Day 6

NEWTON’S LAWS

1 Introduction to Newton’s Principia

It is sarcely an exaggeration to say that from the eighteenth century up until thestart of the twentieth century, physics was Newton’s physics. By this I meanthat everything was based upon Newton’s laws, the subject of today’s session.Newton’s pubication of his book, the “Principia” in 1686 is a marker event isWestern history. It seemed to provide all the tools needed to understand every-thing about the physical world. Ocean tides, the paths of planets and cometsin space, the flight of arrows and artillary, the motion of billiard balls, spinningtops, everything seemed understandable in terms of Newton’s laws. In fact,some people went a bit overboard and began to think of nature, of the wholecosmos, as a vast “world machine” that obeyed Newton’s laws. Neweton, adeeply religious person, felt that by uncovering nature’s laws he was gaining adeeper understanding of the deity that rules the universe. The mathematicianLaplace wrote that if someone could know at any instant the speeds and po-sitions of all the particles in the universe and all of the forces acting on all ofthose particles, she would be able to use Newton’s laws to predict the entirehistory and future of the universe. Wow.

Karl Marx called his economic laws “laws of motion” and while Freud didn’tgo quite as far as that in his psychoanalytical theories, it was clear that hethought of the mind as a mechanism that was understandable if only the rightlaws could be formulated. Indeed this tremendous optimism about the capacityof humans to understand the world thorugh the application of Newton’s lawsengendered confidence that ultimately gave rise to the Englightenment. AndIsaac is indeed a hero of the Enlightment.

But, alas, along came Max Planck in 1900 and Einstein in 1905, and withthem Newtonian physics began to take its licks. In particular quantum mechan-ics showed that Newton’s laws simply fail to work when very small things (thesize of molecules or smaller) are being studied; Einstein’s theories of relativityshoweed that Newton’s laws become less and less accuate as the speed of theobjects under study gets closer and closer to the speed of light (3×108meter

sec ) oras one predicts the behavior of objects near massses (Even the mass of the earthis enough to upset very accurate Newtonian predictions). Alas, poor Isaac. Iknew him well. So will you. And the reason I am bothering you with seven-

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teenth centruy physics that has been superceded is that Newton’s laws remainextremely accurate and useful under a wide variety of circumstances. In all butthe most extreme cases (very small objects or calculations out to many decimalpoitns of accuracy) Newton still does a good job. So I still use Newton and sodo most physical scientists. Even when it is recognized thast Newton’s laws nolonger hold, physicists often still use Newton as a way of making a “first cut” ata poblem to understand what is going on in broad terms. Based on the Newton-ain results I can then decide whether the rough conditions are appropriate fora re-calculation using quantum physics or relativity. In other words, the stuff Iam going to present is old, it is not the most accurate physical theory avaialblebut it is extremely useful and important. 1

2 Statement of Newton’s Laws

What follows are translations of Newton’s laws from the original Latin text.2

1. Law 1 Every body continues in its state of rest or of uniform [unacceler-ated] motion in a right [straight] line unless it is compelled to change thatstate of motion by forces impressed upon it.

2. Law 2 The change in motion [momentum] is proportional to the motiveforce impressed; and is made in the direction of the right [straight] line inwhich that force is impressed.

3. Law 3 To every action there is always opposed an equal reaction: or,the mutual actions of two bodies upon each other are always equal anddirected to contrary parts.

2.1 Force

Take a book and set it down on a clear space on your desk. The book doesnot move on the desk (if your desk is level and there is no gale blowing in yourroom, that is). This profound fact needs discussion—at length. Suppose youwanted your book to move. How would you do it most simply? Go ahead, tryit. Unless you were being a smarty-pants, you probably pushed the book withyour finger and/or your thumb. If you pushed just a little, the book wouldn’tmove. You have to push hard enough.

Just what is a “push?” A physicist calls it a “force.” A force, by definition,is anything that causes an object to change its state of motion. By “state of

1For a stunning and brilliant discussion of Newton’s laws and the advent of relativity theorydo not miss Inside Relativity by D. E. Mook and Thomas Vargish (Princeton: PrincetonUniversity Press, 1985). Don’t wait for the movie version of the Broadway musical based onthe original. Despite all hopes of the authors neither of these seems likely to happen. You’llhave to read the book.

