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I'm Walter Lewin.

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I will be your lecturer
this term.

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In physics, we explore the
very small to the very large.

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The very small is a small
fraction of a proton

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and the very large is
the universe itself.

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They span 45 orders
of magnitude--

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a 1 with 45 zeroes.

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To express measurements
quantitatively

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we have to introduce units.

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And we introduce for the unit
of length, the meter;

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for the unit of time,
the second;

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and for the unit of mass,
the kilogram.

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Now, you can read in your book
how these are defined

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and how the definition
evolved historically.

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Now, there are
many derived units

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which we use in our daily life
for convenience

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and some are tailored
toward specific fields.

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We have centimeters,
we have millimeters

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kilometers.

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We have inches, feet, miles.

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Astronomers even use
the astronomical unit

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which is the mean distance
between the Earth and the sun

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and they use light-years

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which is the distance that
light travels in one year.

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We have milliseconds,
we have microseconds

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we have days, weeks, hours,
centuries, months--

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all derived units.

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For the mass, we have
milligrams, we have pounds

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we have metric tons.

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So lots of derived units exist.

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Not all of them are
very easy to work with.

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I find it extremely difficult
to work with inches and feet.

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It's an extremely
uncivilized system.

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I don't mean to insult you,
but think about it--

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12 inches in a foot,
three feet in a yard.

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Could drive you nuts.

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I work almost
exclusively decimal,

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and I hope you will do the same
during this course

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but we may make some exceptions.

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I will now first show you
a movie,

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which is called
The Powers of Ten.

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It covers 40 orders
of magnitude.

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It was originally conceived
by a Dutchman named Kees Boeke

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in the early '50s.

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This is the second-generation
movie, and you will hear

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the voice of Professor Morrison,
who is a professor at MIT.

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The Power of Ten--
40 Orders of Magnitude.

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Here we go.

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I already introduced,
as you see there

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length, time and mass

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and we call these

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the three fundamental quantities
in physics.

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I will give this the symbol
capital L for length

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capital T for time,
and capital M for mass.

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All other quantities in physics
can be derived

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from these fundamental
quantities.

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I'll give you an example.

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I put a bracket around here.

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I say speed, and that means
the dimensions of speed.

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The dimensions of speed is
the dimension of length

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divided by the dimension
of time.

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So I can write for that:
[L] divided by [T].

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Whether it's meters per second
or inches per year

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that's not what matters.

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It has the dimension
length per time.

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Volume would have
the dimension

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of length to the power three.

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Density would have
the dimension

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of mass per unit volume

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so that means
length to the power three.

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All-important in our course
is acceleration.

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We will deal a lot
with acceleration.

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Acceleration, as you will see,
is length per time squared.

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The unit is meters
per second squared.

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So you get length
divided by time squared.

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So all other quantities
can be derived

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from these three fundamental.

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So now that we have agreed
on the units--

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we have the meter,
the second and the kilogram--

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we can start making
measurements.

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Now, all-important
in making measurements

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which is always ignored
in every college book

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is the uncertainty
in your measurement.

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Any measurement that you make

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without any knowledge
of the uncertainty

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is meaningless.

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I will repeat this.

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I want you to hear it tonight
at 3:00 when you wake up.

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Any measurement that you make

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without the knowledge
of its uncertainty

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is completely meaningless.

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My grandmother used
to tell me that...

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at least she believed it...

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that someone who is lying in bed

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is longer than
someone who stands up.

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And in honor of my grandmother

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I'm going to bring
this today to a test.

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I have here a setup where I can
measure a person standing up

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and a person lying down.

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It's not the greatest bed,
but lying down.

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I have to convince you

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about the uncertainty
in my measurement

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because a measurement without
knowledge of the uncertainty

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is meaningless.

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And therefore, what I will do
is the following.

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I have here an aluminum bar

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and I make the reasonable,
plausible assumption

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that when this aluminum bar
is sleeping--

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when it is horizontal--

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that it is not longer than
when it is standing up.

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If you accept that,
we can compare

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the length of this aluminum bar
with this setup

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and with this setup.

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At least we have some kind
of calibration to start with.

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I will measure it.

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You have to trust me.

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During these three months,
we have to trust each other.

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So I measure here,
149.9 centimeters.

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However, I would think
that the...

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so this is the aluminum bar.

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This is in vertical position.

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149.9.

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But I would think that the
uncertainty of my measurement

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is probably 1 millimeter.

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I can't really guarantee you

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that I did it accurately
any better.

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So that's the vertical one.

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Now we're going to measure
the bar horizontally

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for which we have a setup here.

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Oops!

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The scale is on your side.

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So now I measure
the length of this bar.

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150.0 horizontally.

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150.0, again, plus or minus
0.1 centimeter.

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So you would agree with me
that I am capable of measuring

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plus or minus 1 millimeter.

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That's the uncertainty
of my measurement.

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Now, if the difference
in lengths

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between lying down
and standing up

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if that were one foot

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we would all know it,
wouldn't we?

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You get out of bed
in the morning

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you lie down and you get up
and you go, clunk!

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And you're one foot shorter.

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And we know that that's
not the case.

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If the difference were
only one millimeter

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we would never know.

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Therefore, I suspect that
if my grandmother was right

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then it's probably only
a few centimeters,

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maybe an inch.

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And so I would argue that
if I can measure

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the length of a student
to one millimeter accuracy

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that should settle the issue.

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So I need a volunteer.

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You want to volunteer?

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You look like you're very tall.

