The Last Stoic

Learning Maxwell’s Rope

Posted in Uncategorized by munty13 on September 21, 2010

“A person needs a little madness, or else they never dare cut the rope and be free.”
~~Nikos Kazantzakis

In his 1861 paper, “On Lines of Physical Force,” Maxwell elaborates on his desire to investigate “the mechanical results of certain states of tension and motion in a medium,” so that they might be compared with “the observed phenomena of magnetism and electricity.” A copy of the paper, and fascinating it is too, can be seen here, thanks to Wikisource:
http://en.wikisource.org/wiki/On_Physical_Lines_of_Force

Maxwell, just as Faraday had done before him, treated the lines of magnetic force (as revealed by iron filings spilled over paper held above a magnet,) as being real entities. Maxwell describes each filing as being “magnetized by induction,” and that they unite to form “fibres,” and that these fibres will then “indicate the direction of the lines of force.” Maxwell considers the lines of magnetic force as “existing in the form of some kind of pressure or tension, or, more generally, of stress in the medium.” He imagines the stress evoked by the lines of magnetic force as representative of tension, “like that of a rope.” Rope? I wonder what exactly was Maxwell getting at?

If you look at a piece of string under a magnifying glass as you pull on the ends more and more strongly, you will see the fibers straightening and becoming taut. Different parts of the string are apparently exerting forces on each other. For instance, if we think of the two halves of the string as two objects, then each half is exerting a force on the other half. If we imagine the string as consisting of many small parts, then each segment is transmitting a force to the next segment, and if the string has very little mass, then all the forces are equal in magnitude. We refer to the magnitude of the forces as the tension in the string, T.
http://www.vias.org/physics/bk1_07_04.html

Tension usually arises in the use of ropes or cables to transmit a force. It is the opposite of compression. Tension is the force with which a rope or line pulls. If we hang a length of rope from the ceiling, and add a weight to the end of it, the pull force created by the weight is called the tension force. The tension force on the rope is equal to the weight of the object.

In general, low-mass objects can be treated approximately as if they simply transmitted forces from one object to another. This can be true for strings, ropes, and cords, and also for rigid objects such as rods and sticks.
http://www.lightandmatter.com/html_books/1np/ch05/ch05.html#Section5.4

Tension force, or tensile force, is an example of a pulling force, and is typically measured in pounds (Ibs) or newtons (N.) Tension force will act on opposite ends of the rope and pull it tight. The force is applied in the direction of the rope. Objects on both ends of the rope will experience a pulling force equal to the tension force. Tension force in the rope is of equal magnitude throughout the rope.

If you think about it, a rope without tension is remarkably useless when you need to get something moved (unless of course, you use it as a whip and order someone else to move it!) For example, a length of rope is not much use in pushing an object across a flat surface – it can’t be used like a rod or stick. However, if the rope is tied round said object, and I pull on it, then I stand a chance of shifting the thing. Tension generated in the rope thus allows me to transmit force. Strings, ropes, cables and chains can only be used in instances where there is pull force. Essentially, somebody, or something has to pull the rope.

If we translate these ideas back to lines of magnetic force, and imagine them as ropes or chains that are fashioned in a series of continous loops, then something must be pulling at them. What is required is a mechanism of some description, one that resides in the material of the magnet, and acts upon these “ropes” by pulling on them. What emerges is something that is highly reminiscent of a pulley system.

A pulley is a simple machine consisting of a string (or rope) wrapped around a wheel (sometimes with a groove) with one end of the string attached to an object and the other end attached to a person or a motor. Pulleys may seem simple, but they can provide a powerful mechanical advantage so lifting tasks may be done easily.
http://www.teachengineering.org/view_lesson.php?url=http://www.teachengineering.org/collection/cub_/lessons/cub_simple/cub_simple_lesson05.xml

Rope-pulley systems are used when there is a need to transmit rotary motion. The advantage of ropes and chains is that they can transmit force without their performance being affected by length. A common misconception holds that simple machines, such as levers and pulleys, increase forces – but this is not quite correct; levers and pulleys transmit forces, and if the masses of the levers or pulleys are negligible, the input and output forces are equal. In other words, simple machines make it possible to lift heavy objects because they reduce the magnitude of the required input force, but it shall be seen that regardless of whether we use a pulley system or not, the amount of over-all effort needed to move an object always remains the same. A box that weighs 100 Newtons is always going to need an upward force greater than 100 Newtons to lift it.

