Newton's Smart Ass (The Donkey and Cart Paradox)

 "If I have seen a little further it is by standing on the shoulders of Giants."

~~Sir Isaac Newton

http://images.travelpod.com/users/chris-marianne/bigchina.1117263240.kashgar_-_donkey_cart.jpg

Much like mass or volume, energy is a property of an object. Energy is the potential for an object to exert force. Therefore, it can be said that a simple machine, such as a lever or pulley system, stores energy which then gives it the "potential" to do something. Here it seems that we have also given ourselves a pretty good description of the term "potential energy." That is, potential energy is energy stored in matter.

Energy differs from force, in that I need energy to exert a force. Energy is needed to exert force. If I don't have any energy, I will not be able to do anything. It is energy which is the currency for performing work. You need energy to do work. If I have no energy, I can't use force to make something move. Which thus brings me to the somewhat unsatisfying conclusion that energy can exist without force, but that force cannot exist without energy. Or, to put it another way, energy is not force exactly, but force on the other hand, could be described as being energy in one form or other. There. Clear as mud. To try and build a better picture, let's take a closer look at what is meant by the term "force" exactly.

A force is a push or pull upon an object resulting from the object's interaction with another object. Whenever there is an interaction between two objects, there is a force upon each of the objects. When the interaction ceases, the two objects no longer experience the force. Forces only exist as a result of an interaction.
http://www.physicsclassroom.com/class/newtlaws/u2l2a.cfm

Our modern understanding in the workings of force and motion is derived from observations made by Sir Isaac Newton (1642-1727.) His most outstanding contribution to physics was in formulising the exact forces which act on objects, and how these define an object's motion. Essentially, it amounted to a mathematical representation of how energy was involved in moving objects, and indeed, stationary objects. Thus, Newton was able to arrange his findings to develop a mechanical model of the Universe, giving us the expression "classical mechanics."

The term classical mechanics was coined in the early 20th century to describe the system of mathematical physics begun by Isaac Newton and many contemporary 17th century natural philosophers, building upon the earlier astronomical theories of Johannes Kepler, which in turn were based on the precise observations of Tycho Brahe and the studies of terrestrial projectile motion of Galileo, but before the development of quantum physics and relativity. Therefore, some sources exclude so-called "relativistic physics" from that category. However, a number of modern sources do include Einstein's mechanics, which in their view represents classical mechanics in its most developed and most accurate form.
http://wapedia.mobi/en/Classical_mechanics

What Newton was trying to establish was a set of equations which illustrated how motion is governed by the Universe. In theory, these formulas amounted to a firm set of rules which could be applied to motion anywhere in the Universe - from everyday objects to distant planets. In the following three hundred years, it should be duly noted that the Universe, being something of a stickler for rules and regulations, has always been seen to obey them. In 1686, Newton set-out the formulas in his paper, "Principia Mathematica Philosophiae Naturalis," and presented them as the Three Laws of Motion.

Newtonian concept of force is given in three simple mechanical laws:

1) A body remains in its kinetic state, either at rest or in motion, unless there is an external force acting on it; which is known as the law of inertia.

2) If a (net) force F is applied to an object with mass m, then the acceleration a of the object caused by F follows the relation F = ma

3) If object A exerts a force F on object B then object B, then object B exerts a force -F on object A, equal in magnitude and opposite in direction to F
http://www.thecatalyst.org/physics/chapter-two.html

The definition of "force" can be found in Newton's Third Law, which states that for "every action there is always opposed an equal reaction." According to the Third Law, a force is simply a mutual interaction between two objects that results in an equal and opposite push or pull upon those objects. This illustrates the fact that forces always occur in pairs. Forces never act in isolation, and when two objects interact, action and reaction forces of equal magnitude are always paired and act on the objects in opposite directions.

Newton's 3rd Law states that for every action, there is an equal and opposite reaction. This basically means that forces always occur in pairs. In any situation where one object is exerting a force on the other, then the other object is also exerting a force that is equal in magnitude but opposite in direction on the initial object. So for example, The Sun exerts a gravitational pull on Earth, and likewise, the Earth exerts the same gravitational pull on the Sun. But since the mass of the Earth is so much smaller than that of the Sun, the effects of the force on the Earth is much greater, and so we orbit around the Sun, while the Sun merely does a small wobble in reaction to the forces exerted by the Earth.
http://physicsstash.blogspot.com/2007/07/horse-and-cart-in-syllabus.html

The most important provision is that for forces to occur there must be TWO OBJECTS. Forces only exist as a result of an interaction between two objects. Therefore, if energy can be percieved as a property of one object, then perhaps we can think of force as being a property of two objects. Force is never found to be the property of one object. To develop a greater understanding of force, it is important to know how to apply the Third Law correctly.

