Download 1 Introduction to Newton`s Principia

Day 6
NEWTON’S LAWS
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Introduction to Newton’s Principia
It is sarcely an exaggeration to say that from the eighteenth century up until the
start of the twentieth century, physics was Newton’s physics. By this I mean
that 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 is
Western history. It seemed to provide all the tools needed to understand everything about the physical world. Ocean tides, the paths of planets and comets
in space, the flight of arrows and artillary, the motion of billiard balls, spinning
tops, 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 whole
cosmos, as a vast “world machine” that obeyed Newton’s laws. Neweton, a
deeply religious person, felt that by uncovering nature’s laws he was gaining a
deeper understanding of the deity that rules the universe. The mathematician
Laplace wrote that if someone could know at any instant the speeds and positions of all the particles in the universe and all of the forces acting on all of
those particles, she would be able to use Newton’s laws to predict the entire
history and future of the universe. Wow.
Karl Marx called his economic laws “laws of motion” and while Freud didn’t
go quite as far as that in his psychoanalytical theories, it was clear that he
thought of the mind as a mechanism that was understandable if only the right
laws could be formulated. Indeed this tremendous optimism about the capacity
of humans to understand the world thorugh the application of Newton’s laws
engendered confidence that ultimately gave rise to the Englightenment. And
Isaac is indeed a hero of the Enlightment.
But, alas, along came Max Planck in 1900 and Einstein in 1905, and with
them Newtonian physics began to take its licks. In particular quantum mechanics showed that Newton’s laws simply fail to work when very small things (the
size of molecules or smaller) are being studied; Einstein’s theories of relativity
showeed that Newton’s laws become less and less accuate as the speed of the
objects under study gets closer and closer to the speed of light (3 × 108 meter
sec ) or
as one predicts the behavior of objects near massses (Even the mass of the earth
is enough to upset very accurate Newtonian predictions). Alas, poor Isaac. I
knew 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 remain
extremely accurate and useful under a wide variety of circumstances. In all but
the most extreme cases (very small objects or calculations out to many decimal
poitns of accuracy) Newton still does a good job. So I still use Newton and so
do most physical scientists. Even when it is recognized thast Newton’s laws no
longer hold, physicists often still use Newton as a way of making a “first cut” at
a poblem to understand what is going on in broad terms. Based on the Newtonain results I can then decide whether the rough conditions are appropriate for
a re-calculation using quantum physics or relativity. In other words, the stuff I
am going to present is old, it is not the most accurate physical theory avaialble
but it is extremely useful and important. 1
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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 [unaccelerated] motion in a right [straight] line unless it is compelled to change that
state of motion by forces impressed upon it.
2. Law 2 The change in motion [momentum] is proportional to the motive
force impressed; and is made in the direction of the right [straight] line in
which 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 and
directed to contrary parts.
2.1
Force
Take a book and set it down on a clear space on your desk. The book does
not move on the desk (if your desk is level and there is no gale blowing in your
room, that is). This profound fact needs discussion—at length. Suppose you
wanted your book to move. How would you do it most simply? Go ahead, try
it. Unless you were being a smarty-pants, you probably pushed the book with
your finger and/or your thumb. If you pushed just a little, the book wouldn’t
move. 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
1 For a stunning and brilliant discussion of Newton’s laws and the advent of relativity theory
do not miss Inside Relativity by D. E. Mook and Thomas Vargish (Princeton: Princeton
University Press, 1985). Don’t wait for the movie version of the Broadway musical based on
the original. Despite all hopes of the authors neither of these seems likely to happen. You’ll
have to read the book.
2 From: Sir Isaac Newton’s Mathematical Principles of Natural Philosophy and His System
of 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 in
more than one dimension I’ll have a slightly more broad definition of “state of
motion,” but “speed” will do nicely for now). For example, if the object were
initially at rest, after a force acts the object will begin to move, it’s speed will
change, and its state of motion will change. Or, if the object were initially
moving 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 more
general 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 you
moved the book on your desk you “exerted” a force on the book; a force (due
to your finger) “acted” on the book.
It’s interesting that while the word “force” and even the idea of a force
are pretty common (they have become a part of our everyday language), the
concept 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 and
fondle a force. You can’t weigh a force. You can’t really even imagine just what
a 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 of
equally abstract quantities. And because they are less familiar than “force,”
they can seem pretty weird and even phoney. But every time I introduce a
strange notion that you can’t see or weigh, just think about force.
