I'm Walter Lewin. I will be your lecturer
this term. In physics, we explore the
very small to the very large. The very small is a small
fraction of a proton and the very large is
the universe itself. They span 45 orders
of magnitude-- a 1 with 45 zeroes. To express measurements
quantitatively we have to introduce units. And we introduce for the unit
of length, the meter; for the unit of time,
the second; and for the unit of mass,
the kilogram. And you can read in your book
how these are defined and how the definition
evolved historically. Now, there are
many derived units which we use in our daily life
for convenience and some are tailored
toward specific fields. We have centimeters,
we have millimeters kilometers. We have inches, feet, miles. Astronomers even use
the astronomical unit which is the mean distance
between the Earth and the sun and they use light-years which is the distance that
light travels in one year. We have milliseconds,
we have microseconds we have days, weeks, hours,
centuries, months-- all derived units. For the mass, we have
milligrams, we have pounds we have metric tons. So lots of derived units exist. Not all of them are
very easy to work with. I find it extremely difficult
to work with inches and feet. It's an extremely
uncivilized system. I don't mean to insult you,
but think about it-- 12 inches in a foot,
three feet in a yard. Could drive you nuts. I work almost
exclusively decimal, and I hope you will do the same
during this course but we may make some exceptions. I will now first show you
a movie, which is called
The Powers of Ten. It covers 40 orders
of magnitude. It was originally conceived
by a Dutchman named Kees Boeke in the early '50s. This is the second-generation
movie, and you will hear the voice of Professor Morrison,
who is a professor at MIT. The Powers of Ten--
40 Orders of Magnitude. Here we go. I already introduced,
as you see there length, time and mass and we call these the three fundamental quantities
in physics. I will give this the symbol
capital L for length capital T for time,
and capital M for mass. Many other quantities in physics
can be derived from these fundamental
quantities. I'll give you an example. I put a bracket around here. I say speed, and that means
the dimensions of speed. The dimensions of speed is
the dimension of length divided by the dimension
of time. So I can write for that:
[L] divided by [T]. Whether it's meters per second
or inches per year that's not what matters. It has the dimension
length per time. Volume would have
the dimension of length to the power three. Density would have
the dimension of mass per unit volume so that means
length to the power three. All-important in our course
is acceleration. We will deal a lot
with acceleration. Acceleration, as you will see,
is length per time squared. The unit is meters
per second squared. So you get length
divided by time squared. So all other quantities
can be derived from these three fundamental. So now that we have agreed
on the units-- we have the meter,
the second and the kilogram-- we can start making
measurements. Now, all-important
in making measurements which is always ignored
in every college book is the uncertainty
in your measurement. Any measurement that you make without any knowledge
of the uncertainty is meaningless. I will repeat this. I want you to hear it tonight
at 3:00 when you wake up. Any measurement that you make without the knowledge
of its uncertainty is completely meaningless. My grandmother used
to tell me that... at least she believed it... that someone who is lying in bed is longer than
someone who stands up. And in honor of my grandmother I'm going to bring
this today to a test. I have here a setup where I can
measure a person standing up and a person lying down. It's not the greatest bed,
but lying down. I have to convince you about the uncertainty
in my measurement because a measurement without
knowledge of the uncertainty is meaningless. And therefore, what I will do
is the following. I have here an aluminum bar and I make the reasonable,
plausible assumption that when this aluminum bar
is sleeping-- when it is horizontal-- that it is not longer than
when it is standing up. If you accept that,
we can compare the length of this aluminum bar
with this setup and with this setup. At least we have some kind
of calibration to start with. I will measure it. You have to trust me. During these three months,
we have to trust each other. So I measure here,
149.9 centimeters. However, I would think
that the... so this is the aluminum bar. This is in vertical position. 149.9. But I would think that the
uncertainty of my measurement is probably 1 millimeter. I can't really guarantee you that I did it accurately
any better. So that's the vertical one. Now we're going to measure
the bar horizontally for which we have a setup here. Oops! The scale is on your side. So now I measure
the length of this bar. 150.0 horizontally. 150.0, again, plus or minus
0.1 centimeter. So you would agree with me
that I am capable of measuring plus or minus 1 millimeter. That's the uncertainty
of my measurement. Now, if the difference
in lengths between lying down
and standing up if that were one foot we would all know it,
wouldn't we? You get out of bed
in the morning you lie down and you get up
and you go, clunk! And you're one foot shorter. And we know that that's
not the case. If the difference were
only one millimeter we would never know. Therefore, I suspect that
if my grandmother was right then it's probably only
a few centimeters, maybe an inch. And so I would argue that
if I can measure the length of a student
to one millimeter accuracy that should settle the issue. So I need a volunteer. You want to volunteer? You look like you're very tall. I hope that... yeah, I hope that
