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8.01x - Lect 1 - Powers of 10, Units, Dimensions, Uncertainties, Scaling Arguments — Transcript

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0:00I'm Walter Lewin.
0:01I will be your lecturer this term.
0:04In physics, we explore the very small to the very large.
0:10The very small is a small fraction of a proton
0:13and the very large is the universe itself.
0:16They span 45 orders of magnitude--
0:20a 1 with 45 zeroes.
0:24To express measurements quantitatively
0:28we have to introduce units.
0:31And we introduce for the unit of length, the meter;
0:37for the unit of time, the second;
0:41and for the unit of mass, the kilogram.
0:46And you can read in your book how these are defined
0:49and how the definition evolved historically.
0:54Now, there are many derived units
0:56which we use in our daily life for convenience
0:59and some are tailored toward specific fields.
1:02We have centimeters, we have millimeters
1:05kilometers.
1:06We have inches, feet, miles.
1:10Astronomers even use the astronomical unit
1:13which is the mean distance between the Earth and the sun
1:15and they use light-years
1:17which is the distance that light travels in one year.
1:21We have milliseconds, we have microseconds
1:24we have days, weeks, hours, centuries, months--
1:27all derived units.
1:29For the mass, we have milligrams, we have pounds
1:34we have metric tons.
1:36So lots of derived units exist.
1:41Not all of them are very easy to work with.
1:44I find it extremely difficult to work with inches and feet.
1:48It's an extremely uncivilized system.
1:50I don't mean to insult you, but think about it--
1:5212 inches in a foot, three feet in a yard.
1:56Could drive you nuts.
1:57I work almost exclusively decimal,
2:01and I hope you will do the same during this course
2:03but we may make some exceptions.
2:06I will now first show you a movie,
2:08which is called The Powers of Ten.
2:11It covers 40 orders of magnitude.
2:13It was originally conceived by a Dutchman named Kees Boeke
2:17in the early '50s.
2:19This is the second-generation movie, and you will hear
2:23the voice of Professor Morrison, who is a professor at MIT.
2:30The Powers of Ten-- 40 Orders of Magnitude.
2:37Here we go.
2:40I already introduced, as you see there
2:42length, time and mass
2:45and we call these
2:46the three fundamental quantities in physics.
2:51I will give this the symbol capital L for length
2:55capital T for time, and capital M for mass.
2:59Many other quantities in physics can be derived
3:02from these fundamental quantities.
3:05I'll give you an example.
3:07I put a bracket around here.
3:10I say speed, and that means the dimensions of speed.
3:14The dimensions of speed is the dimension of length
3:16divided by the dimension of time.
3:19So I can write for that: [L] divided by [T].
3:24Whether it's meters per second or inches per year
3:27that's not what matters.
3:28It has the dimension length per time.
3:31Volume would have the dimension
3:37of length to the power three.
3:42Density would have the dimension
3:47of mass per unit volume
3:51so that means length to the power three.
3:54All-important in our course is acceleration.
3:59We will deal a lot with acceleration.
4:02Acceleration, as you will see, is length per time squared.
4:06The unit is meters per second squared.
4:08So you get length divided by time squared.
4:17So all other quantities can be derived
4:19from these three fundamental.
4:22So now that we have agreed on the units--
4:25we have the meter, the second and the kilogram--
4:28we can start making measurements.
4:30Now, all-important in making measurements
4:33which is always ignored in every college book
4:36is the uncertainty in your measurement.
4:40Any measurement that you make
4:43without any knowledge of the uncertainty
4:45is meaningless.
4:47I will repeat this.
4:49I want you to hear it tonight at 3:00 when you wake up.
4:52Any measurement that you make
4:55without the knowledge of its uncertainty
4:57is completely meaningless.
5:01My grandmother used to tell me that...
5:05at least she believed it...
5:07that someone who is lying in bed
5:09is longer than someone who stands up.
5:12And in honor of my grandmother
5:14I'm going to bring this today to a test.
5:19I have here a setup where I can measure a person standing up
5:23and a person lying down.
5:26It's not the greatest bed, but lying down.