2From: Sir Isaac Newton’s Mathematical Principles of Natural Philosophy and His Systemof the World translated by Florian Cajori (Berkeley: University of California Press, 1966), p.13.

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motion,” I mean the speed of the object (once I start talking about things inmore than one dimension I’ll have a slightly more broad definition of “state ofmotion,” but “speed” will do nicely for now). For example, if the object wereinitially at rest, after a force acts the object will begin to move, it’s speed willchange, and its state of motion will change. Or, if the object were initiallymoving and a force acts, the object will speed up or slow down or even stop.Again, its speed changes; its state of motion changes. In fact (and in moregeneral terms), any time the speed of something changes, a force is said to“act” on the object to change the state of motion of the object. When youmoved the book on your desk you “exerted” a force on the book; a force (dueto your finger) “acted” on the book.

It’s interesting that while the word “force” and even the idea of a forceare pretty common (they have become a part of our everyday language), theconcept of a force is pretty darn abstract. After all, you don’t ever see a force;all you can “see” are the results of a force acting. You cannot pick up andfondle a force. You can’t weigh a force. You can’t really even imagine just whata force exactly is. As I say, it’s pretty darn abstract, but because it is familiar,it doesn’t seem to be as abstract as it really is. I’ll be talking about a lot ofequally abstract quantities. And because they are less familiar than “force,”they can seem pretty weird and even phoney. But every time I introduce astrange notion that you can’t see or weigh, just think about force.

Forces are measured in units called newtons (abbreviation: N) in the systemof units I am using. If you exert a force with enough newtons, your book onthe desk will begin to move (change its state of motion) on your desk. If youpush harder, you will exert a force of more newtons on your book and it willmove more rapidly (its state of motion will change even more rapidly). Goahead. Try this. Push with enough force to make the book move, then pusha lot harder. The book responds with a greater change in motion under theinfluence of a greater force. By messing around with your muscles for a longtime, you have acquired the ability to instruct them to exert forces of varyingmagnitudes—usually without even thinking about it.

Isaac Newton studied all of this and formulated a mathematical relationbetween the change in the state of motion (which I’ll define carefully in the nextsection) and the force that acts on an object. This relation is called “Newton’ssecond law,” and it bears careful discussion. But first laws first.

2.2 Newton’s First Law

When I first studied physics, I wondered why Newton had bothered with hisfirst law. Put simply, the law says that unless a force acts, an object will notchange its state of motion. If the object started out at rest, it will remain atrest unless a force acts on it. If the object started out moving at a certain speed,it will continue to move with that same speed unless and until a force acts. Allof this is duplicated in more detail (and with quantitative precision about whatis meant by a “change” in the state of motion) in the second law. So why thefirst law?

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Now Big Isaac was pretty smart and I knew deep down that he wouldn’t juststick in a first law unless it were really necessary, but it wasn’t until I studiedrelativity theory for the first time that I began to understand what the firstlaw is all about and I don’t know why someone didn’t tell me long before that.Permit me to try to enlighten you.

What it’s all about is a statement of the sort of reference frame in whichNewton’s laws will be valid. They are not valid in all reference frames. Let megive you a simple example that you can try for yourself with a pair of ice skatesand a stretch of ice.

Place some medium-sized object (say, your winter hat) somewhere in themiddle of the ice. Now skate away, stand still, and observe the cap for a while.You left it at rest and it stays at rest (echoes of the first law). In fact, if youwatch the cap and see it suddenly begin to move, you will “know” that someforce has acted on it to make it move. Maybe the wind has blown it, or a hockeypuck has struck it, or whatever. But you know that because the hat moved,that is, because it has changed its state of motion, a force had to act.

Now go really far away from the hat and begin to skate with a constantspeed directly toward the hat on the ice. Observe the cap carefully (also watchfor other skaters in your path!). You see the hat moving with respect to you,but the speed of the cap with respect to you does not change; it keeps movingwith the same speed and direction it assumed as soon as you started to skateat a constant speed and in a constant direction with respect to the ice.

