Speaker A
I'm Walter, I will be your lecturer this term.
Walter Lewin introduces units, dimensions, and measurement uncertainty in physics, emphasizing fundamental quantities and scaling arguments.
Key Takeaways
- Fundamental quantities in physics are length, time, and mass, from which all other units derive.
- Measurement without an understanding of uncertainty is meaningless.
- Decimal units are preferred over imperial units for ease of calculation.
- Physical dimensions help describe and relate different physical quantities.
- Scaling arguments can explain natural phenomena, such as size limits in mammals.
Summary
- Introduction to fundamental physical quantities: length (meter), time (second), and mass (kilogram).
- Discussion of derived units and their practical usage, including metric and imperial systems.
- Explanation of dimensions and how physical quantities like speed, volume, density, and acceleration derive from fundamental units.
- Emphasis on the importance of measurement uncertainty and its critical role in meaningful data.
- Demonstration of measurement uncertainty using an aluminum bar and a volunteer to compare vertical and horizontal lengths.
- Validation of the concept that a person is slightly taller lying down than standing up, with precise measurements.
- Historical reference to Galileo Galilei's reasoning on mammal size limitations related to bone strength.
- Encouragement to use decimal-based units for clarity and ease in physics calculations.
- Introduction to the Powers of 10 movie illustrating orders of magnitude in the universe.
- Stress on trust and accuracy in scientific measurement throughout the course.
Full Transcript — Download SRT & Markdown
Speaker A
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 and the very large is the universe itself.
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They spend 45 orders of magnitude, a one with 45 zeros.
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To express measurements quantitatively, we have to introduce units.
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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.
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And you can read in your book how these are defined and how the definition evolved historically.
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Now there are many derived units which we use in our daily life for convenience and some are tailored to its specific fields.
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We have centimeters, we have millimeters, kilometers, we have inches, feet, miles.
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Astronomers even use the astronomical unit, which is the mean distance between the Earth and the Sun, 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, we have days, weeks, hours, centuries, months.
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All the arrived units.
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For the mass we have milligrams, we have pounds, 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.
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But think about it.
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12 inches in a foot, three feet in a yard.
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That drives you nuts.
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I work almost exclusively decimal 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 which is called The Powers of 10.
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It covers 40 orders of magnitude.
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It was originally conceived by a Dutchman named Kees Boeke in the early 50s.
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This is the second generation movie.
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And you will hear the voice of Professor Morrison.
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Who is a professor at MIT.
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The Powers of 10, 40 orders of magnitude.
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There we go.
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I already introduced, as you see there, length, time and mass.
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And we call these the three fundamental quantities in physics.
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I will give this the symbol capital L.
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For length, capital T for time.
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And capital M for mass.
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All other quantities in physics can be derived from these fundamental quantities.
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I give you an example.
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I put a bracket around here.
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I say speed.
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And that means the dimensions of speed.
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The dimensions of speed is the dimension of length divided by the dimension of time.
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So I can write for that bracket L divided by bracket time.
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Whether it's meters per second or inches per year, that's not what matters.
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It has the dimension length per time.
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Volume.
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Would have the dimension of length to the power 3.
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Density.
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Would have the dimension 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 from these three fundamental.
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So now that we have agreed on the units, 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, which is always ignored in every college book, is the uncertainty in your measurement.
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Any measurement that you make without any knowledge of the uncertainty 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 without a knowledge of its uncertainty is completely meaningless.
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My grandmother used to tell me that, at least she believed it.
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That someone who is lying in bed 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 about the uncertainty in my measurements.
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Because a measurement without knowledge of the uncertainty 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 that when this aluminum bar is sleeping, 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 the length of this aluminum bar.
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With this setup 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.
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I would think that the so this is the aluminum bar, this is in vertical position.
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149.9, but I would think that the uncertainty of my measurement is probably 1 millimeter.
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I can't really guarantee you 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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Oh, 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.
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Horizontally.
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150.0, again plus or minus 0.1 cm.
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So you would agree with me that I am capable of measuring plus or minus.
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And standing up if that were one foot, we would all know it, wouldn't we?
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You get out of bed in the morning, you lie down, you get up and you go.
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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, we would never know.
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Therefore, I suspect that if my grandmother was right.
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That 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 a 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 a volunteer.
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Looks like you're very tall.
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I hope that, yeah.
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I hope we can, I hope that we don't run out of.
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You're not taller than 178 or so.
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No.
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What is your name?
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Rick Rider.
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Rick, Rick Rider.
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You're not nervous, right?
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No.
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Man.
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Sit down.
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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.
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What is your name?
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Zack.
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Zack.
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Nice day today, Zack.
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Yeah.
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You feel all right?
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First lecture at MIT.
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Yes.
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No.
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I don't.
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Okay, man.
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Stand there.
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Yeah.
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Okay, 183.2.
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Stay there.
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Stay there.
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Don't move.
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Zack.
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And this is vertical.
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What did I say?
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180.
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Three.