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I hope that... yeah, I hope that
we don't run out of, uh...

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You're not taller
than 178 or so?

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What is your name?

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STUDENT:
Rick Ryder.

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LEWIN:
Rick-- Rick Ryder.

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You're not nervous, right?

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RICK:
No!

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LEWIN:
Man!

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(class laughs )

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Sit down.

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(class laughs )

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I can't have tall guys here.

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Come on.

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We need someone
more modest in size.

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Don't take it personal, Rick.

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Okay, what is your name?

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STUDENT:
Zach.

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LEWIN:
Zach.

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Nice day today, Zach, yeah?

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You feel all right?

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Your first lecture at MIT?

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I don't.

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Okay, man.

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Stand there, yeah.

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Okay, 183.2.

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Stay there, stay there.

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Don't move.

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Zach...

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This is vertical.

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What did I say?

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180?

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Only one person.

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183?

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Come on.

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.2--

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Okay, 183.2.

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Yeah.

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And an uncertainty
of about one...

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Oh, this is centimeters--
0.1 centimeters.

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And now we're going to measure
him horizontally.

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Zach, I don't want you
to break your bones

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so we have a little step
for you here.

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Put your feet there.

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Oh, let me remove
the aluminum bar.

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Watch out for the scale.

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That you don't break that,
because then it's all over.

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Okay, I'll come on your side.

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I have to do that-- yeah, yeah.

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Relax.

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Think of this
as a small sacrifice

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for the sake of science, right?

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Okay, you good?

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ZACH:
Yeah.

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LEWIN:
You comfortable?

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(students laugh )

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You're really comfortable,
right?

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ZACH:
Wonderful.

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LEWIN:
Okay.

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You ready?

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ZACH:
Yes.

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LEWIN:
Okay.

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Okay.

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185.7.

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Stay where you are.

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185.7.

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I'm sure... I want to first
make the subtraction, right?

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185.7, plus or minus
0.1 centimeter.

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Oh, that is five...

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that is 2.5 plus or minus
0.2 centimeters.

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You're about one inch taller
when you sleep

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than when you stand up.

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My grandmother was right.

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She's always right.

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Can you get off here?

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I want you to appreciate
that the accuracy...

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Thank you very much, Zach.

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That the accuracy
of one millimeter

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was more than sufficient
to make the case.

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If the accuracy
of my measurements

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would have been much less

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this measurement would not
have been convincing at all.

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So whenever you make
a measurement

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you must know the uncertainty.

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Otherwise, it is meaningless.

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Galileo Galilei asked himself
the question:

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Why are mammals as large as
they are and not much larger?

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He had a very clever reasoning
which I've never seen in print.

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But it comes down to the fact
that he argued

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that if the mammal
becomes too massive

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that the bones will break

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and he thought that that
was a limiting factor.

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Even though I've never
seen his reasoning in print

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I will try to reconstruct it

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what could have gone
through his head.

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Here is a mammal.

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And this is one of the
four legs of the mammal.

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And this mammal has a size S.

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And what I mean by that is

255
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a mouse is yay big
and a cat is yay big.

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That's what I mean by size--
very crudely defined.

257
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The mass of the mammal is M

258
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and this mammal has
a thigh bone

259
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which we call the femur,
which is here.

260
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And the femur of course carries
the body, to a large extent.

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And let's assume that the femur
has a length l

262
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and has a thickness d.

263
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Here is a femur.

264
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This is what a femur
approximately looks like.

265
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So this will be the length
of the femur...

266
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and this will be
the thickness, d

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and this will be
the cross-sectional area A.

268
00:13:01 --> 00:13:07

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I'm now going to take you
through what we call in physics

270
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a scaling argument.

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I would argue that the length
of the femur

272
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must be proportional
to the size of the animal.

273
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That's completely plausible.

274
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If an animal is four times
larger than another

275
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you would need four times
longer legs.

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And that's all this is saying.

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It's very reasonable.

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It is also very reasonable
that the mass of an animal

279
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is proportional
to the third power of the size

280
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because that's related
to its volume.

281
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And so if it's related
to the third power of the size

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it must also be proportional

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to the third power of the length
of the femur

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because of this relationship.

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Okay, that's one.

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Now comes the argument.

287
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Pressure on the femur
is proportional

288
00:14:01 --> 00:14:07
to the weight of the animal
divided by the cross-section A

289
00:14:07 --> 00:14:13
of the femur.

290
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That's what pressure is.

291
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And that is the mass
of the animal

292
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that's proportional

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to the mass of the animal
divided by d squared

294
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because we want the area here,
it's proportional to d squared.

295
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Now follow me closely.

296
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If the pressure is higher
than a certain level

297
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the bones will break.

298
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Therefore, for an animal
not to break its bones

299
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when the mass goes up
by a certain factor

300
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let's say a factor of four

301
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in order for the bones
not to break

302
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d squared must also go up
by a factor of four.

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That's a key argument
in the scaling here.

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00:14:48 --> 00:14:54
You really have to think
that through carefully.

305
00:14:51 --> 00:14:57
Therefore, I would argue

306
00:14:53 --> 00:14:59
that the mass must be
proportional to d squared.

307
00:14:56 --> 00:15:02
This is the breaking argument.

308
00:14:59 --> 00:15:05
Now compare these two.

309
00:15:01 --> 00:15:07
The mass is proportional
to the length of the femur

310
00:15:04 --> 00:15:10
to the power three

311
00:15:05 --> 00:15:11
and to the thickness of
the femur to the power two.