In fact, if you were to skate with a constant speed and direction with respectto the ice and suddenly saw the hat jump or change its speed with respect toyou, you would conclude that some force had to act on the object (the first lawagain).

So Newton’s first law makes perfect sense as long as you and the hat are atrest on the ice or you are moving at a constant speed with respect to the ice.

But now skate so that you undergo a constant acceleration directly towardthe hat, observing it as you skate. Now the cap accelerates with respect to you,it changes its state of motion with respect to you—but no force is acting on thehat. Newton’s first law is not valid in this case. And the invalidity of the firstlaw is due to the fact that you have chosen to make your observations of thecap from an accelerating coordinate system. That is the real content of the firstlaw. You must make your observations from a system in which the objects understudy do not change their states of motion (do not accelerate) unless and until aforce acts on the bodies. In simpler terms, you should not apply Newton’s lawsin accelerated reference systems. An unaccelerated reference system, in whichNewton’s laws are valid, is called an inertial reference frame. Thus, the firstlaw really says, “OK folks, all the laws of motion will hold in inertial referenceframes. If the reference frame you pick isn’t inertial, all bets are off!”

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2.3 Newton’s Second Law

2.3.1 What’s Momentum

When Newton formulated his second law in the 17th century, he stated it asa direct proportionality between the force acting on an object and somethinghe called the “quantity of motion.” Newton’s “quantity of motion” is whatphysicists today call momentum.

I will have a lot more to say about momentum later because it is a veryuseful quantity for describing how processes take place in the world. For now,however, I only need to understand the mathematical definition of momentumin one dimension to see how it relates to the force acting on an object.

The momentum of an object in one-dimensional motion is defined as theproduct of the object’s speed and a property of the object called its inertialmass:

Momentum = ms

The inertial mass m is something new. The inertial mass of an object can bethought about very roughly as a measure of the “amount of matter” presentin the object. More accurately, the inertial mass measures the reluctance of anobject to undergo a change in its state of motion or its momentum.

So the momentum is defined as the product of the inertial mass of an objectand the speed of the object. I will use the symbol p for the momentum. That’sthe standard symbol. p for Momentum. Well, hey there’s p for pneumatic andp for pseudo and p for psychology, so why not p for momentum? What do youthink of that?

I’m now ready to look at Newton’s second law.

2.3.2 Newton’s Second Law

Here’s the big second:

F =dp

dt(1)

where F is the net force acting on an object and p is the object’s momentum.So according to Newton, the net force acting on an object is equal to its timerate of momentum change. This is Newton’s second law. For many purposes Ican use a more simple version of this law, as I will show in the next section.

2.3.3 The Usual and Customary Thing

I’m going to use the quantitative expression of Newton’s second law and sub-stitute the definition of momentum into that expression:

F =dp

dt=

d(ms)

dt(2)

F =dm

dts+m

ds

dt=

dm

dts +ma (3)

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and in situations where the mass of the system does not change,

dm

dt= 0 (4)

and soF = ma (5)

for this special case. This is the form of Newton’s second law that I will usemost frequently, but I always remember (well, I try to remember) that this is aspecial case of Newton’s law, in which the mass of the system does not change.

2.3.4 Have I Just Perpetrated a Swindle?

Ignoring mass variation isn’t really a sleazy trick. For most situations I willencounter in engineering and in most physics problems involving large (largerthan molecules) objects, variation in mass will not be an issue. (One importantexception to this is the analysis of rocket flight.) For the present, bear in mindthat I am making an assumption whenever I use the

F = ma

form of Newton’s law. Once you have studied relativity and quantum theoryyou will see that even in ordinary situations it is not quite correct to say thatthe mass of an object is constant in ordinary situations. But the amount ofmass variation is so tiny that it will make little or no difference to any usualcalculation.

2.4 Newton’s Third Law

I don’t know about other people, but the first time I saw this law I thought“ho-hum” and it all sounded pretty lame to me. However, as I have gone on tostudy more and more physics and since I’ve had time to think carefully aboutit, this law is the most mysterious of all to me (why is it that when you are“studying” something in school, you never seem to have enough time to reallythink about it at all? That’s crazy). It is also the one law I most easily forgetto apply when I am analyzing a situation. In other words, this one is slipperyand seems deceptively lame.