Forces are measured in units called newtons (abbreviation: N) in the system
of units I am using. If you exert a force with enough newtons, your book on
the desk will begin to move (change its state of motion) on your desk. If you
push harder, you will exert a force of more newtons on your book and it will
move more rapidly (its state of motion will change even more rapidly). Go
ahead. Try this. Push with enough force to make the book move, then push
a lot harder. The book responds with a greater change in motion under the
influence of a greater force. By messing around with your muscles for a long
time, you have acquired the ability to instruct them to exert forces of varying
magnitudes—usually without even thinking about it.
Isaac Newton studied all of this and formulated a mathematical relation
between the change in the state of motion (which I’ll define carefully in the next
section) and the force that acts on an object. This relation is called “Newton’s
second 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 his
first law. Put simply, the law says that unless a force acts, an object will not
change its state of motion. If the object started out at rest, it will remain at
rest 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. All
of this is duplicated in more detail (and with quantitative precision about what
is meant by a “change” in the state of motion) in the second law. So why the
first law?
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Now Big Isaac was pretty smart and I knew deep down that he wouldn’t just
stick in a first law unless it were really necessary, but it wasn’t until I studied
relativity theory for the first time that I began to understand what the first
law 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 which
Newton’s laws will be valid. They are not valid in all reference frames. Let me
give you a simple example that you can try for yourself with a pair of ice skates
and a stretch of ice.
Place some medium-sized object (say, your winter hat) somewhere in the
middle 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 you
watch the cap and see it suddenly begin to move, you will “know” that some
force has acted on it to make it move. Maybe the wind has blown it, or a hockey
puck 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 constant
speed directly toward the hat on the ice. Observe the cap carefully (also watch
for 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 moving
with the same speed and direction it assumed as soon as you started to skate
at 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 respect
to the ice and suddenly saw the hat jump or change its speed with respect to
you, you would conclude that some force had to act on the object (the first law
again).
So Newton’s first law makes perfect sense as long as you and the hat are at
rest 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 toward
the 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 the
hat. Newton’s first law is not valid in this case. And the invalidity of the first
law is due to the fact that you have chosen to make your observations of the
cap from an accelerating coordinate system. That is the real content of the first
law. You must make your observations from a system in which the objects under
study do not change their states of motion (do not accelerate) unless and until a
force acts on the bodies. In simpler terms, you should not apply Newton’s laws
in accelerated reference systems. An unaccelerated reference system, in which
Newton’s laws are valid, is called an inertial reference frame. Thus, the first
law really says, “OK folks, all the laws of motion will hold in inertial reference
frames. If the reference frame you pick isn’t inertial, all bets are off!”
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2.3
2.3.1
Newton’s Second Law
What’s Momentum
When Newton formulated his second law in the 17th century, he stated it as
a direct proportionality between the force acting on an object and something
he called the “quantity of motion.” Newton’s “quantity of motion” is what
physicists today call momentum.
I will have a lot more to say about momentum later because it is a very
useful quantity for describing how processes take place in the world. For now,
however, I only need to understand the mathematical definition of momentum
in 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 the
product of the object’s speed and a property of the object called its inertial
mass:
M omentum = ms
The inertial mass m is something new. The inertial mass of an object can be
thought about very roughly as a measure of the “amount of matter” present
in the object. More accurately, the inertial mass measures the reluctance of an
object 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 object
and the speed of the object. I will use the symbol p for the momentum. That’s
the standard symbol. p for Momentum. Well, hey there’s p for pneumatic and
p for pseudo and p for psychology, so why not p for momentum? What do you
think 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:
dp
(1)
dt
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 time
rate of momentum change. This is Newton’s second law. For many purposes I
can use a more simple version of this law, as I will show in the next section.
F =
2.3.3
The Usual and Customary Thing
I’m going to use the quantitative expression of Newton’s second law and substitute the definition of momentum into that expression:
dp
d(ms)
=
dt
dt
(2)
dm
ds
dm
s+m
=
s + ma
dt
dt
dt
(3)
F =
F =
5
and in situations where the mass of the system does not change,
dm
=0
dt
(4)
F = ma
(5)
and so
for this special case. This is the form of Newton’s second law that I will use
most frequently, but I always remember (well, I try to remember) that this is a
special 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 will
encounter in engineering and in most physics problems involving large (larger
than molecules) objects, variation in mass will not be an issue. (One important
exception to this is the analysis of rocket flight.) For the present, bear in mind
that I am making an assumption whenever I use the
F = ma
form of Newton’s law. Once you have studied relativity and quantum theory
you will see that even in ordinary situations it is not quite correct to say that
the mass of an object is constant in ordinary situations. But the amount of
mass variation is so tiny that it will make little or no difference to any usual
calculation.