In physics, there is a supposed paradox relating to the Third Law, in-which a wise, but particularly lazy donkey refuses to move because he claims that the action of him pulling the cart, only results in the cart pulling him back with an equal opposing force. Never mind how much he strains to pull away, the cart pulls him back with the same magnitude of force, thereby leaving him stuck to the spot. There's nothing for the donkey to do, except shrug his shoulders and ask for another carrot.

In truth though, there is no paradox. That's because the donkey and the cart are not a straight forward action-reaction pair. There is also a third object that is conspicuous by its absence from the donkey's reasoning - the ground! If you imagine the donkey and cart floating in the vacuum of space, with donkey kicking his heels in vain, it might reveal just how vital the ground really is for this type of motion.

To get moving, the donkey has to get the cart moving too. Strictly speaking, it is not as simple as saying that the cart opposes the motion of the donkey. Rather, it is the forces which the donkey generates which are in opposition to the forces generated by the cart. In trying to explain this process a little better, it is perhaps best to think of the donkey and cart as consisting of two conflicting systems. One system is the donkey and the ground making up one action-reaction pair, while the other system consists of the cart and ground in another action-reaction pair.

If we begin with the system of the donkey and ground, we have the donkey which is applying an action force downward and backward, while the ground has a reaction force which is acting forward and upward. The donkey resists the ground moving forward, and the ground resists the donkey moving backward. It is noticeable that the applied backward push on the ground is dependent on a very significant force in order for it to be converted into forward motion - friction.

In ancient times, Aristotle had maintained that a force is what is required to keep a body in motion. The higher the speed, the larger the force needed. Aristotle's idea of force is not unreasonable and is in fact in accordance with experience from everyday life: It does require a force to push a piece of furniture from one corner of a room to another. What Aristotle failed to appreciate is that everyday life is plagued by friction. An object in motion comes to rest because of friction and thus a force is required if it is to keep moving. This force is needed in order to cancel the force of friction that opposes the motion. In an idealized world with no friction, a body that is set into motion does not require a force to keep it moving. Galileo, 2000 years after Aristotle, was the first to realize that the state of no motion and the state of motion with constant speed in a straight line are indistinguishable from each other. Since no force is present in the case of no motion, no forces are required in the case of motion in a straight line with constant speed either.
http://en.wikibooks.org/wiki/IB_Physics/Mechanics

We can think of friction as a force that impedes motion - it is always a resistance to the motion of things. If the ground was slippery for example, the conversion of the action-reaction forces would be greatly reduced by the lack of friction. Friction fulfils the very definition of force, in that it describes an interaction between two objects, such as the wheel and the ground for example, and that both bodies generates forces which act on the other body.

Friction is not a fundamental force but occurs because of the electromagnetic forces between charged particles which constitute the surfaces in contact. Because of the complexity of these interactions friction cannot be calculated from first principles, but instead must be found empirically.
http://en.wikipedia.org/wiki/Friction

Needless to say, friction is also required by system number two, the action-reaction pair of the cart and the ground, to keep the cart moving along the ground. The force of the wheel pushing backward is dependent on friction to convert it into a reaction force which pushes the cart forward. If friction did not exist between the wheels of the cart and the ground (it might help to imagine the wheel and ground smothered in oil - but never include yourself in the picture - that's just wrong!) the wheels would simply never get a grip on the road (and very likely give up energy in the form of heat, and noise) and the cart would never get off the spot.

Friction helps people convert one form of motion into another. For example, when people walk, friction allows them to convert a push backward along the ground into forward motion. Similarly, when car or bicycle tires push backward along the ground, friction with the ground makes the tires roll forward. Friction allows us to push and slide objects along the ground without our shoes slipping along the ground in the opposite direction.
http://superphysics.netfirms.com/friction.html

The motion in both these systems, and therefore the system over-all, is dependent upon frictional forces. If we overlay these two systems on the over-all system of donkey, cart, and ground, we can see that what we effectively need in-order to see motion, is for the donkey to produce a reaction force which is greater than the cart's resistive force. In other words, the cart will move forward when the frictional force between the horse and ground, is greater than the frictional force between the cart and ground. Because both these reaction forces are so dependent upon friction, it also reveals something of the force which is truly responsible for motion (at least walking motion) - electromagnetic force.