Put simply, the law says that forces never occur in isolation in the universe.They always occur in pairs, and you can’t have one force without another.Hmmm. Still seems lame when I say it. But it really isn’t and I guess the onlyway I came to fully appreciate it was by working lots of problems.

When I exert a force on anything (say, with my finger), the thing I push onor pull on will always exert an exactly equal and opposite force on me (on myfinger). Try this: clean off that spot on your desk again and put a good bookthere. Now push on the book. It matters not whether you push hard enoughto move the book, just push with any force at all, but close your eyes when youdo it and pay close attention to what the nerves in your finger are telegraphing

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to your consciousness. You “feel” the book push on you. No kidding. Try itand see. The book pushed on you, and because you are in a position of greatercontrol of the situation than is the book, you know that the book pushed onyou because you pushed on the book.

Now get up and go over to the wall of your room. Push on it. Again closeyour eyes and feel what your nerves are telling you about the world. The wallis pushing on you. And the harder that you push on the wall, the harder youwill feel the wall push on you.

This is really quite amazing. No matter what force you exert on anything,the “anything” is guaranteed to exert an exactly equal and opposite force onyou. That is what Newton’s third says.

As I say, it’s an easy thing to forget.

3 Force Diagrams or “Free Body” Diagrams

3.1 What is a Free Body Diagram?

Free body diagrams are created to make clear the forces which act on a mass.They are not meant to be realistic renderings of the way an object looks. Theyare highly abstract or “schematic” diagrams showing each mass under consid-eration as a point, and the forces acting on each mass as arrows with their tailsfixed on the point and the arrowhead pointing in the direction of the force.Becoming expert at doing these free body diagrams will be very helpful to yourunderstanding of physics. It is an essential skill. You are going to becomeanexpert at doing free body diagrams. That’s right, an expert. Trust me.

For example, a problem might involve a bird. Figure 1 is a bird standing onthe floor (could this be the bluebird of happiness?

Figure 1: A bird on the floor.

From this I will abstract a free body diagram. I start with a point to rep-resent the bird. This is a one-dimensional problem so I will be concerned onlywith forces along the vertical direction.

Figure 2: A point representing the bird.

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I’ll then draw all of the vertical forces acting on the bird. But what arethe forces acting here? This question must be asked every time you confronta mechanical situation; it is absolutely central to understanding mechanicalsituations. And it is in addressing this question that the free body diagram cando the most good in clarifying the underlying physics of a mechanical system.The free body diagram helps me to remember to account for all of the forceswhich act on an object.

In the present case, I know that the bird is pulled to the floor by gravity,and so I draw the gravitational force on my diagram as an arrow whose tail isanchored in the dot representing the bird and whose head is pointing downward.I won’t ever get stuck by wondering where to start my free body diagrams whenI am dealing with any object on or near the earth because I know that the forceof gravity is acting, and I always begin with that force.

weight

Figure 3: The weight added to the representation of the bird

What I have just done for the force of gravity, I now do for any and all otherforces acting on this bird sitting on the floor. But what other forces act? Needthere be any others? Maybe gravity is the whole story. Yet there must be atleast one additional force acting on this bird, and an experiment you will do inthe next section will explain why.

3.2 The Book on the Table Revisited

3.2.1 Gravity

Time to place your favorite book on a clear spot on your desk again. It willlook something like Figure 4. 3

From this diagram, my job is to create a free body diagram which describesall the forces acting on the book. I begin with the book itself, represented as apoint in Figure 5

Then I add the force of gravity (also called the “weight” of the book) inFigure 6.

3Lies, lies, lies. This book is full of them. Did you really buy into that last sentence? Lookat the page and read that sentence again. The blob of green in the figure looks nothing likeyour table and the bluish blob looks nothing like the book. Once again I have depended onyour cultural background to make you imagine that these colored areas somehow “look like”or “represent” real, tangible, three-dimensional objects. This business of “representation” isat the heart of physics and that’s why I keep coming back to it whenever I can.