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 to
study more and more physics and since I’ve had time to think carefully about
it, 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 really
think about it at all? That’s crazy). It is also the one law I most easily forget
to apply when I am analyzing a situation. In other words, this one is slippery
and 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 only
way 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 on
or pull on will always exert an exactly equal and opposite force on me (on my
finger). Try this: clean off that spot on your desk again and put a good book
there. Now push on the book. It matters not whether you push hard enough
to move the book, just push with any force at all, but close your eyes when you
do 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 it
and see. The book pushed on you, and because you are in a position of greater
control of the situation than is the book, you know that the book pushed on
you because you pushed on the book.
Now get up and go over to the wall of your room. Push on it. Again close
your eyes and feel what your nerves are telling you about the world. The wall
is pushing on you. And the harder that you push on the wall, the harder you
will 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 on
you. That is what Newton’s third says.
As I say, it’s an easy thing to forget.
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3.1
Force Diagrams or “Free Body” Diagrams
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. They
are highly abstract or “schematic” diagrams showing each mass under consideration as a point, and the forces acting on each mass as arrows with their tails
fixed 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 your
understanding of physics. It is an essential skill. You are going to become
anexpert 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 on
the 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 represent the bird. This is a one-dimensional problem so I will be concerned only
with 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 are
the forces acting here? This question must be asked every time you confront
a mechanical situation; it is absolutely central to understanding mechanical
situations. And it is in addressing this question that the free body diagram can
do 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 forces
which 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 is
anchored 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 when
I am dealing with any object on or near the earth because I know that the force
of 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 other
forces acting on this bird sitting on the floor. But what other forces act? Need
there be any others? Maybe gravity is the whole story. Yet there must be at
least one additional force acting on this bird, and an experiment you will do in
the next section will explain why.
3.2
3.2.1
The Book on the Table Revisited
Gravity
Time to place your favorite book on a clear spot on your desk again. It will
look something like Figure 4. 3
From this diagram, my job is to create a free body diagram which describes
all the forces acting on the book. I begin with the book itself, represented as a
point in Figure 5
Then I add the force of gravity (also called the “weight” of the book) in
Figure 6.
3 Lies, lies, lies. This book is full of them. Did you really buy into that last sentence? Look
at the page and read that sentence again. The blob of green in the figure looks nothing like
your table and the bluish blob looks nothing like the book. Once again I have depended on
your cultural background to make you imagine that these colored areas somehow “look like”
or “represent” real, tangible, three-dimensional objects. This business of “representation” is
at 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 drew
for the bird. The bird and the book are two utterly different objects in very
different situations, yet their free body diagrams look identical. This is because
the situations of the forces acting on the book and the bird are identical. The
power of free body diagrams resides in this ability to extract the essence of
the force configuration from a system, no matter what or how complicated the
system may be. I’m going to show you that once the force configuration has
been obtained, the analysis of the motion of the system is a straightforward
mathematical exercise.
Look again at the diagram of force on the book. I have only one force acting
so far: the weight. But this cannot be the only force involved. Why? If the
force of gravity were the only force acting on the book, then that would be the
net 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 net
force. But I bet your book is just sitting there (unless you’re messing with it in
some way, and if you are, please cease and desist so the experiment isn’t screwed
up).
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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, the
net force on the book must be zero, and if net force on the book is zero, there
must be some other force counteracting gravity. I’m going to draw such a force
in 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 (exactly
opposite Fweight ), otherwise the weight would not be cancelled out exactly.
N
weight
Figure 7: The normal force added to the diagram.
By using a free body diagram, I have learned something which was not
obvious before: there must be an upward force acting on the book. Now forces
don’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 book
that 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 the
table top. Try it. Lift the book up in the air a centimeter or so and let the
book go. You’ll see that the book accelerates downward (because the weight is
acting down) but then the book seems to stop accelerating downward as soon
as 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 direction
normal to the plane of contact between the book and the table, this force is also
called 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 say
that the table “exerts” the normal force N on the book. But that really doesn’t
answer the question concerning the origin of this force. How does the table exert
this force on the book? What mechanism is operating to generate the force as
soon as the book touches the table but not before?
The normal force of contact is electrical in nature and it arises from the
clouds of electrons surrounding the molecules of the table surface as they interact
with the clouds of electrons surrounding the molecules in the book surface.