we don't run out of, uh... You're not taller
than 178 or so? What is your name? STUDENT:
Rick Ryder. LEWIN:
Rick-- Rick Ryder. You're not nervous, right? RICK:
No! LEWIN:
Man! (class laughs) Sit down. (class laughs) I can't have tall guys here. Come on. We need someone
more modest in size. Don't take it personal, Rick. Okay, what is your name? STUDENT:
Zach. LEWIN:
Zach. Nice day today, Zach, yeah? You feel all right? Your first lecture at MIT? I don't. Okay, man. Stand there, yeah. Okay, 183.2. Stay there, stay there. Don't move. Zach... This is vertical. What did I say? 180? Only one person. 3? Come on. .2 Okay. 183.2. Yeah. And an uncertainty
of about one... Oh, this is centimeters--
0.1 centimeters. And now we're going to measure
him horizontally. Zach, I don't want you
to break your bones so we have a little step
for you here. Put your feet there. Oh, let me remove
the aluminum bar. Don't... Watch out for the scale. That you don't break that,
because then it's all over. Okay, I'll come on your side. I have to do that-- yeah, yeah. Relax. Think of this
as a small sacrifice for the sake of science, right? It's not...
Okay, you good? ZACH:
Yeah. LEWIN:
You comfortable? (students laugh) You're really comfortable,
right? ZACH:
Wonderful. LEWIN:
Okay. You're ready? ZACH:
Yes. LEWIN:
Okay. Okay. 185.7. Stay where you are.
185.7. I'm sure... I want to first
make the subtraction, right? 185.7, plus or minus
0.1 centimeter. Oh, that is five... that is 2.5 plus or minus
0.2 centimeters. You're about one inch taller
when you sleep than when you stand up. My grandmother was right. She's always right. Can you get off here? I want you to appreciate
that the accuracy... Thank you very much, Zach. That the accuracy
of one millimeter was more than sufficient
to make the case. If the accuracy
of my measurements would have been much less this measurement would not
have been convincing at all. So whenever you make
a measurement you must know the uncertainty. Otherwise, it is meaningless. Galileo Galilei asked himself
the question: Why are mammals as large as
they are and not much larger? He had a very clever reasoning
which I've never seen in print. But it comes down to the fact
that he argued that if the mammal
becomes too massive that the bones will break and he thought that that
was a limiting factor. Even though I've never
seen his reasoning in print I will try to reconstruct it what could have gone
through his head. Here is a mammal. And this is the... one of the
four legs of the mammal. And this mammal has a size S. And what I mean by that is a mouse is yay big
and a cat is yay big. That's what I mean by size--
very crudely defined. The mass of the mammal is M and this mammal has
a thigh bone which we call the femur,
which is here. And the femur of course carries
the body, to a large extent. And let's assume that the femur
has a length l and has a thickness d. Here is a femur. This is what a femur
approximately looks like. So this will be the length
of the femur... and this will be
the thickness, d and this will be
the cross-sectional area A. I'm now going to take you
through what we call in physics a scaling argument. I would argue that the length
of the femur must be proportional
to the size of the animal. That's completely plausible. If an animal is four times
larger than another you would need four times
longer legs. And that's all this is saying. It's very reasonable. It is also very reasonable
that the mass of an animal is proportional
to the third power of the size because that's related
to its volume. And so if it's related
to the third power of the size it must also be proportional to the third power of the length
of the femur because of this relationship. Okay, that's one. Now comes the argument. Pressure on the femur
is proportional to the weight of the animal
divided by the cross-section A of the femur. That's what pressure is. And that is the mass
of the animal that's proportional to the mass of the animal
divided by d squared because we want the area here,
it's proportional to d squared. Now follow me closely. If the pressure is higher
than a certain level the bones will break. Therefore, for an animal
not to break its bones when the mass goes up
by a certain factor let's say a factor of four in order for the bones
not to break d squared must also go up
by a factor of four. That's a key argument
in the scaling here. You really have to think
that through carefully. Therefore, I would argue that the mass must be
proportional to d squared. This is the breaking argument. Now compare these two. The mass is proportional
to the length of the femur to the power three and to the thickness of
the femur to the power two. Therefore, the thickness of
the femur to the power two must be proportional
to the length l and therefore the thickness of
the femur must be proportional to l to the power three-halfs. A very interesting result. What is this result telling you? It tells you that if
I have two animals and one is ten times larger
than the other then S is ten times larger that the lengths of the legs
are ten times larger but that the thickness
of the femur is 30 times larger because it is l to the power
three halves. If I were to compare
a mouse with an elephant an elephant is about a hundred
times larger in size so the length of the femur
of the elephant would be a hundred times larger
than that of a mouse but the thickness of the femur would have to be
1,000 times larger. And that may have convinced
Galileo Galilei that that's the reason why the largest animals are
as large as they are. Because clearly,
if you increase the mass there comes a time that
the thickness of the bones is the same as the length
