis(t)] = [(ð*sin(t))^cos(t)]i + [sin(t)^4]j + 1k

    But here's some trouble for me... I'm not certain on how to differentiate Fis(t).

    Tell me, O somebody-who-is-doubtlessly-wiser-than-I, would

    Fis'(t) = [-sin(t)]i + [cos(t)]j + 0k ?

    If that is so, I'll continue to find the dot product, and then begin the actual integration.
    Last edited: Mar 25, 2004
  2. jcsd
  3. Mar 25, 2004 #2
    Since no one seems willing to help right now, I'll continue the calculation, whether or not it's right so far.

    Cur[Fis(t)] = [(ðsin(t))^(cos(t))]i + [sin(t)^4]j + 1k

    Fis'(t) = [-sin(t)]i + [cos(t)]j + 0k


    Cur[Fis(t)] * Fis'(t) = -sin(t)(ðsin(t))^(cos(t)) + (cos(t))(sin(t)^4)


    ∫ -sin(t)(ðsin(t))^(cos(t)) + cos(t)sin(t)^4 dt

    from 0 to 2ð

    As Shmoe, of my native forum, informed me, the force vector function along the curve Fis has an imaginary i component part of the time.

    The 'force * velocity' function(t) isn't continuous, due to that.

    Well... it's continuous on the domain (0, pi).

    With the help of my computer, I can integrate on that domain and determine half of the work the fish does, I guess.
    It says approximately -2.1927, which is correct as far as I can tell.

    Half-Ass Conclusion:
    During the first half of his journey, the fish is helped by the current. He is helped 2.1927 Newton-meters.

    Now... is there any way to get the rest of the work?
    Last edited: Mar 25, 2004
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