5:29I have to convince you
5:30about the uncertainty in my measurement
5:33because a measurement without knowledge of the uncertainty
5:35is meaningless.
5:36And therefore, what I will do is the following.
5:39I have here an aluminum bar
5:41and I make the reasonable, plausible assumption
5:45that when this aluminum bar is sleeping--
5:47when it is horizontal--
5:49that it is not longer than when it is standing up.
5:52If you accept that, we can compare
5:54the length of this aluminum bar with this setup
5:58and with this setup.
5:59At least we have some kind of calibration to start with.
6:03I will measure it.
6:03You have to trust me.
6:05During these three months, we have to trust each other.
6:08So I measure here, 149.9 centimeters.
6:16However, I would think that the...
6:19so this is the aluminum bar.
6:21This is in vertical position.
6:24149.9.
6:27But I would think that the uncertainty of my measurement
6:30is probably 1 millimeter.
6:32I can't really guarantee you
6:33that I did it accurately any better.
6:36So that's the vertical one.
6:38Now we're going to measure the bar horizontally
6:42for which we have a setup here.
6:43Oops!
6:44The scale is on your side.
6:46So now I measure the length of this bar.
6:49150.0 horizontally.
6:56150.0, again, plus or minus 0.1 centimeter.
7:01So you would agree with me that I am capable of measuring
7:05plus or minus 1 millimeter.
7:06That's the uncertainty of my measurement.
7:10Now, if the difference in lengths
7:14between lying down and standing up
7:16if that were one foot
7:18we would all know it, wouldn't we?
7:20You get out of bed in the morning
7:21you lie down and you get up and you go, clunk!
7:23And you're one foot shorter.
7:24And we know that that's not the case.
7:26If the difference were only one millimeter
7:29we would never know.
7:31Therefore, I suspect that if my grandmother was right
7:35then it's probably only a few centimeters,
7:37maybe an inch.
7:39And so I would argue that if I can measure
7:41the length of a student to one millimeter accuracy
7:45that should settle the issue.
7:47So I need a volunteer.
7:51You want to volunteer?
7:52You look like you're very tall.
7:53I hope that... yeah, I hope that we don't run out of, uh...
7:59You're not taller than 178 or so?
8:02What is your name?
8:03STUDENT: Rick Ryder.
8:04LEWIN: Rick-- Rick Ryder.
8:05You're not nervous, right?
8:06RICK: No!
8:08LEWIN: Man!
8:09(class laughs)
8:11Sit down.
8:12(class laughs)
8:15I can't have tall guys here.
8:16Come on.
8:17We need someone more modest in size.
8:21Don't take it personal, Rick.
8:24Okay, what is your name?
8:27STUDENT: Zach.
8:27LEWIN: Zach.
8:30Nice day today, Zach, yeah?
8:32You feel all right?
8:34Your first lecture at MIT?
8:36I don't.
8:39Okay, man.
8:40Stand there, yeah.
8:44Okay, 183.2.
8:49Stay there, stay there.
8:49Don't move.
8:51Zach...
8:55This is vertical.
8:57What did I say? 180?
9:01Only one person.
9:033?
9:06Come on.
9:09.2 Okay.
9:11183.2.
9:13Yeah.
9:14And an uncertainty of about one...
9:19Oh, this is centimeters-- 0.1 centimeters.
9:24And now we're going to measure him horizontally.
9:29Zach, I don't want you to break your bones
9:31so we have a little step for you here.
9:35Put your feet there.
9:37Oh, let me remove the aluminum bar.
9:39Don't... Watch out for the scale.
9:40That you don't break that, because then it's all over.
9:44Okay, I'll come on your side.
9:45I have to do that-- yeah, yeah.
9:48Relax.
9:51Think of this as a small sacrifice
9:53for the sake of science, right?
9:55It's not... Okay, you good?
9:57ZACH: Yeah.
9:58LEWIN: You comfortable?
10:00(students laugh)
10:01You're really comfortable, right?
10:02ZACH: Wonderful.
10:03LEWIN: Okay. You're ready?
10:06ZACH: Yes.
10:07LEWIN: Okay.
10:10Okay.
