Re: [Phys-L] A force multiplier

[John Denker <jsd@av8n.com>]

On 02/07/2013 07:31 PM, Ludwik Kowalski wrote:
> We all know how to explain this mathematically, deriving the
> F2/F1=1/[2*sin(alpha)] equation.

> Unfortunately, I never verified this counterintuitive relation. How
> close are the experimental and the theoretical F1/F2 ratios? If not
> very close then why?

1) The relationship has been verified eleventeen gajillion times.
 It is verified by every overhead powerline in the world.
 It is verified by every suspension bridge in the world.
 etc. etc. etc.

2) For large alpha the relationship is perfectly intuitive.  It is
 exactly what you would expect.

 So the only remaining task is to understand what happens when alpha
 becomes small.  The theory predicts that the F2/F1 ratio diverges.

 Here's one way to understand what really happens.  As a principle of
 formal logic, any statement of the form
	A implies B
 is equivalent to the contrapositive statement
	not(B) implies not(A)

 In this case, the relevant contrapositive statement is:
	Since the force ratio cannot possibly be infinite,
	the angle alpha cannot possibly be very small.

 In other words, the rope always sags.  In practical situations, the
 angle alpha is not under your direct control.

 If you start out with a strictly straight rope and then let go, it
 will sag.  In order to sag, the rope will stretch a little bit and/or
 the endpoints will move a little bit, under the influence of a large
 but finite force.

 At this point, attempts to move the stretch the rope more and/or to 
 move endpoints farther will fail, because now that the rope has
 sagged it no longer has such a large mechanical advantage.

====

 As a practical matter, rope bridges always sag.  Recently the Mythbusters 
 guys tried to build a bridge out of duct tape, and they were surprised
 by the sag issue, demonstrating once again that they have no clue about
 basic physics.  There are suspension bridges where the deck does not sag, 
 but that's because the bridge has high towers supporting catenary cables, 
 and the catenaries sag.

 Continuing that thought, this explains why all the early airplanes were
 biplanes.  They needed cross-bracing, and the wires needed to be angled
 at a not-too-small angle.  It takes serious engineering and excellent
 materials to build anything that is long and thin and strong, such as
 a monoplane wing.

 This has been central to physics since Day One of modern science.  A
 while back I read a book by some guy named Galileo that talked about
 stiffness and strength, and how such things scaled with length, width,
 and thickness.