312
00:15:09 --> 00:15:15
Therefore, the thickness of
the femur to the power two

313
00:15:13 --> 00:15:19
must be proportional
to the length l

314
00:15:15 --> 00:15:21
and therefore the thickness of
the femur must be proportional

315
00:15:18 --> 00:15:24
to l to the power three-halfs.

316
00:15:22 --> 00:15:28
A very interesting result.

317
00:15:25 --> 00:15:31
What is this result telling you?

318
00:15:28 --> 00:15:34
It tells you that if
I have two animals

319
00:15:31 --> 00:15:37
and one is ten times larger
than the other

320
00:15:34 --> 00:15:40
then S is ten times larger

321
00:15:36 --> 00:15:42
that the lengths of the legs
are ten times larger

322
00:15:40 --> 00:15:46
but that the thickness
of the femur is 30 times larger

323
00:15:46 --> 00:15:52
because it is l to the power
three halves.

324
00:15:48 --> 00:15:54
If I were to compare
a mouse with an elephant

325
00:15:50 --> 00:15:56
an elephant is about a hundred
times larger in size

326
00:15:54 --> 00:16:00
so the length of the femur
of the elephant

327
00:15:56 --> 00:16:02
would be a hundred times larger
than that of a mouse

328
00:15:58 --> 00:16:04
but the thickness of the femur

329
00:16:00 --> 00:16:06
would have to be
1,000 times larger.

330
00:16:05 --> 00:16:11
And that may have convinced
Galileo Galilei

331
00:16:09 --> 00:16:15
that that's the reason

332
00:16:11 --> 00:16:17
why the largest animals are
as large as they are.

333
00:16:14 --> 00:16:20
Because clearly,
if you increase the mass

334
00:16:17 --> 00:16:23
there comes a time that
the thickness of the bones

335
00:16:20 --> 00:16:26
is the same as the length
of the bones.

336
00:16:22 --> 00:16:28
You're all made of bones

337
00:16:24 --> 00:16:30
and that is biologically
not feasible.

338
00:16:27 --> 00:16:33
And so there is
a limit somewhere

339
00:16:29 --> 00:16:35
set by this scaling law.

340
00:16:33 --> 00:16:39
Well, I wanted to bring
this to a test.

341
00:16:36 --> 00:16:42
After all

342
00:16:37 --> 00:16:43
I brought my grandmother's
statement to a test

343
00:16:39 --> 00:16:45
so why not bring Galileo
Galilei's statement to a test?

344
00:16:43 --> 00:16:49
And so I went to Harvard

345
00:16:47 --> 00:16:53
where they have a beautiful
collection of femurs

346
00:16:50 --> 00:16:56
and I asked them for the femur
of a raccoon and a horse.

347
00:16:56 --> 00:17:02
A raccoon is this big

348
00:16:58 --> 00:17:04
a horse is about
four times bigger

349
00:17:02 --> 00:17:08
so the length of the femur
of a horse

350
00:17:05 --> 00:17:11
must be about four times
the length of the raccoon.

351
00:17:09 --> 00:17:15
Close.

352
00:17:11 --> 00:17:17
So I was not surprised.

353
00:17:14 --> 00:17:20
Then I measured the thickness,
and I said to myself, "Aha!"

354
00:17:19 --> 00:17:25
If the length is
four times higher

355
00:17:22 --> 00:17:28
then the thickness has
to be eight times higher

356
00:17:26 --> 00:17:32
if this holds.

357
00:17:28 --> 00:17:34
And what I'm going
to plot for you

358
00:17:29 --> 00:17:35
you will see that shortly
is d divided by l, versus l

359
00:17:35 --> 00:17:41
and that, of course,
must be proportional

360
00:17:37 --> 00:17:43
to l to the power one-half.

361
00:17:38 --> 00:17:44
I bring one l here.

362
00:17:40 --> 00:17:46
So, if I compare the horse
and I compare the raccoon

363
00:17:44 --> 00:17:50
I would argue that the thickness

364
00:17:46 --> 00:17:52
divided by the length
of the femur for the horse

365
00:17:49 --> 00:17:55
must be the square root of four,
twice as much

366
00:17:53 --> 00:17:59
as that of the raccoon.

367
00:17:56 --> 00:18:02
And so I was very anxious
to plot that, and I did that

368
00:18:00 --> 00:18:06
and I'll show you the result.

369
00:18:03 --> 00:18:09
Here is my first result.

370
00:18:09 --> 00:18:15
So we see there, d over l.

371
00:18:12 --> 00:18:18
I explained to you why
I prefer that.

372
00:18:14 --> 00:18:20
And here you see the length.

373
00:18:17 --> 00:18:23
You see here the raccoon
and you see the horse.

374
00:18:19 --> 00:18:25
And if you look carefully,
then the d over l for the horse

375
00:18:22 --> 00:18:28
is only about one and a half
times larger than the raccoon.

376
00:18:25 --> 00:18:31
Well, I wasn't too disappointed.

377
00:18:28 --> 00:18:34
One and a half is not two, but
it is in the right direction.

378
00:18:30 --> 00:18:36
The horse clearly has a larger
value for d over l

379
00:18:33 --> 00:18:39
than the raccoon.

380
00:18:36 --> 00:18:42
I realized I needed more data,
so I went back to Harvard.

381
00:18:39 --> 00:18:45
I said, "Look, I need a smaller
animal, an opossum maybe

382
00:18:43 --> 00:18:49
maybe a rat, maybe a mouse,"
and they said, "okay."