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Figure 4: A book lying on a table.

Figure 5: A point representing the book.

Notice that this free body diagram looks identical to the diagram I drewfor the bird. The bird and the book are two utterly different objects in verydifferent situations, yet their free body diagrams look identical. This is becausethe situations of the forces acting on the book and the bird are identical. Thepower of free body diagrams resides in this ability to extract the essence ofthe force configuration from a system, no matter what or how complicated thesystem may be. I’m going to show you that once the force configuration hasbeen obtained, the analysis of the motion of the system is a straightforwardmathematical exercise.

Look again at the diagram of force on the book. I have only one force actingso far: the weight. But this cannot be the only force involved. Why? If theforce of gravity were the only force acting on the book, then that would be thenet force acting on the book. Yet, Newton’s second law tells me that

Fnet = ma

so there would be a non-zero acceleration corresponding to the non-zero netforce. But I bet your book is just sitting there (unless you’re messing with it insome way, and if you are, please cease and desist so the experiment isn’t screwedup).

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weight

Figure 6: The weight added to the point representing the book.

3.2.2 The Normal Force

Still, gravity acts on the book. Yet if the acceleration of the book is zero, thenet force on the book must be zero, and if net force on the book is zero, theremust be some other force counteracting gravity. I’m going to draw such a forcein the free body diagram, and I’ll label the force N .

If things are arranged so that N = −weight, I have

Fnet = 0

and so a = 0 as observed. Notice that N must be directly upward (exactlyopposite Fweight), otherwise the weight would not be cancelled out exactly.

weight

N

Figure 7: The normal force added to the diagram.

By using a free body diagram, I have learned something which was notobvious before: there must be an upward force acting on the book. Now forcesdon’t just “happen.” Forces are always caused by some agent. In the case of“weight,” I know that it is the gravitational attraction of the earth for the bookthat causes the weight. But what about N ? What “causes” this force?

First observe that N acts only as long as the book is in contact with thetable top. Try it. Lift the book up in the air a centimeter or so and let thebook go. You’ll see that the book accelerates downward (because the weight isacting down) but then the book seems to stop accelerating downward as soonas it hits the table top. In other words, N is “caused” by contact with the table

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top. N is called a “contact” force for this reason. Because it acts in a directionnormal to the plane of contact between the book and the table, this force is alsocalled a normal force of contact, hence the letter “N” I used to name it.

So the table exerts a force on the book sitting on the table. We can saythat the table “exerts” the normal force N on the book. But that really doesn’tanswer the question concerning the origin of this force. How does the table exertthis force on the book? What mechanism is operating to generate the force assoon as the book touches the table but not before?

The normal force of contact is electrical in nature and it arises from theclouds of electrons surrounding the molecules of the table surface as they interactwith the clouds of electrons surrounding the molecules in the book surface.When these electron clouds get close enough to one another they begin to repelone another electrically. I will have much to say about the electrical force later;for now the lesson is this: whenever two objects come into contact with oneanother, a force will act on the objects at the point or points of contact and thisforce will be normal to the area of contact.

By the way, I must point our a rather wonderful aspect of this normal force.Try this: place annother book on top of the one on your desk. What ahppens?Nothing, right? So the normal force has now changed. Instead of being equalto the weight of the original book it si now equal to the weight of both books.Add a thrid or a fourth book. That normal force will always know just how bighit has to be to exactly balance the weight of stuff you put onyour desk. Thatreally is quite wonderful. Of course you couldn’t take it to extremes, If youput a campus police cruiser on your desk I bet it would not produce a normalforce equal to the crujiser’s weight and hold it up off the floor. But while thenormal force lasts, it is really quite amazing the way it adjusts automatically towhatever you load on the desk. Magic stuff here.

3.3 Walking

3.3.1 The Experiment

Enough of books on desks. Time to try a whole-body, kinesthetic physics ex-perience. Please follow carefully. Stand up and walk across the room. That’sright, just walk. Go ahead, try it!