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Only one person.
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Three.
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Come on.
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0.2.
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Okay.
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183.2.
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Yeah.
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And an uncertainty of about 0.1 cm.
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And now we're going to measure him horizontally.
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Zack.
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I don't want you to break your bones, 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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Don't watch out for this scale, that you don't break that, because then it's all over.
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Okay.
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I'll come on your side.
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I have to do that.
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Yeah.
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Yeah.
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Relax.
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Think of this as a small sacrifice for the sake of science, right?
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Not.
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Okay.
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You good?
Speaker B
Yes.
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You ready?
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Yes.
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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 cm.
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Oh, that is 5, that is 2.5 plus or minus 0.2 cm.
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You're about 1 inch taller when you sleep 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, thank you very much, Zack.
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That the accuracy of 1 millimeter was more than sufficient to make the case.
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If the accuracy in my measurement would have been much less, this measurement would not have been convincing at all.
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So whenever you make a measurement, you must know the uncertainty, otherwise it is meaningless.
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Galileo Galilei.
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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 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, 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 the 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 a mouse is J big and a cat is J big.
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That's what I mean by size.
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Very crudely defined.
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The mass of the mammal is M.
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And this mammal has a thigh bone, which we call the femur.
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Which is here.
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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 and has a thickness D.
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Here is a femur.
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This is what the femur approximately looks like.
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So this would be the length of the femur.
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And this would be the thickness D.
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And this would be the cross-sectional area A.
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I'm now going to take you through.
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What we call in physics a scaling argument.
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I would argue that the length of the femur must be proportional to the size of the animal.
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That's completely plausible.
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If an animal is four times larger than another, 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 is proportional to the third power of the size.
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Because that's related to its volume.
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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.
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Because of this relationship.
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Okay, that's one.
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Now comes the argument.
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Pressure.
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On the femur is proportional to the weight of the animal.
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Divided by the cross section A of the femur.
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That's what pressure is.
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And that is the mass of the animal, that's proportional to the mass of the animal divided by D squared.
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Because we want the area here.
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It's proportional to D squared.
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Now follow me closely.
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If the pressure is higher than a certain level, the bones will break.
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Therefore, for an animal not to break its bones when the mass goes up by a certain factor.
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Say a factor of four.
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In order for the bones not to break, 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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You really have to think that through carefully.
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Therefore, I would argue that the mass must be proportional to D squared.
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This is the breaking argument.
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Now compare these two.
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The mass is proportional to the length of the femur to the power three.
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And to the thickness of the femur to the power two.
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Therefore, the thickness of the femur to the power two must be proportional to the length L.
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And therefore the thickness of the femur must be proportional to L to the power 3/2.
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A very interesting result.
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What is this result telling you?
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It tells you that if I have two animals and one is 10 times larger than the other.
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That S is 10 times larger.
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That the lengths of the legs are 10 times larger.
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But that the thickness of the femur is 30 times larger because it is L to the power 3/2.
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If I were to compare a mouse with an elephant.
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An elephant is about 100 times larger in size.
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So the length of the femur of the elephant would be 100 times larger than that of a mouse.
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But the thickness of the femur would have to be 1,000 times larger.
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And that.
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May have convinced Galileo Galilei that that's the reason why.
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The largest animals are as large as they are.
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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.
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You're all made of bones and that is biologically not feasible.
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And so there is a limit somewhere.
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Set by this scaling law.
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Well, I I wanted to bring this to a test.
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After all, I brought my grandmother's statement to a test.
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So why not bringing Galileo Galilei's statement to a test?
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And so I went to Harvard where they have a beautiful collection of femurs.
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And I asked them for the femur of a raccoon and a horse.
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A raccoon is this big.
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A horse is about four times bigger.
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So the length of the femur of a horse must be about four times the length of the raccoon.
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Close.
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So I was not surprised.
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Then I measured the thickness.
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And I said to myself, ah.
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If the length is four times higher, then the thickness has to be eight times higher if this holds.
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And what I'm going to plot for you.
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You will see that shortly.
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Is D divided by L versus L.
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And that, of course, must be proportional to L to the power 1/2.
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I bring one L here.
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So if I compare the horse and I compare the raccoon.
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I would argue that the thickness divided by the length of the femur for the horse must be the square root of four.
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Twice as much as that of the raccoon.
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And so I was very anxious to plot that.
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And I did that.
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And I show you the result.
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Here is my first result.
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So, we see there D over L.
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I explained to you why I prefer that to plot it.
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And here you see the length.
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You see here the raccoon.
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And you see the horse.
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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.
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Well, I wasn't too disappointed.
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One and a half is not two.
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But it is in the right direction.
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The horse clearly have a larger value for D over L than the raccoon.
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I realized I needed more data.
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So I went back to Harvard.
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I said, look, I need a smaller animal.
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An opossum, maybe, maybe a rat, maybe a mouse.
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And they said, okay.
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They gave me three more bones.
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They gave me an antelope, which is actually a little larger than the raccoon.