of the bones. You're all made of bones and that is biologically
not feasible. And so there is
a limit somewhere set by this scaling law. Well, I wanted to bring
this to a test. After all I brought my grandmother's
statement to a test so why not bring Galileo
Galilei's statement to a test? And so I went to Harvard where they have a beautiful
collection of femurs and I asked them for the femur
of a raccoon and a horse. A raccoon is this big a horse is about
four times bigger so the length of the femur
of a horse must be about four times
the length of the raccoon. Close. So I was not surprised. Then I measured the thickness,
and I said to myself, "Aha!" If the length is
four times higher then the thickness has
to be eight times higher if this holds. And what I'm going
to plot for you you will see that shortly
is d divided by l, versus l and that, of course,
must be proportional to l to the power one-half. I bring one l here. So, if I compare the horse
and I compare the raccoon I would argue that the thickness divided by the length
of the femur for the horse must be the square root of four,
twice as much as that of the raccoon. And so I was very anxious
to plot that, and I did that and I'll show you the result. Here is my first result. So we see there, d over l. I explained to you why
I prefer that. And here you see the length. You see here the raccoon
and you see the horse. And if you look carefully,
then the d over l for the horse is only about one and a half
times larger than the raccoon. Well, I wasn't too disappointed. One and a half is not two, but
it is in the right direction. The horse clearly has a larger
value for d over l than the raccoon. I realized I needed more data,
so I went back to Harvard. I said, "Look, I need a smaller
animal, an opossum maybe maybe a rat, maybe a mouse,"
and they said, "okay." They gave me three more bones. They gave me an antelope which is actually a little
larger than a raccoon and they gave me an opossum
and they gave me a mouse. Here is the bone
of the antelope. Here is the one of the raccoon. Here is the one of the opossum. And now you won't believe this. This is so wonderful,
so romantic. There is the mouse. (students laugh) Isn't that beautiful? Teeny, weeny little mouse? That's only a teeny,
weeny little femur. And there it is. And I made the plot. I was very curious
what that plot would look like. And... here it is. Whew! I was shocked. I was really shocked. Because look-- the horse
is 50 times larger in size than the mouse. The difference in d over l
is only a factor of two. And I expected something more
like a factor of seven. And so, in d over l, where
I expect a factor of seven I only see a factor of two. So I said to myself,
"Oh, my goodness. Why didn't I ask them for an elephant?" The real clincher
would be the elephant because if that goes
way off scale maybe we can still rescue the
statement by Galileo Galilei and so I went back
and they said "Okay, we'll give you
the femur of an elephant." They also gave me one
of a moose, believe it or not. I think they wanted to get
rid of me by that time to be frank with you. And here is the femur
of an elephant. And I measured it. The length and the thickness. And it is very heavy. It weighs a ton. I plotted it, I was
full of expectation. I couldn't sleep all night. And there's the elephant. There is no evidence whatsoever
that d over l is really larger for the elephant
than for the mouse. These vertical bars indicate
my uncertainty in measurements of thickness and the horizontal scale,
which is a logarithmic scale... the uncertainty
of the length measurements is in the thickness
of the red pen so there's no need for me
to indicate that any further. And here you have
your measurements in case you want to check them. And look again at the mouse
and look at the elephant. The mouse has indeed only one
centimeter length of the femur and the elephant is, indeed,
hundred times longer. So the first scaling argument
that S is proportional to l that is certainly
what you would expect because an elephant is about
a hundred times larger in size. But when you go to d over l,
you see it's all over. The d over l for the mouse is really not all that
different from the elephant and you would have expected
that number to be with the square root of 100 so you expect it to be
ten times larger instead of about the same. I now want to discuss with you what we call in physics
dimensional analysis. I want to ask myself
the question: If I drop an apple
from a certain height and I change that height what will happen with the time
for the apple to fall? Well, I drop the apple
from a height h and I want to know what happened
with the time when it falls. And I change h. So I said to myself,
"Well, the time that it takes must be proportional to the height
to some power alpha." Completely reasonable. If I make the height larger we all know that it takes longer
for the apple to fall. That's a safe thing. I said to myself, "Well,
if the apple has a mass m it probably is
also proportional to the mass of that apple to the power beta." I said to myself, "Gee, yeah,
if something is more massive it will probably
take more time." So maybe m to some power beta. I don't know alpha,
I don't know beta. And then I said, "Gee, there's
also something like gravity that is the Earth's
gravitational pull-- the gravitational acceleration
of the Earth." So let's introduce that, too and let's assume that that time
is also proportional to the gravitational
acceleration-- this is an acceleration; we will
learn a lot more about that-- to the power gamma. Having said this, we can now do
what's called in physics a dimensional analysis. On the left we have a time and if we have a left...