383
00:18:47 --> 00:18:53
They gave me three more bones.

384
00:18:51 --> 00:18:57
They gave me an antelope

385
00:18:52 --> 00:18:58
which is actually a little
larger than a raccoon

386
00:18:54 --> 00:19:00
and they gave me an opossum
and they gave me a mouse.

387
00:19:00 --> 00:19:06
Here is the bone
of the antelope.

388
00:19:08 --> 00:19:14
Here is the one of the raccoon.

389
00:19:14 --> 00:19:20
Here is the one of the opossum.

390
00:19:18 --> 00:19:24
And now you won't believe this.

391
00:19:20 --> 00:19:26
This is so wonderful,
so romantic.

392
00:19:26 --> 00:19:32
There is the mouse.

393
00:19:27 --> 00:19:33
(students laugh )

394
00:19:28 --> 00:19:34
Isn't that beautiful?

395
00:19:29 --> 00:19:35
Teeny, weeny little mouse?

396
00:19:31 --> 00:19:37
That's only a teeny,
weeny little femur.

397
00:19:35 --> 00:19:41
And there it is.

398
00:19:38 --> 00:19:44
And I made the plot.

399
00:19:41 --> 00:19:47
I was very curious
what that plot would look like.

400
00:19:44 --> 00:19:50
And...

401
00:19:51 --> 00:19:57
here it is.

402
00:19:53 --> 00:19:59
Whew! I was shocked.

403
00:19:57 --> 00:20:03
I was really shocked.

404
00:19:59 --> 00:20:05
Because look-- the horse
is 50 times larger in size

405
00:20:03 --> 00:20:09
than the mouse.

406
00:20:05 --> 00:20:11
The difference in d over l
is only a factor of two.

407
00:20:08 --> 00:20:14
And I expected something more
like a factor of seven.

408
00:20:14 --> 00:20:20
And so, in d over l, where
I expect a factor of seven

409
00:20:17 --> 00:20:23
I only see a factor of two.

410
00:20:19 --> 00:20:25
So I said to myself,
"Oh, my goodness.

411
00:28:27 --> 00:28:33
Why didn't I ask them
for an elephant?"

412
00:20:24 --> 00:20:30
The real clincher
would be the elephant

413
00:20:27 --> 00:20:33
because if that goes
way off scale

414
00:20:29 --> 00:20:35
maybe we can still rescue the
statement by Galileo Galilei

415
00:20:33 --> 00:20:39
and so I went back
and they said

416
00:20:35 --> 00:20:41
"Okay, we'll give you
the femur of an elephant."

417
00:20:38 --> 00:20:44
They also gave me one
of a moose, believe it or not.

418
00:20:40 --> 00:20:46
I think they wanted to get
rid of me by that time

419
00:20:43 --> 00:20:49
to be frank with you.

420
00:20:44 --> 00:20:50
And here is the femur
of an elephant.

421
00:20:49 --> 00:20:55
And I measured it.

422
00:20:50 --> 00:20:56
The length and the thickness.

423
00:20:53 --> 00:20:59
And it is very heavy.

424
00:20:56 --> 00:21:02
It weighs a ton.

425
00:20:58 --> 00:21:04
I plotted it, I was
full of expectation.

426
00:21:02 --> 00:21:08
I couldn't sleep all night.

427
00:21:05 --> 00:21:11
And there's the elephant.

428
00:21:07 --> 00:21:13
There is no evidence whatsoever
that d over l is really larger

429
00:21:11 --> 00:21:17
for the elephant
than for the mouse.

430
00:21:13 --> 00:21:19
These vertical bars indicate
my uncertainty

431
00:21:15 --> 00:21:21
in measurements of thickness

432
00:21:17 --> 00:21:23
and the horizontal scale,
which is a logarithmic scale...

433
00:21:20 --> 00:21:26
the uncertainty
of the length measurements

434
00:21:23 --> 00:21:29
is in the thickness
of the red pen

435
00:21:24 --> 00:21:30
so there's no need for me
to indicate that any further.

436
00:21:28 --> 00:21:34
And here you have
your measurements

437
00:21:30 --> 00:21:36
in case you want to check them.

438
00:21:33 --> 00:21:39
And look again at the mouse
and look at the elephant.

439
00:21:36 --> 00:21:42
The mouse has indeed only one
centimeter length of the femur

440
00:21:43 --> 00:21:49
and the elephant is, indeed,
hundred times longer.

441
00:21:45 --> 00:21:51
So the first scaling argument
that S is proportional to l

442
00:21:49 --> 00:21:55
that is certainly
what you would expect

443
00:21:51 --> 00:21:57
because an elephant is about
a hundred times larger in size.

444
00:21:54 --> 00:22:00
But when you go to d over l,
you see it's all over.

445
00:21:58 --> 00:22:04
The d over l for the mouse

446
00:21:59 --> 00:22:05
is really not all that
different from the elephant

447
00:22:02 --> 00:22:08
and you would have expected
that number to be

448
00:22:04 --> 00:22:10
with the square root of

449
00:22:09 --> 00:22:15
so you expect it to be
ten times larger

450
00:22:11 --> 00:22:17
instead of about the same.

451
00:22:14 --> 00:22:20
I now want to discuss with you

452
00:22:17 --> 00:22:23
what we call in physics
dimensional analysis.

453
00:22:24 --> 00:22:30
I want to ask myself
the question:

454
00:22:27 --> 00:22:33
If I drop an apple
from a certain height

455
00:22:31 --> 00:22:37
and I change that height

456
00:22:34 --> 00:22:40
what will happen with the time
for the apple to fall?