The effect of a pulley is analogous to the gears of a bicycle; a lower gear makes it easier to turn the pedals, but they must be turned more often for the bicycle to travel the same distance. Likewise, a pulley system makes it easier to lift a load, but the length of cable used to lift the load is greater than the distance the load is lifted. The more pulleys that are used, the easier it is to lift the load, but the longer the length of cable needed to lift the the same distance.
http://wiki.answers.com/Q/Where_is_a_pulley_used

In practise, rope-pulley systems are generally regarded as inefficient due to the force of friction. Ropes tend to slip and stick along pulley wheels, meaning that energy is lost from the system. Another disadvantage with the system is that rope can permanently stretch under tension, robbing the rope of its elasticity and strength. This is why in some installations it is better to use a chain system, whose rigid design allows it to retain tension, and prevents stretching. Chains can also be made to fit on gears so that slipping is not a problem. For these reasons, if I were to try and imagine how lines of force might physically appear, then I would come up with a chain system. I then picture these chains as being pushed and pulled by a multitude of sprockets and gears, all whizzing frantically away inside the magnetic material.

A sprocket is a profiled wheel with teeth that meshes with a chain, track or other perforated or indented material. It is distinguished from a gear in that sprockets are never meshed together directly, and differs from a pulley in that sprockets have teeth and pulleys are smooth.

Sprockets are used in bicycles, motorcycles, cars, tanks, and other machinery either to transmit rotary motion between two shafts where gears are unsuitable or to impart linear motion to a track, tape etc.
http://en.wikipedia.org/wiki/Sprocket

Effectively, what a chain or rope-pulley system allows us to do is “trade” force for distance – which is the exact same principle by which a simple lever works. Pulleys lengthen the distance of the rope, thereby increasing the distance over which the force acts. By increasing distance, the system is able to increase the amount of energy that it stores. I tend to imagine it somewhat as the total force that is required to lift an object, as being broken down into smaller, more manageable units, and then spread over a greater distance.

With four wheels and four ropes, a pulley cuts the lifting force you need to one quarter. But you have to pull the end of the rope four times as far.
http://www.explainthatstuff.com/pulleys.html

It is interesting that as a direct consequence of the increase in distance, we are now seeing that action takes place over a longer time. This is in compliance with the “golden rule” of mechanics, in that the mechanical advantage derived will always be accompanied by a loss in displacement, or in other words, time. This further suggests that force, regardless of the amount, always seems to be transmitted at the same speed. It means that Archimedes claim to be able to lift the world, given a lever that was long enough, is theoretically possible – just as long as we’re prepared to wait a few million years!

Archimedes knew that by applying a lever, one could lift the heaviest of weights by applying even the weakest of forces. One had only to apply this force to the levers longer arm and cause the shorter one to act on the load. He therefore thought that by pressing with his hand on the extremely long arm of a lever he would be able to lift a weight, the mass of which would be equivalent to that of the earth.

Let us imagine for a moment that Archimedes had at his disposal “another earth” and also the point of support he sought. Further imagine that he was even able to manufacture a lever of the required length. I wonder if you can guess the amount of time he would need to lift a load equivalent in mass to that of the earth, by at least a centimeter? Thirty million million years- and no less!!
http://www.buzzle.com/editorials/7-30-2004-57259.asp

When you push down on a lever, the force you push with is multiplied by the length of the lever to produce a torque. The torque of a force is the turning effect of the force about a point. Torque is, by definition, the product of a force applied in a rotational motion or twisting force. It is a turning force. It is the force that produces rotation. Because pulley systems use rotational forces, they too use the principle of torque.

Torque, also called moment or moment of force, is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist.
http://en.wikipedia.org/wiki/Torque

Whenever I think of torque, I’m always reminded of the cordless screwdriver I would wield at work. You may have noticed that most cordless screwdrivers have a torque setting. To remove a screw from the wall, one that was wedged in especially tight, it is necessary for the torque to be switched over to a high setting. If the torque is not high enough, the screw will not budge, because the drill is simply not strong enough. It is torque which is defining the drill’s strength. If the screw is especially tight, and the torque setting is strong enough, you might find that the screw will still not move, but now it is your arm that the drill wants to spin round!