Now, there are some who explain the paradox of the lazy donkey, but still fail to acknowledge that the donkey and cart are NOT an action-reaction pair. If the donkey and cart were an action-reaction pair, motion would only be possible if the action of the donkey was somehow greater than the reaction of the cart. But how is this possible, if according to Newton's Third Law, the opposing force is ALWAYS of the same magnitude as the applied force?

"If a force acts upon a body, then an equal and opposite force must act upon another body."
http://en.wikipedia.org/wiki/Reaction_(physics)

Some commentators still maintain that motion is possible because the reaction force and the action force are working independently from one another. This means the reaction force does not cancel the action force because they are both acting on different bodies - the donkey is acting on the cart, and the cart is reacting to the donkey. Unfortunately, their reasoning blatantly contradicts what defines an action-reaction pair in the first place - that is, they always come in pairs! It seems inconceivable that one can operate with a strength which is totally independent from the other.

As we have seen, motion is not dependent, in anyway, upon an "interaction" between the donkey and cart. Sure, there is something at work between the donkey and cart because they are tied together, meaning that forces must exist between them, but as such, this arrangement is not responsible for the forces which drive motion. Rather, it is the interaction with the ground which is driving the motion of the donkey and cart; the forces which occupy the harness, and wagon tree between the donkey and cart are a response to that interaction.

In an "interaction," forces appear between two objects, and exert forces of the same magnitude in opposite directions. In effect, what we are seeing is each body being repelled in the opposite direction by a force that is exerted by the other body. What we see is one body moving away from the other body - one body goes left while the other goes right. A good way of visualising this is to imagine two ice-skaters who are leaning one against the other with their hands. This presents itself as a good example because the lack of frictional forces between the skaters and the ice, allows us to dismiss the "interaction" between the ice-skaters and the ground, and to concentrate more on the ice-skaters acting as an action-reaction pair.

On one level, the difference between dancing on a floor and skating on ice is the lack of friction. Smooth ice provides very little resistance against objects, like ice skates, being dragged across its surface. Compared to, say, a wooden floor, ice has much less friction.
http://www.foxnews.com/scitech/2010/02/17/physics-figure-skating/

Thus, ice-skater A pushes against ice-skater B, and a force will emerge that also pushes back against A. They "interact." Both A and B are sent in opposite directions AWAY from each other, they repel one another, and they do so with the same magnitude of force. This means that if A and B both have the same mass, then at least in theory, they shall both glide the same distance away from each other.

Using the example of two-ice-skaters, brings to the forefront the importance of the term "interaction." At the conclusion of my last post, I basically said that it was possible to think of force and energy as describing the same entity, even though it seems to break a well-founded tradition in physics that supposes they are not. What is exciting is that by developing an understanding of what exactly is meant by "interaction," the discreet affair between "force" and "energy" is finally exposed.

Energy is the property of one object, whereas force is never the property of one object, but ALWAYS two objects. However, strictly speaking, force is not the property of two objects either. A very distinctive feature of force is that it describes the "interaction" between two objects. In other words, force is the "interaction." Quite what this means, evaded me somewhat, until I came across a post on the blog "Gravity and Levity," and at once, everything was revealed. Some extracts from the post are featured below:

Force and energy: which is more real? This question sounds ridiculous, and maybe it is. So if you’re not in the mood for philosophy right now, you can skip this post.

Nonetheless, I think it makes sense to talk about our general attitude toward the concepts of force and energy. “Which is more real?” may not be a very well-defined question, but I think it is a very natural one that cuts to the heart of how we think about forces and energies. Furthermore, it is one to which my answer has changed over the years. The change was a difficult one: force and energy are such profoundly important concepts in physics that to change your view of them is to change your view of all topics that are built upon them (basically, everything). But for me it has been extremely important. Shifting my position from “force is more real” to “energy is more real” was essential for understanding and enjoying advanced topics in physics.

...If you believe that energy is more “real” than force, then you stop talking about “forces acting on objects” and instead talk about “interactions between objects.” Your starting assumptions for describing the universe must be the “interaction energies”.