457
00:22:40 --> 00:22:46
Well, I drop the apple
from a height h

458
00:22:47 --> 00:22:53
and I want to know what happened
with the time when it falls.

459
00:22:51 --> 00:22:57
And I change h.

460
00:22:54 --> 00:23:00
So I said to myself,
"Well, the time that it takes

461
00:31:02 --> 00:31:08
must be proportional to the
height to some power alpha."

462
00:23:00 --> 00:23:06
Completely reasonable.

463
00:23:01 --> 00:23:07
If I make the height larger

464
00:23:03 --> 00:23:09
we all know that it takes longer
for the apple to fall.

465
00:23:06 --> 00:23:12
That's a safe thing.

466
00:23:08 --> 00:23:14
I said to myself, "Well,
if the apple has a mass m

467
00:23:12 --> 00:23:18
"it probably is
also proportional

468
00:31:20 --> 00:31:26
to the mass of that apple
to the power beta."

469
00:23:17 --> 00:23:23
I said to myself, "Gee, yeah,
if something is more massive

470
00:23:21 --> 00:23:27
it will probably
take less time."

471
00:23:23 --> 00:23:29
So maybe m to some power beta.

472
00:23:26 --> 00:23:32
I don't know alpha,
I don't know beta.

473
00:23:28 --> 00:23:34
And then I said, "Gee, there's
also something like gravity

474
00:23:31 --> 00:23:37
that is the Earth's
gravitational pull--

475
00:23:33 --> 00:23:39
the gravitational acceleration
of the Earth."

476
00:23:36 --> 00:23:42
So let's introduce that, too

477
00:23:38 --> 00:23:44
and let's assume that that time
is also proportional

478
00:23:41 --> 00:23:47
to the gravitational
acceleration--

479
00:23:43 --> 00:23:49
this is an acceleration; we will 
learn a lot more about that--

480
00:23:46 --> 00:23:52
to the power gamma.

481
00:23:49 --> 00:23:55
Having said this, we can now do
what's called in physics

482
00:23:53 --> 00:23:59
a dimensional analysis.

483
00:23:57 --> 00:24:03

484
00:23:59 --> 00:24:05
On the left we have a time

485
00:24:03 --> 00:24:09
and if we have a left...
on the left side a time

486
00:24:05 --> 00:24:11
on the right side
we must also have time.

487
00:24:08 --> 00:24:14
You cannot have coconuts on one
side and oranges on the other.

488
00:24:12 --> 00:24:18
You cannot have seconds
on one side

489
00:24:14 --> 00:24:20
and meters per second
on the other.

490
00:24:17 --> 00:24:23
So the dimensions left and right
have to be the same.

491
00:24:20 --> 00:24:26
What is the dimension here?

492
00:24:22 --> 00:24:28
That is [T] to the power one.

493
00:24:25 --> 00:24:31
That T... that must be the same
as length to the power alpha

494
00:24:33 --> 00:24:39
times mass to the power beta,
times acceleration--

495
00:24:42 --> 00:24:48
remember, it is still there
on the blackboard--

496
00:24:44 --> 00:24:50
that's dimension [L]
divided by time squared

497
00:24:49 --> 00:24:55
and the whole thing
to the power gamma

498
00:24:51 --> 00:24:57
so I have a gamma here
and I have a gamma there.

499
00:24:54 --> 00:25:00
This side must have the same
dimension as that side.

500
00:24:57 --> 00:25:03
That is nonnegotiable
in physics.

501
00:25:00 --> 00:25:06
Okay, there we go.

502
00:25:01 --> 00:25:07
There is no M here,
there is only one M here

503
00:25:04 --> 00:25:10
so beta must be zero.

504
00:25:08 --> 00:25:14
There is here [L] to the power
alpha, [L] to the power gamma

505
00:25:12 --> 00:25:18
there is no [L] here.

506
00:25:13 --> 00:25:19
So [L] must disappear.

507
00:25:15 --> 00:25:21
So alpha plus gamma
must be zero.

508
00:25:19 --> 00:25:25
There is [T]
to the power one here

509
00:25:22 --> 00:25:28
and there is here
[T] to the power -2 gamma.

510
00:25:25 --> 00:25:31
It's minus because
it's downstairs.

511
00:25:27 --> 00:25:33
So one must be equal
to -2 gamma.

512
00:25:31 --> 00:25:37
That means gamma must be
minus one half.

513
00:25:35 --> 00:25:41
That if gamma is minus one half,
then alpha equals plus one half.

514
00:25:43 --> 00:25:49
End of my dimensional analysis.

515
00:25:45 --> 00:25:51
I therefore conclude that
the time that it takes

516
00:25:49 --> 00:25:55
for an object to fall

517
00:25:51 --> 00:25:57
equals some constant,
which I do not know

518
00:25:55 --> 00:26:01
but that constant
has no dimension--

519
00:25:57 --> 00:26:03
I don't know what it is--

520
00:25:59 --> 00:26:05
times the square root
of h divided by g.

521
00:26:07 --> 00:26:13
Beta is zero,
there is no mass

522
00:26:10 --> 00:26:16
h to the power one half--
you see that here--

523
00:26:13 --> 00:26:19
and g to the power
minus one half.

524
00:26:15 --> 00:26:21
This is proportional
to the square root of h

525
00:26:19 --> 00:26:25
because g is a given
and c is a given

526
00:26:21 --> 00:26:27
even though I don't know c.