10:13185.7.
10:15Stay where you are. 185.7.
10:19I'm sure... I want to first make the subtraction, right?
10:22185.7, plus or minus 0.1 centimeter.
10:28Oh, that is five...
10:30that is 2.5 plus or minus 0.2 centimeters.
10:35You're about one inch taller when you sleep
10:37than when you stand up.
10:37My grandmother was right.
10:39She's always right.
10:40Can you get off here?
10:42I want you to appreciate that the accuracy...
10:45Thank you very much, Zach.
10:46That the accuracy of one millimeter
10:48was more than sufficient to make the case.
10:51If the accuracy of my measurements
10:53would have been much less
10:54this measurement would not have been convincing at all.
10:59So whenever you make a measurement
11:00you must know the uncertainty.
11:01Otherwise, it is meaningless.
11:05Galileo Galilei asked himself the question:
11:10Why are mammals as large as they are and not much larger?
11:17He had a very clever reasoning which I've never seen in print.
11:20But it comes down to the fact that he argued
11:23that if the mammal becomes too massive
11:27that the bones will break
11:29and he thought that that was a limiting factor.
11:32Even though I've never seen his reasoning in print
11:35I will try to reconstruct it
11:37what could have gone through his head.
11:39Here is a mammal.
11:43And this is the... one of the four legs of the mammal.
11:48And this mammal has a size S.
11:55And what I mean by that is
11:57a mouse is yay big and a cat is yay big.
12:01That's what I mean by size-- very crudely defined.
12:06The mass of the mammal is M
12:09and this mammal has a thigh bone
12:13which we call the femur, which is here.
12:17And the femur of course carries the body, to a large extent.
12:22And let's assume that the femur has a length l
12:25and has a thickness d.
12:27Here is a femur.
12:34This is what a femur approximately looks like.
12:37So this will be the length of the femur...
12:45and this will be the thickness, d
12:49and this will be the cross-sectional area A.
12:57I'm now going to take you through what we call in physics
13:01a scaling argument.
13:04I would argue that the length of the femur
13:07must be proportional to the size of the animal.
13:10That's completely plausible.
13:11If an animal is four times larger than another
13:14you would need four times longer legs.
13:16And that's all this is saying.
13:18It's very reasonable.
13:21It is also very reasonable that the mass of an animal
13:24is proportional to the third power of the size
13:28because that's related to its volume.
13:31And so if it's related to the third power of the size
13:34it must also be proportional
13:36to the third power of the length of the femur
13:39because of this relationship.
13:42Okay, that's one.
13:45Now comes the argument.
13:48Pressure on the femur is proportional
13:54to the weight of the animal divided by the cross-section A
13:59of the femur.
14:01That's what pressure is.
14:03And that is the mass of the animal
14:05that's proportional
14:06to the mass of the animal divided by d squared
14:09because we want the area here, it's proportional to d squared.
14:14Now follow me closely.
14:18If the pressure is higher than a certain level
14:22the bones will break.
14:25Therefore, for an animal not to break its bones
14:29when the mass goes up by a certain factor
14:31let's say a factor of four
14:33in order for the bones not to break
14:35d squared must also go up by a factor of four.
14:38That's a key argument in the scaling here.
14:40You really have to think that through carefully.
14:43Therefore, I would argue
14:45that the mass must be proportional to d squared.
14:48This is the breaking argument.
14:51Now compare these two.
14:53The mass is proportional to the length of the femur
14:56to the power three
14:57and to the thickness of the femur to the power two.
15:00Therefore, the thickness of the femur to the power two
15:05must be proportional to the length l
15:07and therefore the thickness of the femur must be proportional
15:10to l to the power three-halfs.
15:13A very interesting result.
15:16What is this result telling you?
15:19It tells you that if I have two animals
15:23and one is ten times larger than the other
15:26then S is ten times larger
15:28that the lengths of the legs are ten times larger
15:31but that the thickness of the femur is 30 times larger
15:38because it is l to the power three halves.
15:39If I were to compare a mouse with an elephant
15:42an elephant is about a hundred times larger in size
15:46so the length of the femur of the elephant
15:48would be a hundred times larger than that of a mouse
15:50but the thickness of the femur
15:52would have to be 1,000 times larger.
