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Re: [Phys-L] A force multiplier



1) In the first message of this thread (2/7/2013) 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) John D. responded (2/7/2013)

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

3) Yes indeed. (F2/F1=> 1/2, when alpha=> 90 degrees) is intuitively obvious; nearly one half of the pulling force acts on the car and another half acts on the car.

4) Yesterday (2/8/2013) I asked: " 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. "

5) For the large (alpha==>90 degr) the answer is also intuitively obvious. One half of the applied force F1=24 N acts on the hook at point D while another half is used to pull the 12 N weight. This again is in agreement with the above theoretical formula.

6) But for small angle (alpha =5 degr) the theoretical prediction F2/F1=5.7, conflicts with my intuitive expectation. How can a force F1=2 N overcome the force F2 equal to 12 N?

7) What is wrong with my intuition?

8) Can someone perform the conceived two-pulleys experiment, pulling the rope down with a force meter (to control the angle alpha and to measure the corresponding values of F1)? I would very much like to see experimental data (for angles between 5 and 50 degrees. Perhaps something can be learned from such data.

9) I would not be surprised to learn that someone is performing such experiments routinely. If so then please share what you know and think?

10) Here is the ending of the message posted by John D.:

"... 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 [between the two pulleys in my figure L.K.] 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.