527
00:26:22 --> 00:26:28
I make no pretense that I can
predict how long it will take

528
00:26:26 --> 00:26:32
for the apple to fall.

529
00:26:27 --> 00:26:33
All I'm saying is, I can compare
two different heights.

530
00:26:31 --> 00:26:37
I can drop an apple
from eight meters

531
00:26:33 --> 00:26:39
and another one from two meters

532
00:26:35 --> 00:26:41
and the one from eight meters
will take two times longer

533
00:26:39 --> 00:26:45
than the one from two meters.

534
00:26:41 --> 00:26:47
The square root of h to two,
four over two

535
00:26:45 --> 00:26:51
will take two times longer,
right?

536
00:26:47 --> 00:26:53
If I drop one from eight meters

537
00:26:49 --> 00:26:55
and I drop another one
from two meters

538
00:26:52 --> 00:26:58
then the difference in time will
be the square root of the ratio.

539
00:26:56 --> 00:27:02
It will be twice as long.

540
00:26:57 --> 00:27:03
And that I want to bring
to a test today.

541
00:27:03 --> 00:27:09
We have a setup here.

542
00:27:05 --> 00:27:11
We have an apple there
at a height of three meters

543
00:27:08 --> 00:27:14
and we know the length to an
accuracy... the height

544
00:27:11 --> 00:27:17
of about three millimeters,
no better.

545
00:27:14 --> 00:27:20
And here we have a setup
whereby the apple

546
00:27:16 --> 00:27:22
is about one and a half meters
above the ground.

547
00:27:18 --> 00:27:24
And we know that to about
also an accuracy

548
00:27:21 --> 00:27:27
of no better than
about three millimeters.

549
00:27:26 --> 00:27:32
So, let's set it up.

550
00:27:29 --> 00:27:35
I have here...

551
00:27:34 --> 00:27:40
something that's going
to be a prediction--

552
00:27:37 --> 00:27:43
a prediction of the time that
it takes for one apple to fall

553
00:27:43 --> 00:27:49
divided by the time
that it takes

554
00:27:44 --> 00:27:50
for the other apple to fall.

555
00:27:47 --> 00:27:53
H one is three meters

556
00:27:51 --> 00:27:57
but I claim there is
an uncertainty

557
00:27:54 --> 00:28:00
of about three millimeters.

558
00:27:56 --> 00:28:02
Can't do any better.

559
00:27:57 --> 00:28:03
And h 2 equals 1.5 meters

560
00:28:02 --> 00:28:08
again with an uncertainty
of about three millimeters.

561
00:28:09 --> 00:28:15
So the ratio h one over h two...

562
00:28:14 --> 00:28:20
is 2

563
00:28:15 --> 00:28:21
and now I have to come up
with an uncertainty

564
00:28:20 --> 00:28:26
which physicists sometimes call
an error in their measurements

565
00:28:23 --> 00:28:29
but it's really an uncertainty.

566
00:28:25 --> 00:28:31
And the way you find
your uncertainty is

567
00:28:27 --> 00:28:33
that you add the three here
and you subtract the three here

568
00:28:31 --> 00:28:37
and you get the largest value
possible.

569
00:28:33 --> 00:28:39
You can never get
a larger value.

570
00:28:35 --> 00:28:41
And you'll find
that you get 2.006.

571
00:28:38 --> 00:28:44
And so I would say
the uncertainty is then .006.

572
00:28:44 --> 00:28:50
This is a dimensionless number

573
00:28:46 --> 00:28:52
because it's length
divided by length.

574
00:28:50 --> 00:28:56
And so the time t1
divided by t2

575
00:28:56 --> 00:29:02
would be the square root
of h1 divided by h2.

576
00:28:59 --> 00:29:05
That is the dimensional
analysis argument

577
00:29:02 --> 00:29:08
that we have there.

578
00:29:04 --> 00:29:10
And we find if we take
the square root of this number

579
00:29:07 --> 00:29:13
we find 1.414,
plus or minus 0.0

580
00:29:12 --> 00:29:18
and I think that is a two.

581
00:29:15 --> 00:29:21
That is correct.

582
00:29:16 --> 00:29:22
So here is a firm prediction.

583
00:29:22 --> 00:29:28
This is a prediction.

584
00:29:25 --> 00:29:31
And now we're going to make
an observation.

585
00:29:31 --> 00:29:37
So we're going to measure t1
and there's going to be a number

586
00:29:37 --> 00:29:43
and then we're going
to measure t2

587
00:29:40 --> 00:29:46
and there's going
to be a number.

588
00:29:42 --> 00:29:48
I have done this experiment
ten times

589
00:29:44 --> 00:29:50
and the numbers always reproduce
within about one millisecond.

590
00:29:49 --> 00:29:55
So I could just adopt an
uncertainty of one millisecond.

591
00:29:51 --> 00:29:57
I want to be a little bit
on the safe side.

592
00:29:54 --> 00:30:00
Occasionally it differs
by two milliseconds.

593
00:29:56 --> 00:30:02
So let us be conservative

594
00:29:59 --> 00:30:05
and let's assume that I can
measure this to an accuracy

595
00:30:02 --> 00:30:08
of about two milliseconds.

596
00:30:06 --> 00:30:12
That is pretty safe.

597
00:30:08 --> 00:30:14
So now we can measure
these times

598
00:30:12 --> 00:30:18
and then we can take
the ratio

599
00:30:15 --> 00:30:21
and then we can see whether
we actually confirm

600
00:30:19 --> 00:30:25
that the time that it takes
is proportional to the height

601
00:30:24 --> 00:30:30
to the square root
of the height.