on the left side a time on the right side
we must also have time. You cannot have coconuts on one
side and oranges on the other. You cannot have seconds
on one side and meters per second
on the other. So the dimensions left and right
have to be the same. What is the dimension here? That is [T] to the power one. That T... that must be the same
as length to the power alpha times mass to the power beta,
times acceleration-- remember, it is still there
on the blackboard-- that's dimension [L]
divided by time squared and the whole thing
to the power gamma so I have a gamma here
and I have a gamma there. This side must have the same
dimension as that side. That is nonnegotiable
in physics. Okay, there we go. There is no M here,
there is only one M here so beta must be zero. There is here [L] to the power
alpha, [L] to the power gamma there is no [L] here. So [L] must disappear. So alpha plus gamma
must be zero. There is [T]
to the power one here and there is here
[T] to the power -2 gamma. It's minus because
it's downstairs. So one must be equal
to -2 gamma. That means gamma must be
minus one half. That if gamma is minus one half,
then alpha equals plus one half. End of my dimensional analysis. I therefore conclude that
the time that it takes for an object to fall equals some constant,
which I do not know but that constant
has no dimension-- I don't know what it is-- times the square root
of h divided by g. Beta is zero,
there is no mass h to the power one half--
you see that here-- and g to the power
minus one half. This is proportional
to the square root of h because g is a given
and c is a given even though I don't know c. I make no pretense that I can
predict how long it will take for the apple to fall. All I'm saying is, I can compare
two different heights. I can drop an apple
from eight meters and another one from two meters and the one from eight meters
will take two times longer than the one from two meters. The square root of h to two,
four over two will take two times longer,
right? If I drop one from eight meters and I drop another one
from two meters then the difference in time will
be the square root of the ratio. That will be twice as long. And that I want to bring
to a test today. We have a setup here. We have an apple there
at a height of three meters and we know the length to an
accuracy... the height of about three millimeters,
no better. And here we have a setup
whereby the apple is about one and a half meters
above the ground. And we know that to about
also an accuracy of no better than
about three millimeters. So, let's set it up. I have here... something that's going
to be a prediction-- a prediction of the time that
it takes for one apple to fall divided by the time
that it takes for the other apple to fall. h1 is three meters but I claim there is
an uncertainty of about three millimeters. Can't do any better. And h2 equals 1.5 meters again with an uncertainty
of about three millimeters. So the ratio h1 over h2... is 2.000 and now I have to come up
with an uncertainty which physicists sometimes call
an error in their measurements but it's really an uncertainty. And the way you find
your uncertainty is that you add the three here and you subtract the three here and you get the largest value
possible. You can never get
a larger value. And you'll find
that you get 2.006. And so I would say
the uncertainty is then .006. This is a dimensionless number because it's length
divided by length. And so the time t1
divided by t2 would be the square root
of h1 divided by h2. That is the dimensional
analysis argument that we have there. And we find if we take
the square root of this number we find 1.414,
plus or minus 0.0 and I think that is a two. That is correct. So here is a firm prediction. This is a prediction. And now we're going to make
an observation. So we're going to measure t1
and there's going to be a number and then we're going
to measure t2 and there's going
to be a number. I have done this experiment
ten times and the numbers always reproduce
within about one millisecond. So I could just adopt an
uncertainty of one millisecond. I want to be a little bit
on the safe side. Occasionally it differs
by two milliseconds. So let us be conservative and let's assume that I can
measure this to an accuracy of about two milliseconds. That is pretty safe. So now we can measure
these times and then we can take
the ratio and then we can see whether
we actually confirm that the time that it takes
is proportional to the height to the square root
of the height. So I will make it a little more
comfortable for you in the lecture hall. That's all right. We have the setup here. We first do the experiment
with the... three meters. There you see the three meters. And the time... the moment
that I pull this string the apple will fall, the contact
will open, the clock will start. The moment that it hits the
floor, the time will stop. I have to stand on that side. Otherwise the apple
will fall on my hand. That's not the idea. I'll stand here. You ready? Okay, then I'm ready. Everything set? Make sure that I've
zeroed that properly. Yes, I have. Okay. Three, two, one, zero. 781 milliseconds. So this number...