Now do it again, only this time pay close attention to the forces your body“feels” as it is accelerated from a state of rest to one of motion, and then againfrom its state of motion to one of rest as you come to a stop. Since we aredoing physics in just one dimension for a while, I will be concerned only withhorizontal forces, the ones that make you walk across the floor.

This will take several free-body diagrams to describe because there are sev-eral forces acting and at least two distinct processes.

3.3.2 Phase I: The Start-Up

I begin with the free body diagram describing the situation when you are stand-ing but still at rest on the floor.

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weight

N

Figure 8: The weight and normal force added to point representing you standingon the floor.

This one should be looking pretty familiar by now. It’s identical to everyother free body diagram I have drawn for an object at rest on some surface.This redundancy is correct because in terms of the forces acting, the situationsare identical.

But now I am going to forget what is going on in the vertical dimension andconcentrate only on the horizontal direction (one-dimensional physics, right?).

So start walking. I’ll assume that the direction you walk is to the right in mydiagram. You were initially at rest; you then began to walk, so your momentumchanged.

According to Newton’s second law, a force had to act in the direction of youracceleration. I’ll call this force a “push.”

push

Figure 9: The “pushing” force added to the diagram.

This diagram accounts for the fact that you accelerate across the floor (thepush acts on your mass to accelerate it).4 Please don’t make a mistake commonin people like me just starting to use free body diagrams: do not add arrows tothe diagram that are not forces known to act on the body. For example, somefolks might like to draw an arrow on the diagram to indicate the “motion” ofthe object, or the direction of motion, or something like that. Don’t you do it!Not only is it bad form, it quickly leads to error. I have drawn an arrow onmy free body diagram of you walking that I called “push” meaning that thisarrow represents a force acting on the object. I know that some force acts inthis direction because the object is accelerating in this direction.

4To be completely accurate in my diagram, I would have to add in a backward force dueto friction between your body and the air as you move through the air, but the air frictionforce is tiny and I will ignore it.

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Another point. I am greatly simplifying the act of walking in this discussion.In reality, your body is lifted up and dropped down a bit as you walk. Thereare rotational motions involved, too—all very complicated. But for a first cutat understanding the act of walking, and especially since I am presently doingphysics in one dimension, I will only pay attention to the horizontal force I amcalling “push” that causes you to accelerate from a state of rest.

3.3.3 Detour for a Bit of Wisdom, Part I : On Ignoring Things

Wait a minute! Not so fast. A little sneaky item I just tried to slip by you inthe last footnote bears discussion (“To be completely accurate in my diagram,I would have to add in a backward force due to friction between your body andthe air as you move through the air, but the air friction force is tiny and I willignore it”). Notice that with great cunning I burried this in a footnote to makeit easier to slip by you. Heh, heh. Gtta watch me—sneaky all the way to the lab.Many physics books sneak this fact by, the way I tried to do, because they don’twant to “interrupt the flow” of the presentation. I feel that the interruption isnot only important, it is downright necessary—but not necessarily on the firstreading. If you’re inclined, read on, otherwise go ahead and finish the rest ofthis section and then come back to review this bit of wisdom.

Notice again what was said in the parentheses: “. . . but the air friction forceis tiny and I will ignore it.” The assertion of “tininess” is the key. Ignoring theforce seems to be a perfectly acceptable thing to do if, by the force being “tiny,”I mean that it is so small that its inclusion in the calculated acceleration wouldhave no noticeable effect on my analysis, given the accuracy of my calculation(given the number of decimal places I choose to use in the arithmetic). Buthow can this assumption of tininess be justified? The simple answer is thatexperience has taught me that even if I were to include the air friction force ina case like this, my final result will be no different than if I had left it out. So Ichoose to ignore it.

In this case, experience has given me justification for ignoring somethingwhich I know to be present in the physical situation, because I also know itwill not greatly alter my conclusions if I include it. This sort of experience isvaluable, but it can also be a bit tricky, because sometimes I ignore somethingon purpose, thinking that it is too small to worry about, only to find later thatit makes all the difference in the world. Other times I ignore things that arevery important because I just forget them. That’s an erro. I try to avoid errors,of course, but alas . . . .