602
00:30:26 --> 00:30:32
So I will make it a little more
comfortable for you

603
00:30:33 --> 00:30:39
in the lecture hall.

604
00:30:35 --> 00:30:41
That's all right.

605
00:30:37 --> 00:30:43
We have the setup here.

606
00:30:39 --> 00:30:45
We first do the experiment
with the... three meters.

607
00:30:47 --> 00:30:53
There you see the three meters.

608
00:30:50 --> 00:30:56
And the time... the moment
that I pull this string

609
00:30:53 --> 00:30:59
the apple will fall, the contact
will open, the clock will start.

610
00:30:57 --> 00:31:03
The moment that it hits the
floor, the time will stop.

611
00:31:02 --> 00:31:08
I have to stand on that side.

612
00:31:04 --> 00:31:10
Otherwise the apple
will fall on my hand.

613
00:31:06 --> 00:31:12
That's not the idea.

614
00:31:08 --> 00:31:14
I'll stand here.

615
00:31:10 --> 00:31:16
You ready?

616
00:31:12 --> 00:31:18
Okay, then I'm ready.

617
00:31:15 --> 00:31:21
Everything set?

618
00:31:17 --> 00:31:23
Make sure that I've
zeroed that properly.

619
00:31:19 --> 00:31:25
Yes, I have.

620
00:31:20 --> 00:31:26
Okay.

621
00:31:21 --> 00:31:27
Three, two, one, zero.

622
00:31:23 --> 00:31:29

623
00:31:27 --> 00:31:33
781 milliseconds.

624
00:31:30 --> 00:31:36
So this number...
you should write it down

625
00:31:34 --> 00:31:40
because you will need it
for your second assignment.

626
00:31:37 --> 00:31:43
781 milliseconds, with an
uncertainty of two milliseconds.

627
00:31:41 --> 00:31:47
You ready for the second one?

628
00:31:47 --> 00:31:53
You ready?

629
00:31:50 --> 00:31:56
You ready?

630
00:31:52 --> 00:31:58
Okay, nothing wrong.

631
00:31:54 --> 00:32:00
Ready.

632
00:31:58 --> 00:32:04
Zero, zero, right?

633
00:32:01 --> 00:32:07
Thank you.

634
00:32:02 --> 00:32:08
Okay.

635
00:32:04 --> 00:32:10
Three, two, one, zero.

636
00:32:06 --> 00:32:12

637
00:32:08 --> 00:32:14
551 milliseconds.

638
00:32:14 --> 00:32:20
Boy, I'm nervous because
I hope that physics works.

639
00:32:21 --> 00:32:27
So I take my calculator

640
00:32:25 --> 00:32:31
and I'm now going to take
the ratio t1 over t2.

641
00:32:32 --> 00:32:38
The uncertainty you can find
by adding the two here

642
00:32:35 --> 00:32:41
and subtracting the two there

643
00:32:38 --> 00:32:44
and that will
then give you an uncertainty

644
00:32:40 --> 00:32:46
of, I think, .0... mmm, .08.

645
00:32:46 --> 00:32:52
Yeah, .08.

646
00:32:48 --> 00:32:54
You should do that
for yourself-- .008.

647
00:32:51 --> 00:32:57
Dimensionless number.

648
00:32:52 --> 00:32:58
This would be the uncertainty.

649
00:32:55 --> 00:33:01
This is the observation.

650
00:32:58 --> 00:33:04
781 divided by 551.

651
00:33:04 --> 00:33:10
One point...

652
00:33:05 --> 00:33:11
Let me do that once more.

653
00:33:07 --> 00:33:13
Seven eight one,
divided by five five one...

654
00:33:11 --> 00:33:17
One four one seven.

655
00:33:16 --> 00:33:22
Perfect agreement.

656
00:33:19 --> 00:33:25
Look, the prediction says
1

657
00:33:24 --> 00:33:30
but it could be 1 point...
it could be two higher.

658
00:33:27 --> 00:33:33
That's the uncertainty
in my height.

659
00:33:29 --> 00:33:35
I don't know any better.

660
00:33:31 --> 00:33:37
And here I could even
be off by an eight

661
00:33:34 --> 00:33:40
because that's the uncertainty
in my timing.

662
00:33:36 --> 00:33:42
So these two measurements
confirm.

663
00:33:38 --> 00:33:44
They are in agreement
with each other.

664
00:33:40 --> 00:33:46
You see, uncertainties in
measurements are essential.

665
00:33:45 --> 00:33:51
Now look at our results.

666
00:33:53 --> 00:33:59
We have here a result
which is striking.

667
00:33:59 --> 00:34:05
We have demonstrated that
the time that it takes

668
00:34:01 --> 00:34:07
for an object to fall
is independent of its mass.

669
00:34:07 --> 00:34:13
That is an amazing
accomplishment.

670
00:34:11 --> 00:34:17
Our great-grandfathers must
have worried about this

671
00:34:17 --> 00:34:23
and argued about this
for more than 300 years.

672
00:34:22 --> 00:34:28
Were they so dumb

673
00:34:24 --> 00:34:30
to overlook this simple
dimensional analysis?

674
00:34:31 --> 00:34:37
Inconceivable.

675
00:34:34 --> 00:34:40
Is this dimensional analysis
perhaps not quite kosher?

676
00:34:39 --> 00:34:45
Maybe.