15:57And that may have convinced Galileo Galilei
16:01that that's the reason
16:02why the largest animals are as large as they are.
16:06Because clearly, if you increase the mass
16:09there comes a time that the thickness of the bones
16:12is the same as the length of the bones.
16:14You're all made of bones
16:16and that is biologically not feasible.
16:18And so there is a limit somewhere
16:20set by this scaling law.
16:25Well, I wanted to bring this to a test.
16:28After all
16:29I brought my grandmother's statement to a test
16:31so why not bring Galileo Galilei's statement to a test?
16:35And so I went to Harvard
16:38where they have a beautiful collection of femurs
16:42and I asked them for the femur of a raccoon and a horse.
16:48A raccoon is this big
16:50a horse is about four times bigger
16:54so the length of the femur of a horse
16:57must be about four times the length of the raccoon.
17:01Close.
17:03So I was not surprised.
17:05Then I measured the thickness, and I said to myself, "Aha!"
17:11If the length is four times higher
17:14then the thickness has to be eight times higher
17:18if this holds.
17:20And what I'm going to plot for you
17:21you will see that shortly is d divided by l, versus l
17:27and that, of course, must be proportional
17:28to l to the power one-half.
17:30I bring one l here.
17:32So, if I compare the horse and I compare the raccoon
17:36I would argue that the thickness
17:38divided by the length of the femur for the horse
17:41must be the square root of four, twice as much
17:45as that of the raccoon.
17:47And so I was very anxious to plot that, and I did that
17:52and I'll show you the result.
17:55Here is my first result.
18:01So we see there, d over l.
18:03I explained to you why I prefer that.
18:07And here you see the length.
18:08You see here the raccoon and you see the horse.
18:11And if you look carefully, then the d over l for the horse
18:14is only about one and a half times larger than the raccoon.
18:17Well, I wasn't too disappointed.
18:20One and a half is not two, but it is in the right direction.
18:22The horse clearly has a larger value for d over l
18:25than the raccoon.
18:28I realized I needed more data, so I went back to Harvard.
18:31I said, "Look, I need a smaller animal, an opossum maybe
18:35maybe a rat, maybe a mouse," and they said, "okay."
18:39They gave me three more bones.
18:42They gave me an antelope
18:43which is actually a little larger than a raccoon
18:46and they gave me an opossum and they gave me a mouse.
18:51Here is the bone of the antelope.
18:59Here is the one of the raccoon.
19:06Here is the one of the opossum.
19:09And now you won't believe this.
19:12This is so wonderful, so romantic.
19:17There is the mouse.
19:18(students laugh)
19:20Isn't that beautiful?
19:21Teeny, weeny little mouse?
19:23That's only a teeny, weeny little femur.
19:27And there it is.
19:29And I made the plot.
19:33I was very curious what that plot would look like.
19:36And...
19:42here it is.
19:46Whew! I was shocked.
19:48I was really shocked.
19:51Because look-- the horse is 50 times larger in size
19:55than the mouse.
19:56The difference in d over l is only a factor of two.
20:00And I expected something more like a factor of seven.
20:06And so, in d over l, where I expect a factor of seven
20:09I only see a factor of two.
20:11So I said to myself, "Oh, my goodness.
20:13Why didn't I ask them for an elephant?"
20:16The real clincher would be the elephant
20:18because if that goes way off scale
20:21maybe we can still rescue the statement by Galileo Galilei
20:25and so I went back and they said
20:28"Okay, we'll give you the femur of an elephant."
20:30They also gave me one of a moose, believe it or not.
20:32I think they wanted to get rid of me by that time
20:34to be frank with you.
20:36And here is the femur of an elephant.
20:41And I measured it.
20:42The length and the thickness.
20:45And it is very heavy.
20:48It weighs a ton.
20:50I plotted it, I was full of expectation.
20:54I couldn't sleep all night.
20:56And there's the elephant.
20:59There is no evidence whatsoever that d over l is really larger
21:03for the elephant than for the mouse.