Another point about ignoring things: often I have to analyze a problemwhich is completely new to me, and I have no direct experience with whataspects of the problem may be negligible. In that case, I might consult sometexts or research papers or even a colleague or six to see if anyone else has anyexperience I can use to simplify the problem in a realistic way.

Then there are situations where no experience is available, whether directlyor indirectly. Indeed I can be working on a problem which nobody has evertackled before, perhaps in some realm of physics totally beyond direct human

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experience (astrophysicists do this all the time when they apply physics to su-pernovae, the structure of galaxies or the national debt). So in such cases, do Iinclude every conceivable force in my free body diagram? Often to do so wouldbe the kiss of death for solving the problem. As I will show before too long,if I add enough forces to the diagram, the mathematics can quickly becomenightmarish—or even downright insoluble. When this is a possibility, or whenI want to make a “first pass” at a problem to see how big various forces canbe, I use what is called “intuition” to guide the process of simplification. This“intuition” is hard to define; I continue to develop it over a period of time byworking lots of physics problems and reading lots about other people’s solutionsof problems. Some people are extremely good at intuition—that is, at knowingjust what I can ignore and what I can’t. Indeed some physicists are very famousfor it. I’m not famous for anything.

There will be many situations in which my intuition is already just fine toguide me—for example, you were probably not deeply offended by my suggestionthat I ignore air friction as I analyze your walk across the floor. You and I havedone lots of walking and you know that the air friction (the “wind resistance”)you encounter when walking in a room is very small; you can literally “feel” thiswith the nerves in your skin. In such cases, after a little thought, you will havea pretty good idea of what to ignore or what not to ignore. But there will besome other cases where you may not be at all sure. I will try to be good aboutsupplying some suggestions about what you can and cannot ignore. Unless Imess up.

3.3.4 Back to Phase I

Back to my discussion of the free body diagram of your walk. What about thatforce I have labeled “push?” What is the origin of this force? This is a veryimportant point, so please read carefully and think about it as much as youneed to.

Whatever the force that causes the “push” on your body, I know two thingsabout it right away:

1. it acts on your body

2. it acts in the direction in which you want to walk (that is, the directionof your acceleration).

Clearly you have some control over this force. You can change the directionof the force by changing the way your leg muscles act. You can also changethe magnitude of the force by changing the degree to which you exert your legmuscles—so you might be tempted to think that it is your leg muscles that arepushing you. In other words, you might think that the “push” is a “self-push.”You are literally “forcing yourself” to walk.

But it’s not that simple. Try this: stand up and grab your right wrist withyour left hand and pull your right hand foreward with your left hand. Clearlyin this case your right hand is being pulled by your left. Do you move forward?

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Your right arm does, but not your whole body. Indeed it seems that only byexerting a force with your legs can you walk forward. So what is so magic aboutyour legs that they can make you move forward? Tugging on any part of yourbody with your hands won’t do it.

Next try this: get down on your hands and knees. Now you can move forwardby either exerting your leg or your arm muscles. So when you are standing, yourleg muscles are the only ones capable of moving you; when you are down on allfours, your arms or your legs are capable of moving you along.

I’m not going to ask you to stand on your hands, but if you can, you shouldtry it. If you can’t, take my word for it,5 you can move across the floor byexerting your arm muscles; your leg muscles are useless as far as moving yourbody across the floor is concerned when you are standing on your hands.

So now I know something else about the force I have called “push” in thefree body diagram: it is associated with contact with the floor by some partof your body. Only by exerting a force in some limb which is in contact withthe floo can you accelerate yourself along the floor. So the floor is key to youracceleration.

If it’s winter outside when you are reading this, the next experiment will befun to try; if it isn’t winter, you’ll just have to wait for some ice to show up onthe sidewalk or be satisfied with what I am about to tell you. If you try to walkon perfectly smooth ice when you are wearing perfectly smooth shoes, it won’twork.

You can exert all the forces you want with your legs; you’ll only flail aroundon the ice and you won’t be able to move forward.