677
00:34:42 --> 00:34:48
Is this dimensional analysis

678
00:34:45 --> 00:34:51
perhaps one that could have
been done differently?

679
00:34:50 --> 00:34:56
Yeah, oh, yeah.

680
00:34:52 --> 00:34:58
You could have done it
very differently.

681
00:34:55 --> 00:35:01
You could have said
the following.

682
00:34:59 --> 00:35:05
You could have said,
"The time for an apple to fall

683
00:43:09 --> 00:43:15
"is proportional to the height
that it falls from

684
00:43:14 --> 00:43:20
to a power alpha."

685
00:35:09 --> 00:35:15
Very reasonable.

686
00:35:11 --> 00:35:17
We all know, the higher it is,
the more it will take--

687
00:35:13 --> 00:35:19
the more time it will take.

688
00:35:15 --> 00:35:21
And we could have said,

689
00:35:16 --> 00:35:22
"Yeah, it's probably
proportional

690
00:43:24 --> 00:43:30
"to the mass somehow.

691
00:43:25 --> 00:43:31
If the mass is more, it will
take a little bit less time."

692
00:35:23 --> 00:35:29
Turns out to be not so,
but you could think that.

693
00:35:26 --> 00:35:32
But you could have said

694
00:35:27 --> 00:35:33
"Well, let's not take the
acceleration of the Earth

695
00:43:35 --> 00:43:41
but let's take the mass
of the Earth itself."

696
00:35:32 --> 00:35:38
Very reasonable, right?

697
00:35:33 --> 00:35:39
I would think if I increased
the mass of the Earth

698
00:35:36 --> 00:35:42
that the apple will fall faster.

699
00:35:39 --> 00:35:45
So now I will put in the math
of the Earth here.

700
00:35:43 --> 00:35:49
And I start
my dimensional analysis

701
00:35:46 --> 00:35:52
and I end up dead in the waters.

702
00:35:49 --> 00:35:55
Because, you see,
there is no mass here.

703
00:35:54 --> 00:36:00
There is a mass
to the power beta here

704
00:35:56 --> 00:36:02
and one to the power gamma

705
00:35:58 --> 00:36:04
so what you would have found
is beta plus gamma equals zero

706
00:36:02 --> 00:36:08
and that would be end of story.

707
00:36:05 --> 00:36:11
Now you can ask yourself
the question

708
00:36:08 --> 00:36:14
well, is there something wrong
with the analysis that we did?

709
00:36:12 --> 00:36:18
Is ours perhaps better
than this one?

710
00:36:15 --> 00:36:21
Well, it's a different one.

711
00:36:17 --> 00:36:23
We came to the conclusion

712
00:36:18 --> 00:36:24
that the time that it takes
for the apple to fall

713
00:36:20 --> 00:36:26
is independent of the mass.

714
00:36:22 --> 00:36:28
Do we believe that?

715
00:36:25 --> 00:36:31
Yes, we do.

716
00:36:27 --> 00:36:33
On the other hand, there are
very prestigious physicists

717
00:36:32 --> 00:36:38
who even nowadays do
very fancy experiments

718
00:36:36 --> 00:36:42
and they try to demonstrate that
the time for an apple to fall

719
00:36:39 --> 00:36:45
does depend on its mass

720
00:36:41 --> 00:36:47
even though it probably is
only very small, if it's true

721
00:36:45 --> 00:36:51
but they try to prove that.

722
00:36:46 --> 00:36:52
And if any of them succeeds
or any one of you succeeds

723
00:36:49 --> 00:36:55
that's certainly worth
a Nobel Prize.

724
00:36:52 --> 00:36:58
So we do believe that it's
independent of the mass.

725
00:36:55 --> 00:37:01
However, this, what I did
with you, was not a proof

726
00:37:00 --> 00:37:06
because if you do it this way,
you get stuck.

727
00:37:04 --> 00:37:10
On the other hand, I'm
quite pleased with the fact

728
00:37:06 --> 00:37:12
that we found that the time
is proportional

729
00:37:08 --> 00:37:14
with the square root of h.

730
00:37:10 --> 00:37:16
I think that's very useful.

731
00:37:11 --> 00:37:17
We confirmed that
with experiment

732
00:37:13 --> 00:37:19
and indeed it came out that way.

733
00:37:16 --> 00:37:22
So it was not a complete
waste of time.

734
00:37:18 --> 00:37:24
But when you do a dimensional
analysis, you better be careful.

735
00:37:25 --> 00:37:31
I'd like you to think this over,
the comparison between the two

736
00:37:31 --> 00:37:37
at dinner and maybe at breakfast

737
00:37:34 --> 00:37:40
and maybe even while
you are taking a shower

738
00:37:37 --> 00:37:43
whether it's needed or not.

739
00:37:39 --> 00:37:45
It is important that you
digest and appreciate

740
00:37:43 --> 00:37:49
the difference between
these two approaches.

741
00:37:46 --> 00:37:52
It will give you an insight
in the power

742
00:37:49 --> 00:37:55
and also into the limitations
of dimensional analysis.

743
00:37:53 --> 00:37:59
This goes to the very heart

744
00:37:55 --> 00:38:01
of our understanding
and appreciation of physics.

745
00:37:59 --> 00:38:05
It's important that
you get a feel for this.

746
00:38:02 --> 00:38:08
You're now at MIT.

747
00:38:04 --> 00:38:10
This is the time.

748
00:38:05 --> 00:38:11
Thank you.

749
00:38:06 --> 00:38:12
See you Friday.

750
00:38:08 --> 00:38:14.000