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work in cyclic motion

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  • work in cyclic motion

    Hello friends:

    I am sorry to be entering into this so late, and I must confess I
    have no knowledge of what started this. However, I have seen several
    well-intentioned postings the last few days which have reflected some
    of the confusion which commonly arises from considerations of cyclic
    motion. It's very easy to get confused with this, but I hope to try
    to clarify some of the points that seem to confuse people most
    frequently. In so doing I hope to avoid offending anybody; if I
    point out a (perceived?) error please don't think I am singling
    somebody out.

    The most common misconception that arises is that if the final
    position is the same as the initial position, no work has been done.
    This stems from an (sorry, every word I think of for here sounds
    unfairly critical) interpretation of the definition of work, which
    most people take to be W = F delta X. However, a more illustrative
    definition in my opinion is that W = the integral of FdX. This
    seemingly simplistic (even semantic) distinction makes a profound

    Rather than insult your intelligence with a review of this (because I
    can hear the lightbulbs clicking on all over the world I know it
    isn't necessary) I'll instead risk the wrath of a few courageous
    folks who unfortunately overlooked this in their attempts to explain.
    Friends, this isn't to draw attention to your errors, but is done
    because those errors are the ones we ALL are inclined to make, and
    because the examples will clarify the necessity of integration over
    the path rather than mere attention to the endpoints.

    If a hand presses against a spring and compresses it some distance,
    then the hand performs work on the spring. If the hand relaxes and
    the spring presses the hand back to the original position, then the
    spring perofms work on the hand. Now that they are returned to the
    original position, the temptation is to say that the net wok done on
    the spring is zero. But it isn't; in fact the return to the original
    position in no way reduces the work done on the spring. If you wish
    to see this mathematically you can graph it; for a constant rate
    spring you will get a triangle whose area is a measure of the work
    done. An "easier" way is to make the situation initially more
    complex. Imagine that we had a thingamajig (all inclusive term for
    mechanical translation device - e.g. a gear) to transform that spring
    compression (the motion thereof) into the movement of yet another
    piece. When the spring is allowed to return to its initial position,
    that other movement would still have taken place, right? And in
    fact, this principle is, I am sure you will agree, rather closely
    related to the principle behind the Carnot cycle (and the internal
    combustion engine).

    Back to our example. Upon return to the starting point, the net work
    done on the spring is the work done in compressing it. The net work
    done on the hand by the spring is the work done in moving the hand
    back to the starting position. These do not "sum" to zero.

    The swimmer will be the second example. If we cling to the idea that
    work is defined by F delta X, then one rotation of an arm would seem
    to yield zero. We can't do that. In fact, we can't directly apply
    the definition I gave, as it is "phrased" for direct application to
    rectilinear motion. For rotational motion it becomes W = integral of
    tau d theta, where tau is the torque and theta the angular
    displacement. And this will give the type of results that are
    desired; if we rotate the arms twice with a constant stroke we get
    twice the work, not a return to zero. Now admittedly the tethered
    swimmer problem is fairly complex (how to determine tau?) but this is
    the approach that will give the mechanical work done by the swimmer
    on the water.

    Thank you for your attention. I hope I haven't bored or offended to
    many of you; if any clarification is desired I will do what I can.