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Re: [Phys-L] A force multiplier



1) Yesterday I wrote:


Nearly every introductory physics textbook has an illustration showing a car being pulled out of mud. 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. The F2/F1 ratio is 6 when the angle is 5 degree.

We all know how to explain this mathematically, deriving the F2/F1=1/[2*sin(alpha)] equation.

And one does not need a car to experimentally verify this theoretical expectation. The only instruments needed are two pulleys, hanging weights and a large protractor. Unfortunately, I never verified this counterintuitive relation. How close are the experimental and the theoretical F2/F1 ratios? If not very close then why?
2) Is the situation shown at:

http://pages.csam.montclair.edu/~kowalski/forces.jpg

conceptually similar to the one I described yesterday (pulling a car from the mud)?

A hook at the point D is instead of a tree (on the other side of the road); the 12 N weight is instead of the car; the point C is where the man is pulling the stretched cable toward himself. Does the theoretically derived formula (in red above) apply to the illustrated situation (where the force F1=2 N is overcoming the force F2=12 N)? If not then why not ? The pulleys and the cable, as usual in such problems, are nearly ideal.

Ludwik Kowalski
http://csam.montclair.edu/~kowalski/life/intro.html

==========================================================

On Feb 7, 2013, at 10:48 PM, John Denker wrote:

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.
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