you should write it down because you will need it
for your second assignment. 781 milliseconds, with an
uncertainty of two milliseconds. You ready for the second one? You ready? You ready? Okay, nothing wrong. Ready. Zero, zero, right? Thank you. Okay. Three, two, one, zero. 551 milliseconds. Boy, I'm nervous because
I hope that physics works. So I take my calculator and I'm now going to take
the ratio t1 over t2. The uncertainty you can find
by adding the two here and subtracting the two there and that will
then give you an uncertainty of, I think, .0... mmm, .08. Yeah, .08. You should do that
for yourself-- .008. Dimensionless number. This would be the uncertainty. This is the observation. 781 divided by 551. One point... Let me do that once more. Seven eight one,
divided by five five one... One four one seven. Perfect agreement. Look, the prediction says
1.414 but it could be 1 point...
it could be two higher. That's the uncertainty
in my height. I don't know any better. And here I could even
be off by an eight because that's the uncertainty
in my timing. So these two measurements
confirm. They are in agreement
with each other. You see, uncertainties in
measurements are essential. Now look at our results. We have here a result
which is striking. We have demonstrated that
the time that it takes for an object to fall
is independent of its mass. That is an amazing
accomplishment. Our great-grandfathers must
have worried about this and argued about this
for more than 300 years. Were they so dumb to overlook this simple
dimensional analysis? Inconceivable. Is this dimensional analysis
perhaps not quite kosher? Maybe. Is this dimensional analysis perhaps one that could have
been done differently? Yeah, oh, yeah. You could have done it
very differently. You could have said
the following. You could have said,
"The time for an apple to fall "is proportional to the height
that it falls from to a power alpha." Very reasonable. We all know, the higher it is,
the more it will take-- the more time it will take. And we could have said, "Yeah, it's probably
proportional "to the mass somehow. If the mass is more, it will
take a little bit less time." Turns out to be not so,
but you could think that. But you could have said "Well, let's not take the
acceleration of the Earth but let's take the mass
of the Earth itself." Very reasonable, right? I would think if I increased
the mass of the Earth that the apple will fall faster. So now I will put in the math
of the Earth here. And I start
my dimensional analysis and I end up dead in the waters. Because, you see,
there is no mass here. There is a mass
to the power beta here and one to the power gamma so what you would have found
is beta plus gamma equals zero and that would be end of story. Now you can ask yourself
the question well, is there something wrong
with the analysis that we did? Is ours perhaps better
than this one? Well, it's a different one. We came to the conclusion that the time that it takes
for the apple to fall is independent of the mass. Do we believe that? Yes, we do. On the other hand, there are
very prestigious physicists who even nowadays do
very fancy experiments and they try to demonstrate that
the time for an apple to fall does depend on its mass even though it probably is
only very small, if it's true but they try to prove that. And if any of them succeeds
or any one of you succeeds that's certainly worth
a Nobel Prize. So we do believe that it's
independent of the mass. However, this, what I did
with you, was not a proof because if you do it this way,
you get stuck. On the other hand, I'm
quite pleased with the fact that we found that the time
is proportional with the square root of h. I think that's very useful. We confirmed that
with experiment and indeed it came out that way. So it was not a complete
waste of time. But when you do a dimensional
analysis, you better be careful. I'd like you to think this over,
the comparison between the two at dinner and maybe at breakfast and maybe even while
you are taking a shower whether it's needed or not. It is important that you
digest and appreciate the difference between
these two approaches. It will give you an insight
in the power and also into the limitations
of dimensional analysis. This goes to the very heart of our understanding
and appreciation of physics. It's important that
you get a feel for this. You're now at MIT. This is the time. Thank you, see you Friday.
