Summary: Spot the Fallacy [longish]

[Mike Taylor <mike@indexdata.com>]

I am a bit hesitant to post this summary, because it doesn't really
represent a convincing consensus.  I fear that it will spark a Pure
Mechanics thread which M&M may consider off-topic.  Nevertheless, I
did promise to summarise responses to the list, so there they are.

You'll remember I exhibited the following, surely faulty, bit of
reasoning:

        If a sauropod has no ligament support for its neck, it
        has to hold it up using muscle-power entirely.  How
        much energy does this use?  Well, energy is the
        capacity to do work, so doing work is the thing that
        uses up energy.  But work done is force applied times
        distance moved; and since all the sauropod is doing is
        holding the neck steady (not raising it), the distance
        moved is zero whatever the force involved -- so the
        work done is zero, and the energy used is also zero!

>From one or two of the responses, it seems I didn't make it clear that
this steaming pile of nonsense is all my own work.  I wasn't citing a
published work -- if I had been, I'd have given a reference.

So here are the responses, some of which are ... shall we say,
amusing.  Names withheld to protect both the guilty and the innocent.
(If I just withheld the names of these who wrote the replies I
consider guilty, then they'd who who they were!)

The convincing replies all seemed to be variations on the following:

        It is NOT right.  Work in the world of Newtonian
        physics is not the same as work in normal English.
        Basically, if an object moves one place and moves back
        to its original place, the net work is zero.  Energy
        IS used however, the rather blurry relationship
        between work and energy "explained" in the statement
        is not telling the whole story there.

And similarly,

        No Mike, that's not correct.  It's true that the
        potential energy of the neck system is not increasing,
        so no "work" is being done in the strict physics-based
        sense (work is defined as increasing the potential
        energy of a system, like pushing a crate up an
        incline, or using a pully to hoist something,
        therefore pushing a crate along a level surface is not
        work, despite the energy expended to overcome kinetic
        frictional forces and impart momentum).  The neck is
        being restrained against the acceleration of gravity,
        however, so energy IS being expended even though no
        work is being done.  The author appears to have
        confused the common usge of "work" with the physics
        based definition.

(I pause here to note that this definition of work -- increasing the
potential energy of a system -- seems incompatible with the classic
definiton of force applied x distance moved.  Pushing a brick
horizontally one meter with a force of one newton expends one joule in
the classic system, but not according to potential-energy definition.
Curiouser and curiouser.)

What both these explanations leave unexplained is where the energy
_goes_ in a zero-work system in which energy is expended.  There are
some hints of answers to this in the next explanation:

        The mistake is that work done equalling zero doesn't
        imply that energy used is also zero.  This is the
        physics definition of "work", all right, but nothing
        says you can't expend energy if you don't do physics
        definition work -- measured in Joules, Newtons per
        meter [should be Newton-meters -- Ed.] Newtons are kg
        m/s^2, mass times acceleration.  So you get kg m^2/s^2.

        Our rigidly still sauropod isn't moving, so no work --
        no mass is moving.

        Consider trying to push a large truck loaded with
        cement yourself; it won't move -- so you do no physics
        work in moving the truck -- but you can expend lots
        and lots of energy trying.

        It's getting expended chemically, in your muscles,
        where it's producing heat, which is _also_ measured in
        Joules.  Considered at that finer scale of detail, our
        sauropod has to provide a balancing force against
        gravity *somehow*, and that force has to persist over
        time, of course course energy is expended.

So this posits and answer to the question of where the energy _goes_
(since it can be neither created nor destroyed) -- it's converted to
heat.  Is that convincing?  I'm not sure ... I guess I don't see any
alternatives, but purely intuitively this _seems_ like a hell of a lot
of heat to lose.

Next reply was short and sweet:

        Work has to be done against gravity. Holding it steady
        means counteracting gravity to keep it in place.