When these electron clouds get close enough to one another they begin to repel
one 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 one
another, a force will act on the objects at the point or points of contact and this
force 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 equal
to 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 bigh
it has to be to exactly balance the weight of stuff you put onyour desk. That
really is quite wonderful. Of course you couldn’t take it to extremes, If you
put a campus police cruiser on your desk I bet it would not produce a normal
force equal to the crujiser’s weight and hold it up off the floor. But while the
normal force lasts, it is really quite amazing the way it adjusts automatically to
whatever you load on the desk. Magic stuff here.
3.3
3.3.1
Walking
The Experiment
Enough of books on desks. Time to try a whole-body, kinesthetic physics experience. Please follow carefully. Stand up and walk across the room. That’s
right, 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 again
from its state of motion to one of rest as you come to a stop. Since we are
doing physics in just one dimension for a while, I will be concerned only with
horizontal forces, the ones that make you walk across the floor.
This will take several free-body diagrams to describe because there are several 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 standing but still at rest on the floor.
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N
weight
Figure 8: The weight and normal force added to point representing you standing
on the floor.
This one should be looking pretty familiar by now. It’s identical to every
other 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 situations
are identical.
But now I am going to forget what is going on in the vertical dimension and
concentrate 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 my
diagram. You were initially at rest; you then began to walk, so your momentum
changed.
According to Newton’s second law, a force had to act in the direction of your
acceleration. 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 (the
push acts on your mass to accelerate it). 4 Please don’t make a mistake common
in people like me just starting to use free body diagrams: do not add arrows to
the diagram that are not forces known to act on the body. For example, some
folks might like to draw an arrow on the diagram to indicate the “motion” of
the 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 on
my free body diagram of you walking that I called “push” meaning that this
arrow represents a force acting on the object. I know that some force acts in
this direction because the object is accelerating in this direction.
4 To be completely accurate in my diagram, I would have to add in a backward force due
to friction between your body and the air as you move through the air, but the air friction
force 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. There
are rotational motions involved, too—all very complicated. But for a first cut
at understanding the act of walking, and especially since I am presently doing
physics in one dimension, I will only pay attention to the horizontal force I am
calling “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 in
the 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 and
the air as you move through the air, but the air friction force is tiny and I will
ignore it”). Notice that with great cunning I burried this in a footnote to make
it 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’t
want to “interrupt the flow” of the presentation. I feel that the interruption is
not only important, it is downright necessary—but not necessarily on the first
reading. If you’re inclined, read on, otherwise go ahead and finish the rest of
this section and then come back to review this bit of wisdom.
Notice again what was said in the parentheses: “. . . but the air friction force
is tiny and I will ignore it.” The assertion of “tininess” is the key. Ignoring the
force 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 would
have 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). But
how can this assumption of tininess be justified? The simple answer is that
experience has taught me that even if I were to include the air friction force in
a case like this, my final result will be no different than if I had left it out. So I
choose to ignore it.
In this case, experience has given me justification for ignoring something
which I know to be present in the physical situation, because I also know it
will not greatly alter my conclusions if I include it. This sort of experience is
valuable, but it can also be a bit tricky, because sometimes I ignore something
on purpose, thinking that it is too small to worry about, only to find later that
it makes all the difference in the world. Other times I ignore things that are
very 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 problem
which is completely new to me, and I have no direct experience with what
aspects of the problem may be negligible. In that case, I might consult some
texts or research papers or even a colleague or six to see if anyone else has any
experience I can use to simplify the problem in a realistic way.
Then there are situations where no experience is available, whether directly
or indirectly. Indeed I can be working on a problem which nobody has ever
tackled 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 supernovae, the structure of galaxies or the national debt). So in such cases, do I
include every conceivable force in my free body diagram? Often to do so would
be 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 become
nightmarish—or even downright insoluble. When this is a possibility, or when
I want to make a “first pass” at a problem to see how big various forces can
be, 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 by
working lots of physics problems and reading lots about other people’s solutions
of problems. Some people are extremely good at intuition—that is, at knowing
just what I can ignore and what I can’t. Indeed some physicists are very famous
for it. I’m not famous for anything.
There will be many situations in which my intuition is already just fine to
guide me—for example, you were probably not deeply offended by my suggestion
that I ignore air friction as I analyze your walk across the floor. You and I have
done 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” this
with the nerves in your skin. In such cases, after a little thought, you will have
a pretty good idea of what to ignore or what not to ignore. But there will be
some other cases where you may not be at all sure. I will try to be good about
supplying some suggestions about what you can and cannot ignore. Unless I
mess up.
3.3.4
Back to Phase I
Back to my discussion of the free body diagram of your walk. What about that
force I have labeled “push?” What is the origin of this force? This is a very
important point, so please read carefully and think about it as much as you
need to.