So it’s not your leg muscles (or arm muscles or whatever muscles) alone that“force” you to walk. The force I call “push,” which I know must be there inthe free body diagram when you accelerate, has to come from something else.It is related to what you do with your muscles, but unless things are right withthe floor, you won’t walk. The floor itself plays a key role. Understandingthis point and the content of the next section is absolutely essential for yourunderstanding of Newtonian physics. Reread it and think about it and talkabout it with others as much as you need to until you understand the pointclearly.

3.3.5 Detour for a Bit of Wisdom, Part II : Newton’s Third Law

I promise that I’ll get back to the problem of walking on the floor soon, but tomake what I am about to tell you as simple as possible, I first want to go backto the experiment with the book on the table. Place the book on the table andpush with your finger—one finger please. Now close your eyes and repeat theexperiment and note carefully what your nerves in your finger tell you is goingon.

5I can’t stand on my hands. I’ve never even been able to turn a cartwheel. But I stillbeleive what I am saying here even though I have never tried it myself. I beleive and trustthe physics so that I am sure that what I am saying here is so.

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You feel the book pushing on your finger as you push the book. Furthermore,notice that as you push the book one way, your nerves feel the book pushingthe opposite way on your finger. We could summarize this by saying:

If you decide to push on the book, then the book will push on you in theopposite direction.

And that, in the high falutin’ terminology of physics, is the first part ofNewton’s third law. The second part of the law isn’t so easy for you to detectwith your fingers, but if you were to instrument your finger and the book sothat you could measure the magnitude of the forces involved, you’d find thathowever strongly you push on the book, the book will push back on you withan exactly equal but opposite force.

Now try walking again. If you carefully pay attention to your feet, you’llfind that however strongly you push on the floor with your leg muscles, the floorwill push in the opposite direction on your feet. It is this backward force bythe floor on your feet that is the “push” force in the free body diagram of youwalking. It is the floor that makes you walk forward by pushing on you. If youtry to walk on ice, there is no friction between your feet and the ice, so theice can not push back on your feet and you do not push on the ice—you don’tmove.

According to Newton’s third law, there is no such thing as a single isolatedforce. Forces, any forces, always occur in couples. Whenever any object inthe universe exerts a force on a second object, the second exerts an equal andopposite force on the first.

Why this is so is not clear. The fact is that Newton’s third law does describewhat we observe to be the case in the world, so, like Newton’s second law, weaccept it as “true” for the purposes of doing physics.

3.3.6 Phase II: The Slow-Down

This phase of the walk will be a piece of cake after my discussion of the firstphase. I’ll go through it, however, because it will be good practice at applyingNewton’s laws and free body diagrams.

So now you’re walking along the floor (to the right in my diagram) and youexert the necessary leg muscle to slow yourself down. This means that accordingto Newton’s second law, the force that the floor exerts on you must act in theleftward direction.

This new push slows you down. Again it is a force exerted by the floor onyou.

push

Figure 10: The push from the floor that slows you down.

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3.4 Bootstrapping

When I was a kid, I used to wonder why, if I could lift up a book and carryit home from school, I couldn’t also lift myself up and carry myself home fromschool without touching the ground. I don’t know whether I tried it, althoughI suspect I did. It didn’t work. I do know that I couldn’t understand why for along time.

The key to understanding why I can’t bootstrap myself up in the air can befound in the free body diagram of me. I begin with me standing on the floor:It’s just the same old diagram I have seen so often whenever I have an object

weight

N

Figure 11: The normal force and weight acting on me as I stand on the floor.

sitting on the floor (or on anything else for that matter.Now I’ll add a force showing that my right hand has reached around and is

pulling upward on my left arm pit, so as to lift me up in the air. But wait a

weight

Npull

react

Figure 12: My pull upward and the reaction force to this pull added to thefigure.

minute. What’s that downward force that suddenly appeared in the diagram?

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That downward force has to be there. If my arm exerts an upward force onmy arm pit, according to Newton’s third law, my arm pit must exert an equaland opposite (downward) force on my hand. And since both my arm pit andmy hand are parts of my body, the net result is that equal and opposite forcesare exerted on my body and they cancel each other out. I can’t pull myself upin the air by exerting an upward force on my body. Shucks. I think it wouldbe really neat to carry myself home aboout a foot above the ground, especiallywhen it is rainy and there are lots of puddles.

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