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.
Now, 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 Power 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.
All 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.
183?
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.
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?
Okay, you good?
ZACH: Yeah.
LEWIN: You comfortable?
(students laugh )
You're really comfortable, right?
ZACH: Wonderful.
LEWIN: Okay.
You ready?
ZACH: Yes.
LEWIN: 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 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
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 less 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.
It 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.
H one is three meters
but I claim there is an uncertainty
of about three millimeters.
Can't do any better.
And h 2 equals 1.5 meters
again with an uncertainty of about three millimeters.
So the ratio h one over h two...
is 2
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?
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
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.