Whatever the force that causes the “push” on your body, I know two things
about it right away:
1. it acts on your body
2. it acts in the direction in which you want to walk (that is, the direction
of your acceleration).
Clearly you have some control over this force. You can change the direction
of the force by changing the way your leg muscles act. You can also change
the magnitude of the force by changing the degree to which you exert your leg
muscles—so you might be tempted to think that it is your leg muscles that are
pushing 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 with
your left hand and pull your right hand foreward with your left hand. Clearly
in 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 by
exerting a force with your legs can you walk forward. So what is so magic about
your legs that they can make you move forward? Tugging on any part of your
body with your hands won’t do it.
Next try this: get down on your hands and knees. Now you can move forward
by either exerting your leg or your arm muscles. So when you are standing, your
leg muscles are the only ones capable of moving you; when you are down on all
fours, 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 should
try it. If you can’t, take my word for it,5 you can move across the floor by
exerting your arm muscles; your leg muscles are useless as far as moving your
body 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 the
free body diagram: it is associated with contact with the floor by some part
of your body. Only by exerting a force in some limb which is in contact with
the floo can you accelerate yourself along the floor. So the floor is key to your
acceleration.
If it’s winter outside when you are reading this, the next experiment will be
fun to try; if it isn’t winter, you’ll just have to wait for some ice to show up on
the sidewalk or be satisfied with what I am about to tell you. If you try to walk
on perfectly smooth ice when you are wearing perfectly smooth shoes, it won’t
work.
You can exert all the forces you want with your legs; you’ll only flail around
on 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 in
the 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 with
the floor, you won’t walk. The floor itself plays a key role. Understanding
this point and the content of the next section is absolutely essential for your
understanding of Newtonian physics. Reread it and think about it and talk
about it with others as much as you need to until you understand the point
clearly.
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 to
make what I am about to tell you as simple as possible, I first want to go back
to the experiment with the book on the table. Place the book on the table and
push with your finger—one finger please. Now close your eyes and repeat the
experiment and note carefully what your nerves in your finger tell you is going
on.
5 I can’t stand on my hands. I’ve never even been able to turn a cartwheel. But I still
beleive what I am saying here even though I have never tried it myself. I beleive and trust
the 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 pushing
the 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 the
opposite direction.
And that, in the high falutin’ terminology of physics, is the first part of
Newton’s third law. The second part of the law isn’t so easy for you to detect
with your fingers, but if you were to instrument your finger and the book so
that you could measure the magnitude of the forces involved, you’d find that
however strongly you push on the book, the book will push back on you with
an exactly equal but opposite force.
Now try walking again. If you carefully pay attention to your feet, you’ll
find that however strongly you push on the floor with your leg muscles, the floor
will push in the opposite direction on your feet. It is this backward force by
the floor on your feet that is the “push” force in the free body diagram of you
walking. It is the floor that makes you walk forward by pushing on you. If you
try to walk on ice, there is no friction between your feet and the ice, so the
ice can not push back on your feet and you do not push on the ice—you don’t
move.
According to Newton’s third law, there is no such thing as a single isolated
force. Forces, any forces, always occur in couples. Whenever any object in
the universe exerts a force on a second object, the second exerts an equal and
opposite force on the first.
Why this is so is not clear. The fact is that Newton’s third law does describe
what we observe to be the case in the world, so, like Newton’s second law, we
accept 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 first
phase. I’ll go through it, however, because it will be good practice at applying
Newton’s laws and free body diagrams.
So now you’re walking along the floor (to the right in my diagram) and you
exert the necessary leg muscle to slow yourself down. This means that according
to Newton’s second law, the force that the floor exerts on you must act in the
leftward direction.
This new push slows you down. Again it is a force exerted by the floor on
you.
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 carry
it home from school, I couldn’t also lift myself up and carry myself home from
school without touching the ground. I don’t know whether I tried it, although
I suspect I did. It didn’t work. I do know that I couldn’t understand why for a
long time.
The key to understanding why I can’t bootstrap myself up in the air can be
found 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
N
weight
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
pull
react
N
weight
Figure 12: My pull upward and the reaction force to this pull added to the
figure.
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 on
my arm pit, according to Newton’s third law, my arm pit must exert an equal
and opposite (downward) force on my hand. And since both my arm pit and
my hand are parts of my body, the net result is that equal and opposite forces
are exerted on my body and they cancel each other out. I can’t pull myself up
in the air by exerting an upward force on my body. Shucks. I think it would
be really neat to carry myself home aboout a foot above the ground, especially
when it is rainy and there are lots of puddles.
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