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And they gave me an opossum.
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And they gave me a mouse.
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Here is the bone of the antelope.
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Here is the one of the raccoon.
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Here is the one of the opossum.
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And now you won't believe this.
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This is so wonderful.
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So romantic.
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There is the mouse.
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Isn't that beautiful?
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Teeny, weeny, little mouse, it's only a teeny, weeny, little femur.
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And there it is.
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And I, um.
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I made the plot.
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I was very curious what that plot would look like.
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And.
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Here it is.
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I was shocked.
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I was really shocked.
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Because look.
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The horse is 50 times larger in size than the mouse.
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The difference in D over L is only a factor of two.
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And I expected something more like a factor of seven.
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And so in D over L where I expect a factor of seven, I only see a factor of two.
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So I said to myself, oh my goodness.
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Why didn't I ask them for an elephant?
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The real clincher would be the elephant.
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Because if that goes way off scale, maybe we can still rescue the statement by Galileo Galilei.
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And so I went back.
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And they said, okay, we'll give you the femur of an elephant.
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They also gave me one of a moose, believe it or not.
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I think they wanted to get rid of me by that time, to be frank of you.
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And here is the femur of an elephant.
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And I measured it.
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The length and the thickness.
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And it is very heavy.
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It weighs a ton.
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I plotted it.
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I was full of expectation.
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I couldn't sleep all night.
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And there's the elephant.
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There is no evidence whatsoever that D over L is really larger for the elephant than for the mouse.
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These vertical bars indicate my uncertainty.
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In measurements of thickness and the horizontal scale, which is a logarithmic scale, the uncertainty of the length measurements.
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Is in the thickness of the red pen, so there's no need for me to indicate that any further.
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And here you have your measurements.
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In case you want to check them.
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And look again at the mouse.
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And look at the elephant.
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The mouse has indeed only 1 cm length of the femur and the elephant is indeed 100 times longer.
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So the first scaling argument that S is proportional to L, that is certainly what you expect.
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Because elephant is about 100 times larger in size.
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But when you go to D over L, you see it's all over.
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The D over L for the mouth is really not all that different from the elephant.
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And you would have expected that number to be with the square root of.
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100, so you expect it to be 10 times larger instead of about the same.
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I now want to discuss with you.
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What we call in physics dimensional analysis.
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I want to ask myself the question.
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If I drop an apple from a certain height.
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And I change that height.
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What will happen with the time for the apple to fall?
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Well.
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I drop the apple from a height H.
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And I want to know what happens with the time when it falls.
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And I change H.
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So I said to myself, well.
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The time that it takes must be proportional to the height to some power alpha.
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Completely reasonable.
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If I make the height larger, we all know that it takes longer for the apple to fall.
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But that's a safe thing.
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I said to myself, well.
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If the apple has a mass M, it probably is also proportional to the mass of that apple to the power beta.
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Turns out to be not so.
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But you could think that.
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But you could have said, well, let's not take the acceleration of the Earth.
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But let's take the mass of the Earth itself.
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Very reasonable, right?
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I would think if I increase the mass of the Earth, that the apple will fall faster.
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So now I would put in the mass of the Earth here.
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And I start my dimensional analysis and I end up dead in the waters.
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Because you see, there is no mass here.
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There is a mass to the power beta here and one to the power gamma.
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So what you would have found is beta plus gamma equals zero.
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And that would be end of story.
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Now you can ask yourself the question.
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Well, is there something wrong with the analysis that we did?
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Is ours perhaps better than this one?
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Well, it's a different one.
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We came to the conclusion that the time that it takes for the apple to fall is independent of the mass.
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Do we believe that?
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Yes, we do.
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On the other hand, there are very prestigious physicists.
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Who even nowadays do very fancy experiments and they try to demonstrate.
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That the time for an apple to fall does depend on its mass.
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Even though it probably is only very small if it's true, but they try to prove that.
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If any of them succeeds or any one of you succeeds, that's certainly worth a Nobel Prize.
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So we do believe that it's independent of the mass.
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However.
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This what I did with you was not a proof.
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Because if you do it this way.
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You get stuck.
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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.
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I think that's very useful.
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We confirm that with experiment.
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And indeed it came out that way.
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So it was not a complete waste of time.
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But when you do a dimensional analysis.
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You better be careful.
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I like you to think this over.
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The comparison between the two.
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At dinner and maybe at breakfast and maybe even while you're taking a shower.
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Whether it's needed or not.
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It is important that you digest and appreciate the difference between these two approaches.
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It will give you an insight in the power.
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And also into the limitations of dimensional analysis.
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This goes to the very heart of our understanding and appreciation of physics.
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It's important that you get a feel for this.
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You're now at MIT.
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This is the time.
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Thank you, see you Friday.
Topics:unitsdimensionsmeasurement uncertaintyfundamental quantitiesscaling argumentsphysicsWalter Lewinclassical mechanicsPowers of 10Galileo Galilei