...So which viewpoint is more correct? In a sense, it doesn’t matter: both give results that are perfectly consistent with our observations of nature. The second one seems a little crazier, but in fact it requires fewer assumptions. It also manages to explain why all forces come in pairs: between two objects there is only one interaction energy, to which both objects will respond. In my mind, the “energy is more real” viewpoint is much more compatible with advanced physics concepts like thermodynamics (where energy is really the only consideration), quantum mechanics (where our force laws are no longer strictly obeyed, but energy remains absolute) and field theory (where we are given a way of picturing where the energy is really stored). Perhaps most importantly, I find the energy viewpoint much more conducive to wonder.
http://gravityandlevity.wordpress.com/2009/04/13/force-and-energy-which-is-more-real/

The question of whether force is simply another term for describing energy, though presenting itself as a bit of a stumbling block for me, is something which seems to be already familiar with both world-weary physicists and philosophers (and bloggers!) alike. The authors of "Gravity and Levity" (unfortunately, I couldn't find their real names anywhere on the blog) are to be congratulated for such a great post. They have approached the problem in such an effective way, that they have succeeded in producing an expression of the relationship between energy and force in one turn of phrase, and it is one that is so astute, so sublime, that it becomes, quite frankly, life-changing:

"...Between two objects there is only one interaction energy, to which both objects will respond."

There. Isn't that just beautiful? Force is an interaction energy. But of course there's more. Do you see? The interaction energy is describing something that exists OUTSIDE the two objects. It is describing the existence of energy ouside bodies, and it is this energy which acts on bodies, meaning that there must be energy outside all bodies, meaning that we may aswell go the whole hog and say that energy exists EVERYWHERE.

If you recall the definition of potential energy given at the top of this page, it describes energy as being the property of an object. Saying that energy can exist outside objects, or bodies, is not without controversy in today's climate. Heavily influenced by relativity, and for too many reasons to divulge in this post, the entire field of physics is sold on the concept that all energy is contained inside matter. Even the photon, which is essentially massless, is still considered to be a particle of some description. The void of a vacuum is supposed to be just that - void! There's not supposed to be anything there, least of all energy! The idea that energy can exist outside matter would present all sorts of problems for modern theory, because it would be forced to admit to the existence of energy, not as an abstract mathematical concept, but as a real, physical substance.

If energy was a real substance that the Universe was immersed in, it would mean, at least in theory, it should be possible to reach out and grab it and use it, and do so, (here comes the dirty word) for "free." That's right, I'm talking about "free energy." No-one respectable in physics likes to discuss free energy. Talking about the possibility of free energy amounts to breaking one of the greatest taboos in science. It's the definitive "no-no." If you demand to talk to a physicist about free energy, they're likely to grab your wrist and give you a nasty chinese burn. Free energy, they will tell you, is impossible. And they'll probably call you a "crackpot" too. For the moment though, there's no need for us to enter the forray, and so we'll stick more closely to the term "interaction energy."

In comparing the two ice-skaters to the donkey and cart, we are seeing some obvious differences in the way energy interacts. With the ice-skaters, energy acts upon them and repels one from the other in opposite directions. Something quite different happens to the donkey and cart, because unlike the two ice-skaters, these two move together in the same direction. The forces are moving in opposite directions, but now they pass one another, like trains in the night, so to speak. Forces which move from the donkey to the cart, appear to combine in the harness and wagon tree, with those forces moving in the opposite direction from the cart to the donkey. There is an intimate dance of energy, where one force can be seen moving from left to right, while the other moves from right to left, as it were.

Between the donkey and cart we find there exists a transmission of forces, and with our new-found wisdom, we might also describe it as a transmission of energy. In other words, energy is being transmitted. It is the harness and wagon tree which is responsible for transmitting forces from the donkey to the cart, and from the cart to the donkey. Remember, these forces are not directly derived from an action-reaction pair, which means that they are not the result of an "interaction" between the donkey and the cart. No, the forces are derived from two seperate systems, namely: the donkey and the ground, and the cart and the ground. What we have are two independent sources for the forces, which means that the magnitude of the forces can differ - something that is not possible with a simple action-reaction pair. I think this gives a much more rounded explanation as to how the donkey is able to overcome the resistive force of the cart.

(I should point out this is not a new post. It's a few years old now. I have not re-edited it since then, even though, after just re-reading it, I feel that it could probably do with a bit of a polish. My sympathies to anyone seeking mathematical solutions to the problem, as truth be told, I simply do not possess the tools to describe them. (Must learn more math) But I might make the important distinction that even at zero acceleration, the donkey and cart should never be combined and viewed as an action-reaction pair. It remains to be seen that they will always be two seperate systems, that is the donkey/ground system and the cart/ground system, and that they just so happen to share zero acceleration when they are both at rest.)