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Re: [Phys-L] A force multiplier



I understand what you are saying, but in real life, I've seen this work. Several years ago, about 10 years into my teaching career, my family and I visited family in Michigan for our annual Labor Day weekend get together. We were helping my uncle in building a bridge so that he could take his tractor to the other side of a creek. Since it was so hot, we built this bridge under the shade of a tree with the hopes that we'll just drag it down to the creek and set it up there. This bridge was built with a bunch of 4x4 and 4x8...it was very heavy. When finished, we strapped a bunch of ropes onto it and we were going to manually drag it down. Nearly 20 adults couldn't make it budge. They got the tractor and the tires just spun. So my dad, a farmer and steel worker with only a high school education, wrapped a chain around a tree and the other end to the bridge, tightened it up the chain, and lifted up on the chain. The bridge moved, but just a little bit. He retightened the chain and repeated the whole process again. We made some plywood sleds and he continued to use that process to slide the bridge on the sled so that we could use the tractor the rest of the way.

Yes, the bridge didn't move much, and even less the further along it moved, but the accumulated effort does add up. My dad did make it move a good 8 feet, much further than what a group of people or a tractor could do.

Dwight Souder


This email was sent using 100% recycled electrons.

On Feb 17, 2013, at 12:13 AM, "John Denker" <jsd@av8n.com> wrote:

On 02/16/2013 08:54 PM, Ludwik Kowalski wrote:
Suppose the maximum force a person can pull is F1= 200 lb while the
force needed to move the car from the mud is F2=4000 lb. The angle
alpha, needed, when F2/F1=20, according to the above formula, is
about 1.4 degrees. Suppose the original horizontal distance from the
bumper to the middle of the road is 10 meters, and that the rope is
originally not stretched. The man pulls it up. The needed 1.4 degree
is reached when the middle of the rope is only 0.24 meters above the
road. The distance to the car (along the rope) changes from 10 meters
to 10.24 meters. Such 2.4 % change, in the length of the rope, is not
large enough to generate the needed F2/F1 ratio. The "spring
constant" of a typical rope, in other words, is not large enough to
satisfy the theoretical formula. That formula, however, would be
reliable if the rope were replaced by a thick steel cable, as
indicated by John. D.

When I run the numbers, I get an even more discouraging result.

The origin of the mechanical advantage lies in the geometry and
trigonometry of the situation. The transverse deflection is
proportional to sin(theta), while the longitudinal motion is
proportional to 1-cos(theta). For small theta, the former is
first order, while the latter is second order! You can't use
simple proportional reasoning. (Maybe that's the answer to the
original question as to why this mechanism is counterintuitive
in the small-angle case.)

rope length 20 m
F2 4000 lbf
F1 200 lbf

ratio 0.05
half 0.025

theta 0.0250 radian i.e. 1.43 degree
transverse 0.25 m (agrees with LK)
deflection

cos theta 0.999687
1-cos theta 0.000313 i.e. 0.03%
motion of end 0.006251 m i.e 6.25 mm

In other words, I calculate that even with an infinitely non-stretchy
rope, the car moves only 6.25 mm (which is 40 times less than the LK
result).

Even with a steel cable 1 cm in diameter, the putative motion is an order
of magnitude less than the stretch in the cable.

modulus of 65000000000 Pa i.e. 0.64 megabar
wire rope
diameter 0.01 m
area 7.854E-05 m^2
force-scale 5105088.062 N
relative exten 0.0035
total stretch 0.0698 m i.e 69.81 mm

The spreadsheet I used to calculate all this is at
http://www.av8n.com/physics/cable-mech-adv.xls

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

It is amusing to reflect on what works and what doesn't:
-- Dimensional analysis fails in this case. Proportional
reasoning fails.
-- Non-dimensional scaling analysis works fine. You have to
work out the physics in enough detail to realize that the
longitudinal effect is *second order* in the transverse effect
(for small theta).

Take-home lesson: Dimensional analysis is a poor man's approximation
to a scaling analysis.
http://www.av8n.com/physics/dimensional-analysis.htm
http://www.av8n.com/physics/scaling.htm
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