21:04These vertical bars indicate my uncertainty
21:07in measurements of thickness
21:09and the horizontal scale, which is a logarithmic scale...
21:12the uncertainty of the length measurements
21:15is in the thickness of the red pen
21:16so there's no need for me to indicate that any further.
21:20And here you have your measurements
21:22in case you want to check them.
21:24And look again at the mouse and look at the elephant.
21:28The mouse has indeed only one centimeter length of the femur
21:35and the elephant is, indeed, hundred times longer.
21:37So the first scaling argument that S is proportional to l
21:41that is certainly what you would expect
21:43because an elephant is about a hundred times larger in size.
21:46But when you go to d over l, you see it's all over.
21:49The d over l for the mouse
21:51is really not all that different from the elephant
21:54and you would have expected that number to be
21:57with the square root of 100
22:01so you expect it to be ten times larger
22:03instead of about the same.
22:07I now want to discuss with you
22:09what we call in physics dimensional analysis.
22:16I want to ask myself the question:
22:19If I drop an apple from a certain height
22:24and I change that height
22:27what will happen with the time for the apple to fall?
22:34Well, I drop the apple from a height h
22:39and I want to know what happened with the time when it falls.
22:43And I change h.
22:46So I said to myself, "Well, the time that it takes
22:48must be proportional to the height to some power alpha."
22:53Completely reasonable.
22:54If I make the height larger
22:55we all know that it takes longer for the apple to fall.
22:58That's a safe thing.
23:00I said to myself, "Well, if the apple has a mass m
23:04it probably is also proportional
23:06to the mass of that apple to the power beta."
23:09I said to myself, "Gee, yeah, if something is more massive
23:13it will probably take more time."
23:15So maybe m to some power beta.
23:17I don't know alpha, I don't know beta.
23:20And then I said, "Gee, there's also something like gravity
23:23that is the Earth's gravitational pull--
23:25the gravitational acceleration of the Earth."
23:28So let's introduce that, too
23:30and let's assume that that time is also proportional
23:33to the gravitational acceleration--
23:35this is an acceleration; we will learn a lot more about that--
23:38to the power gamma.
23:41Having said this, we can now do what's called in physics
23:45a dimensional analysis.
23:51On the left we have a time
23:55and if we have a left... on the left side a time
23:57on the right side we must also have time.
24:00You cannot have coconuts on one side and oranges on the other.
24:04You cannot have seconds on one side
24:06and meters per second on the other.
24:09So the dimensions left and right have to be the same.
24:12What is the dimension here?
24:14That is [T] to the power one.
24:17That T... that must be the same as length to the power alpha
24:26times mass to the power beta, times acceleration--
24:34remember, it is still there on the blackboard--
24:36that's dimension [L] divided by time squared
24:42and the whole thing to the power gamma
24:43so I have a gamma here and I have a gamma there.
24:46This side must have the same dimension as that side.
24:48That is nonnegotiable in physics.
24:51Okay, there we go.
24:53There is no M here, there is only one M here
24:56so beta must be zero.
24:59There is here [L] to the power alpha, [L] to the power gamma
25:03there is no [L] here.
25:05So [L] must disappear.
25:07So alpha plus gamma must be zero.
25:11There is [T] to the power one here
25:14and there is here [T] to the power -2 gamma.
25:17It's minus because it's downstairs.
25:19So one must be equal to -2 gamma.
25:23That means gamma must be minus one half.
25:27That if gamma is minus one half, then alpha equals plus one half.
25:34End of my dimensional analysis.
25:37I therefore conclude that the time that it takes
25:41for an object to fall
25:43equals some constant, which I do not know
25:47but that constant has no dimension--
25:49I don't know what it is--
25:51times the square root of h divided by g.
25:59Beta is zero, there is no mass
26:02h to the power one half-- you see that here--
26:05and g to the power minus one half.
26:07This is proportional to the square root of h
26:11because g is a given and c is a given
26:12even though I don't know c.
26:14I make no pretense that I can predict how long it will take
26:18for the apple to fall.
26:19All I'm saying is, I can compare two different heights.