Thing is, I already know that.  That's how I knew that the original
description of the situation had to be wrong.  So I don't see this as
an answer so much as a restatement of the problem.  (Reminds me of one
of my favourite quotes: "I don't have any solution but I certainly
admire the problem" -- Ashleigh Brilliant.)

OK, next answer, which is one of the more comprehensive ones:

        no, it is not right - just carry a heavy suitcase for
        a mile or so and see whether it doesn't need some
        energy. (Here the distance is perpendicular to the
        force of gravity, so in a phyiscal sense no energy is
        needed.)

        The point is that muscles need constant energy supply
        to stay in a contracted position - without energy, the
        muscle relaxes. There are molecules inside the muscles
        which do the contraction of the muscle by a change in
        some binding angle (out of my head I don't remember
        the details, but if you need them, I can look them up
        somewhere - it's probably in one of the books by
        McGowen); but to stay in this contracted position, ATP
        molecules are needed (which are the source of cell
        energy).

        Looked at from the outside, no work is done in a
        physical sense. This means that all the energy used is
        converted into heat and is the cause for heating up
        when doing this kind of work (or any other kind as
        well, as the mechanical efficiency of muscles is not
        too good anyway).

        With ligaments, it is different, because they do not
        require energy - they are elastically loaded by the
        wight i.e., the forces between the molecules are what
        provides to force.

And another that seems to cover the bases pretty well:

        > energy is the capacity to do work,
        > [...]
        > so the work done is zero, and the energy used is
        > also zero!

        The problem is that in the last sentence I quoted, you
        haven't quite stuck with the definition in the prior
        sentence that I quoted.  Energy is the *capacity* to
        do work.  The energy burned by sauropod muscles as it
        holds its neck in a particular position *could* be
        used to perform mechanical work (force x distance) but
        it isn't.  It's used in other ways, particularly, it
        is converted into heat.

        [Other, interesting but not directly relevant, stuff
        snipped -- Ed.]

        If you want to ask what is the difference between
        holding your head up with ligaments vs. holding your
        head up with muscles, you can think of ligaments as
        passive springs and muscles as stepper motors.  You
        apply forces to ligaments, and they stretch but then
        more or less stay there.  At the molecular level,
        that's not what muscles do.  They constantly break and
        re-attach bonds, and this -- primarily the breaking of
        the bonds -- requires changes in chemical potential
        energy.  An isometrically contracted muscle is in a
        state of dynamic equilibrium, whereas a stretched
        ligament is (to a good approximation) in static
        equilibrium.

And finally, I present in no particular order all the other replies I
got:

        The fallacy is that while the neck does not appear to
        be moving, it, and the entire sauropod, are
        accelerating downward at a rate of 9.5 meters per
        second per second due to gravity.  So instead of
        imagining a static neck, imagine one that is pulled
        down by gravity, then immediately pulled up by
        muscular force.  So it will take some force to hold
        the neck steady.

And:

        It is right -- in zero gravity. Gravity is a force,
        so, in order to keep things in place, you need a
        counteracting force.

        And why should there be no ligaments in a sauropod
        neck?

And:

        However, if Work = Force X Distance, then technically
        the neck musculature holding the neck upright is doing
        the equivalent 'work' of lifting the mass from the
        ground to the end position - constantly.  Besides: who
        says sauropods didn't have neck ligaments?

And:

        Seems to me you're not factoring in the acceleration
        of gravity, which it holds the neck against. Stop
        holding the neck, and it accelerates to the ground,
        right?

(BTW., to those who asked what makes me thing sauropods had no neck
ligaments: I _don't_ think that, it's a hypothetical to set up the
question.  But I _do_ know sauropod workers who do think that
ligaments played little or no part in maintaining sauropod posture.)

 _/|_    _______________________________________________________________
/o ) \/  Mike Taylor  <mike@indexdata.com>  http://www.miketaylor.org.uk
)_v__/\  "An object in orbit remains in orbit until it hits something"
         -- Bill Keel's law of orbital dynamics in the solar system

--
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