Both levers and the inclined plane lower the force required for a task at the price of having to apply that force over a longer distance. With wheels and axles the same is true: a poweful force and movement of the axle is converted to a greater movement, but less force, at the circumference of the wheel. In a circular geometry, torque is a more useful concept than force and distance. The wheel and axle can be thought of as simply a circular lever, as shown in Figure 5 [below]. Many common items rely on the wheel and axle such as the screwdriver, the steering wheel, the wrench, and the faucet.
http://cnx.org/content/m13594/latest/

If you want to increase the load that a pulley system may carry, then it is necessary to increase torque. The greater the torque, the more energy stored. Torque units contain a distance and a force. Torque is, in effect, the product of the force and the length of the lever arm. Understood in this way, it is clear that there are two ways of increasing torque; either increase the force or increase the length of the lever arm. As torque is a product of force and distance, one may be “traded” for the other.

Torque, like work is measured in pound-feet (lb-ft) However, torque, unlike work, may exist even though no movement occurs. A good example of this is the torque exerted when you try and loosen a very tight nut. As you are pulling on the wrench you are exerting a force, but not until the the nut moves has this torque resulted in work.

Work is done when we use a force (a push or pull) to move something over a distance. Energy is needed to move a force through distance. In calculating work done, two things need to be measured: the amount of force and the distance that it moves. Thus, it is important to make the distinction that work is a measure of what is done, not the effort applied in trying to move it. Where energy is the capacity for doing work, it means that both these quantities are measured by the same unit – in a sense the two are equivalent. Work and energy are merely different ways of looking at the same thing.

Because of the way it is defined, there will be plenty of times in my life (some might say the story of my life!) where I will spend energy, but no work is done. That’s because it’s possible to for me to exert a force where we see nothing move. If I remain still, holding a dumbell above my head, a physics textbook will tell me that I have done no work because my actions do not involve motion – no distance, no work. A physicist might be cheeky enough to tell me that I’m not doing any work – but given a few minutes, it won’t stop my arms from feeling like they want to fall off! The fact is, I am still spending energy even though there is no evidence of “work done.”

People often confuse energy, power, and force. Force is a push or a pull on an object or body. The strength of the force used and the distance through which it moves determine the amount of work done. Two factors determine the amount of work done. One factor is the amount of force applied. The other is the distance the object moves. In physics, work occurs only when the force is sufficient to move the object. In other words, work is a measure of what is done, not the effort applied in attempting to move the object. People do work when lifting, pushing, or sliding an object from one place to another. They do no work when holding an object without moving it, even though they may become tired.
http://www.sci-ctr.edu.sg/ssc/detailed.jsp?artid=4875&type=6&root=5&parent=5&cat=57

If work and energy are both a measure of force times distance, what does this mean if we extract distance from the equation? Of course we don’t even have to go as far as removing it, but simply reduce its quantity to zero. Thus, energy now becomes equal to force times zero distance, surely meaning that force is only describing energy in some way? Looking at this equation again, it is also interesting that distance too emerges as being another means of describing energy.

Returning to the idea that a lever, or pulley system are able to reduce the magnitude of the necessary force by applying force over a longer distance, it seems that this “trade” of force for distance would suggest that the system is storing force in some way. Indeed, if we were to expand on this a little further, it might be said that distance itself is responsible for storing this force.

Now, I must tread carefully in how I explain the idea of how a system might “store force.” As any physics textbook will tell you, it is not possible to “store force” as such – rather, it is energy which is stored, and not force. This is because force, at least in terms of how some physicists describe it, is not energy. Energy and force are treated as being something of different concepts, but this does not mean they are incompatible. I think it’s possible to argue that energy and forces are merely different guises of the same entity.

Now there may well be some who are more familiar with physics, whose sensibilities are more delicate than others, and whom after reading that last sentence, might just have sprayed the wall with coffee. To these persons I apologise profusely for any mess caused, but I believe that there are good grounds for such a theory, and I’m going to discuss them in my next post (those that are consuming drinks have been warned!)

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