26:23I can drop an apple from eight meters
26:25and another one from two meters
26:27and the one from eight meters will take two times longer
26:31than the one from two meters.
26:33The square root of h to two, four over two
26:37will take two times longer, right?
26:38If I drop one from eight meters
26:40and I drop another one from two meters
26:43then the difference in time will be the square root of the ratio.
26:47That will be twice as long.
26:49And that I want to bring to a test today.
26:55We have a setup here.
26:57We have an apple there at a height of three meters
27:00and we know the length to an accuracy... the height
27:03of about three millimeters, no better.
27:05And here we have a setup whereby the apple
27:07is about one and a half meters above the ground.
27:10And we know that to about also an accuracy
27:13of no better than about three millimeters.
27:19So, let's set it up.
27:21I have here...
27:26something that's going to be a prediction--
27:29a prediction of the time that it takes for one apple to fall
27:35divided by the time that it takes
27:37for the other apple to fall.
27:39h1 is three meters
27:43but I claim there is an uncertainty
27:45of about three millimeters.
27:47Can't do any better.
27:49And h2 equals 1.5 meters
27:54again with an uncertainty of about three millimeters.
28:01So the ratio h1 over h2...
28:06is 2.000
28:09and now I have to come up with an uncertainty
28:11which physicists sometimes call an error in their measurements
28:15but it's really an uncertainty.
28:16And the way you find your uncertainty is
28:19that you add the three here
28:21and you subtract the three here
28:23and you get the largest value possible.
28:25You can never get a larger value.
28:27And you'll find that you get 2.006.
28:30And so I would say the uncertainty is then .006.
28:36This is a dimensionless number
28:38because it's length divided by length.
28:42And so the time t1 divided by t2
28:47would be the square root of h1 divided by h2.
28:51That is the dimensional analysis argument
28:54that we have there.
28:55And we find if we take the square root of this number
28:58we find 1.414, plus or minus 0.0
29:04and I think that is a two.
29:06That is correct.
29:08So here is a firm prediction.
29:14This is a prediction.
29:17And now we're going to make an observation.
29:23So we're going to measure t1 and there's going to be a number
29:29and then we're going to measure t2
29:32and there's going to be a number.
29:34I have done this experiment ten times
29:36and the numbers always reproduce within about one millisecond.
29:41So I could just adopt an uncertainty of one millisecond.
29:43I want to be a little bit on the safe side.
29:45Occasionally it differs by two milliseconds.
29:48So let us be conservative
29:50and let's assume that I can measure this to an accuracy
29:55of about two milliseconds.
29:57That is pretty safe.
30:00So now we can measure these times
30:04and then we can take the ratio
30:07and then we can see whether we actually confirm
30:11that the time that it takes is proportional to the height
30:16to the square root of the height.
30:18So I will make it a little more comfortable for you
30:22in the lecture hall.
30:27That's all right.
30:29We have the setup here.
30:31We first do the experiment with the... three meters.
30:39There you see the three meters.
30:41And the time... the moment that I pull this string
30:45the apple will fall, the contact will open, the clock will start.
30:49The moment that it hits the floor, the time will stop.
30:54I have to stand on that side.
30:56Otherwise the apple will fall on my hand.
30:58That's not the idea.
31:00I'll stand here.
31:02You ready?
31:04Okay, then I'm ready.
31:07Everything set?
31:08Make sure that I've zeroed that properly.
31:10Yes, I have.
31:12Okay.
31:13Three, two, one, zero.
31:18781 milliseconds.
31:22So this number... you should write it down
31:26because you will need it for your second assignment.
31:29781 milliseconds, with an uncertainty of two milliseconds.
31:34You ready for the second one?
31:39You ready?
31:42You ready?
31:43Okay, nothing wrong.
31:46Ready.
31:50Zero, zero, right?
31:53Thank you.
31:54Okay.
31:55Three, two, one, zero.
32:00551 milliseconds.
32:05Boy, I'm nervous because I hope that physics works.
32:13So I take my calculator
32:17and I'm now going to take the ratio t1 over t2.
32:24The uncertainty you can find by adding the two here
32:28and subtracting the two there
32:30and that will then give you an uncertainty
32:32of, I think, .0... mmm, .08.
