Re: [Phys-L] expansion to lowest order ... not necessarily first order (was: force multiplier)

[John Denker <jsd@av8n.com>]

On 02/07/2013 07:31 PM, Ludwik Kowalski wrote:
> One end of the rope is attached to the car's bumper while the other
> is attached to a tree, on the other side of the road. A man, standing
> in the middle of the road, pulls the rope upwards, with a force F1.
> The force exerted by the rope on the car, F2, turns out to be several
> times larger, than F1, depending on the angle between the road and
> the rope.

> F2/F1=1/[2*sin(alpha)]                   [1]
>
> ... counterintuitive ....

To me, the physics seems cut-and-dried;  the interesting question is why
some people might find the result counterintuitive.

Here are some thoughts on that subject.

1) First of all, the category of "counterintuitive" is a moving target.  There
is a proverb that says:
   _Education is the process of cultivating your intuition_
In some sense our goal as educators is to get everybody to the point where
this result is completely intuitive.


2) Proportional reasoning is considered a milestone marking the "final" stage
in Piaget's model of intellectual development.  However (!), this cannot 
really be the final stage, because proportional reasoning only works if
there is a valid first-order expansion ... which is not always the case.

As physicists, we operate at a stage far beyond mere proportional reasoning.


3) Expanding things to lowest order is a big part of the stock-in-trade of
any physicist.

That's fine as far as it goes, but it would be a mistake to get overly reliant 
on expanding things to /first/ order, because sometimes the lowest order is 
second order, or higher ... and sometimes there is no Taylor expansion at all.
See below for examples.

In any case where position X1 is second order in position X2, you are going
to have infinite mechanical advantage (or infinite mechanical disadvantage,
depending on how you look at it).

There are two pedagogical issues here:

a) In an algebra-based physics course, you can't expect the students to know
how to think intelligently about infinity.  However, sometimes infinity is 
the right answer, even when the question is relatively simple.

There are of course lots of simple questions for which the answer is outside 
the scope of the introductory course.  The thing that makes the straight-rope
example slightly unusual is that the formula is easy to derive;  it is only
the conceptual interpretation that is troublesome.  Proportional reasoning is
not going to get the job done.

b) Even if somebody understands the theory, if they have never seen an example 
of infinite mechanical advantage they might find it counterintuitive.  Later,
after they have encountered a bazillion examples, including literally hands-on 
examples, it becomes entirely intuitive.  Infinite mechanical advantage is
exactly what you'd expect.  It /has/ to be that way.


=========
There are many categories of examples:

-- It is as easy as π to find situations where the first-order term vanishes 
 by symmetry.  As an example in this category, consider the pedals on a
 bicycle.  When the pedal is at top-dead-center (or bottom-dead-center)
 you can stomp on the pedal as hard as you like, and it's not going to go
 anywhere.  There is infinite mechanical disadvantage.

 A straight rope (such as a rope hanging vertically supporting a heavy load) 
 falls into this category.  Pushing the rope north or south is the same, so 
 the sideways force must vanish to first order by symmetry.

-- As another type of symmetry, consider kinetic energy (a scalar) as a
 function of velocity (a vector).  There cannot possibly any first-order
 term, since you can't have a scalar be proportional to a vector.  The
 lowest term is second order.

 OTOH if you think about things properly, i.e. four-dimensionally, there
 *is* an expansion in terms of rapidity, containing all orders:
   The zeroth-order term is the invariant mass, m.
   The first-order term is the momentum, m v.
   The second-order term is the kinetic energy, m v^2.
 These all contribute to various components of the [Energy, Momentum] 4-vector.
 On the third hand, each component of the [E,p] 4-vector is separately
 conserved, so we are entirely within our rights to focus on the timelike 
 component, and there is no first-order contribution to this term -- only 
 zeroth-order and second-order contributions.
    http://www.av8n.com/physics/spacetime-welcome.htm#sec-low-v-approx

-- It is common to find a system that exhibits stable equilibrium, which 
 means you are likely to find it naturally in the equilibrium state.  Almost
 by definition, at this point the first-order term vanishes.

 Example:  A marble in the bottom of a bowl.  The vertical displacement is 
 second order in horizontal displacement, for any small horizontal displacement. 
 The first order term must vanish, as a condition of equilibrium.

 -- Although stability is common, it is not necessary.  The bike pedal at
 top-dead-center exhibits unstable equilibrium.  Similarly, a straight 
 knee-joint exhibits unstable equilibrium, yet it is common to find the 
 joint in that state.  Knee joints, cams, and related structures exhibiting
 infinite mechanical disadvantage are commonly used in locks and latches,
 so that the bolt can't be forced open.  Other applications include retractable 
 landing-gear mechanisms, the hold-down mechanisms on a rocket launch pad,
 quick-release shackles, et cetera.

 Also, there is usually no stability principle that drives a system to sit 
 at a critical point, as discussed below (although in special cases there is 
 such a thing as self-induced criticality).

-- At a critical point, many system properties exhibit power-law scaling with
 non-integer exponents.  For example, the spontaneous magnetization (M) of a
 ferromagnet scales according to
    M ∝ (Tc - T)^β                             [2]
 where the critical exponent is
    β = 0.3645 ± 0.0025                        [3]
 reference: http://physlab.lums.edu.pk/images/c/cf/Gadolinium_v1.pdf

 If you think there "should" be a first order expansion for equation [2],
 try it sometime.  I claim there is no Taylor expansion to first order
 /or any other order/ that converges at the critical point.  Hint:  Draw
 the graph of equation [2].