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OOPS, Sorry for mistakes below, especially in the referred figure. I will replace the figure shortly, and repost the message.
Ludwik
=================================
On Feb 17, 2013, at 1:53 PM, Ludwik Kowalski wrote:
To discuss the current topic in a class I would use the setup shown at:
http://csam.montclair.edu/~kowalski/force.jpg
The system is at equilibrium; the net force acing at the point C is zero. This leads to the mathematical relation"
F2/F1=1/[2*sin(alpha)]
, where F1=m*g and each F2=M*g. The condition of equilibrium can thus be written as
M/m=1/[2*sin(alpha)]
It is a relation beween three quantities, one angle and two masses. Suppose I decide to decrease the angle by increasing each M. The point C moves upward and the new equilibrium position is established. After demonstrating this fact I would ask several questions:
(a) What happens to the tension in the rope when the angle becomes smaller?
(b) Are vertical displacements of large masses M the same as the vertical displacement of the smaller mass m?
(c) How does the length of the rope change, and what does it depend on?
(d) What can eventually happen to the rope as the angle becomes smaller and smaller.
The question (b) can be answered by measuring dy1 and dy2.
Do these four questions address all aspects of our topic? If not then what else is worth emphasizing?
Ludwik Kowalski
http://csam.montclair.edu/~kowalski/life/intro.html
============================================
F2=On Feb 17, 2013, at 10:23 AM, John Denker wrote:
There definitely *are* useful things that can be accomplished by
pulling (or pushing!) sideways on a rope.
*) Perhaps the simplest example has to do with a heavy weight that
is already hanging from a long rope. You can push the weight sideways
with verrrry little effort.
*) Another example has to do with a block and fall (aka block and
tackle). Suppose there are N parts of rope between the two blocks,
giving a mechanical advantage of N, where N is fairly large (say
N=6). It is fairly common to find that a lot of force is being lost
due to friction in the blocks. In this case, tighten the hauling
part as much as you can. Then pull sideways on each of the parts
between the blocks. This temporarily creates the extra force
required to overcome friction. Then tighten the hauling part some
more, and iterate.
++ Note that in both of the previous examples, all of the slack and
stretch has been taken out /before/ we get around to applying the
sideways deflection. Similarly, in the following story, a critical
piece of the story is the part that speaks of tightening the chain.
On 02/16/2013 11:11 PM, Souder Dwight wrote:
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.
Also note that the tractor did not stall out. The tractor was limited
by slippage. I also suspect that the 20 adults were limited by slippage
rather than by ultimate strength. So ... in addition to the issue of
mechanical advantage, another important part of the story is the fact
that the trick of pushing on a rope turns the direction of force 90
degrees. This allows the person to stomp on the rope or lift vertically,
without risk of slippage.
I suspect that a winch or a block and fall (if available) would have been
a far more efficient way of moving the bridge.
In hilly and/or wooded country, a lot of tractors have winches on them
... partly for dragging stuff, and partly for getting the tractor itself
un-stuck when necessary.
==============
It is also worth keeping in mind the reverse result: If you have a rope
with a transverse load, there is *no* amount of tension that will pull
the rope straight. That's negative statement, but it's still useful,
since it tells you where to focus your attention. If you want to build
a suspension bridge with a flat deck, you don't do it using a straight
cable with infinite tension. Instead you build towers and /allow/ the
catenaries to sag.
Even so, on a typical suspension bridge
http://catenary-project.wikispaces.com/file/view/800px-GoldenGateBridge-001.jpg
at each point where a suspender attaches, the catenary bends through a
very small angle (not much more than one degree or so). That tells you
that the tension in the catenaries vastly exceeds the total weight of
the bridge.
==============
Similar physics occurs in two dimensions: The force required to hold a
sail flat, or even approximately flat, is enormous.
==============
Similar physics ideas can be applied to archery. For starters, when the
bowstring is nearly straight, it imparts only a small force to the arrow
(small compared to the tension in the string). This means that a goodly
part of the draw-length is wasted. What you actually want is the reverse,
i.e. a lot of /let off/. That is, you want a relatively small amount of
force when the bow is fully drawn, so the archer can aim without getting
unduly stressed or tired.
This leads to the invention of recurved bows and compound bows, some of
which contain quite a lot of sophisticated physics and engineering. An
ancient and seemingly-simple yet very clever design is the Wabanaki bow
aka Penobscot bow:
http://www.primitivearcher.com/smf/index.php/topic,2259.msg148508.html#msg148508
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Forum for Physics Educators
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Forum for Physics Educators
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