32:38Yeah, .08.
32:39You should do that for yourself-- .008.
32:43Dimensionless number.
32:44This would be the uncertainty.
32:47This is the observation.
32:49781 divided by 551.
32:56One point...
32:57Let me do that once more.
32:59Seven eight one, divided by five five one...
33:03One four one seven.
33:09Perfect agreement.
33:11Look, the prediction says 1.414
33:16but it could be 1 point... it could be two higher.
33:19That's the uncertainty in my height.
33:21I don't know any better.
33:23And here I could even be off by an eight
33:26because that's the uncertainty in my timing.
33:28So these two measurements confirm.
33:30They are in agreement with each other.
33:32You see, uncertainties in measurements are essential.
33:37Now look at our results.
33:45We have here a result which is striking.
33:50We have demonstrated that the time that it takes
33:53for an object to fall is independent of its mass.
34:00That is an amazing accomplishment.
34:05Our great-grandfathers must have worried about this
34:09and argued about this for more than 300 years.
34:14Were they so dumb
34:16to overlook this simple dimensional analysis?
34:23Inconceivable.
34:26Is this dimensional analysis perhaps not quite kosher?
34:31Maybe.
34:35Is this dimensional analysis
34:38perhaps one that could have been done differently?
34:42Yeah, oh, yeah.
34:44You could have done it very differently.
34:47You could have said the following.
34:51You could have said, "The time for an apple to fall
34:55"is proportional to the height that it falls from
34:59to a power alpha."
35:01Very reasonable.
35:02We all know, the higher it is, the more it will take--
35:04the more time it will take.
35:07And we could have said,
35:08"Yeah, it's probably proportional
35:10"to the mass somehow.
35:11If the mass is more, it will take a little bit less time."
35:15Turns out to be not so, but you could think that.
35:17But you could have said
35:18"Well, let's not take the acceleration of the Earth
35:22but let's take the mass of the Earth itself."
35:24Very reasonable, right?
35:25I would think if I increased the mass of the Earth
35:28that the apple will fall faster.
35:30So now I will put in the math of the Earth here.
35:35And I start my dimensional analysis
35:37and I end up dead in the waters.
35:41Because, you see, there is no mass here.
35:46There is a mass to the power beta here
35:48and one to the power gamma
35:50so what you would have found is beta plus gamma equals zero
35:54and that would be end of story.
35:58Now you can ask yourself the question
36:00well, is there something wrong with the analysis that we did?
36:04Is ours perhaps better than this one?
36:07Well, it's a different one.
36:09We came to the conclusion
36:10that the time that it takes for the apple to fall
36:12is independent of the mass.
36:15Do we believe that?
36:17Yes, we do.
36:20On the other hand, there are very prestigious physicists
36:24who even nowadays do very fancy experiments
36:28and they try to demonstrate that the time for an apple to fall
36:32does depend on its mass
36:33even though it probably is only very small, if it's true
36:37but they try to prove that.
36:38And if any of them succeeds or any one of you succeeds
36:41that's certainly worth a Nobel Prize.
36:44So we do believe that it's independent of the mass.
36:47However, this, what I did with you, was not a proof
36:52because if you do it this way, you get stuck.
36:56On the other hand, I'm quite pleased with the fact
36:58that we found that the time is proportional
37:00with the square root of h.
37:01I think that's very useful.
37:03We confirmed that with experiment
37:05and indeed it came out that way.
37:07So it was not a complete waste of time.
37:09But when you do a dimensional analysis, you better be careful.
37:17I'd like you to think this over, the comparison between the two
37:23at dinner and maybe at breakfast
37:26and maybe even while you are taking a shower
37:29whether it's needed or not.
37:31It is important that you digest and appreciate
37:35the difference between these two approaches.
37:38It will give you an insight in the power
37:41and also into the limitations of dimensional analysis.
37:45This goes to the very heart
37:47of our understanding and appreciation of physics.
37:50It's important that you get a feel for this.
37:54You're now at MIT.
37:56This is the time.
37:58Thank you, see you Friday.

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