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.
What the video covers
- 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- Historical reference to Galileo Galilei's reasoning on mammal size limitations related to bone strength.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.
Speaker A
The very small is a small fraction of a proton and the very large is the universe itself.
Speaker A
They spend 45 orders of magnitude, a one with 45 zeros.
Speaker A
To express measurements quantitatively, we have to introduce units.
Speaker A
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.
Speaker A
And you can read in your book how these are defined and how the definition evolved historically.
Speaker A
Now there are many derived units which we use in our daily life for convenience and some are tailored to its specific fields.
Speaker A
We have centimeters, we have millimeters, kilometers, we have inches, feet, miles.
Speaker A
Astronomers even use the astronomical unit, which is the mean distance between the Earth and the Sun, and they use light years.
Speaker A
Which is the distance that light travels in one year.
Speaker A
We have milliseconds, we have microseconds, we have days, weeks, hours, centuries, months.
Speaker A
All the arrived units.
Speaker A
For the mass we have milligrams, we have pounds, we have metric tons.
Speaker A
So lots of derived units exist.
Speaker A
Not all of them are very easy to work with.
Speaker A
I find it extremely difficult to work with inches and feet.
Speaker A
It's an extremely uncivilized system.
Speaker A
I don't mean to insult you.
Speaker A
But think about it.
Speaker A
12 inches in a foot, three feet in a yard.
Speaker A
That drives you nuts.
Speaker A
I work almost exclusively decimal and I hope you will do the same during this course.
Speaker A
But we may make some exceptions.
Speaker A
I will now first show you a movie which is called The Powers of 10.
Speaker A
It covers 40 orders of magnitude.
Speaker A
It was originally conceived by a Dutchman named Kees Boeke in the early 50s.
Speaker A
This is the second generation movie.
Speaker A
And you will hear the voice of Professor Morrison.
Speaker A
Who is a professor at MIT.
Speaker A
The Powers of 10, 40 orders of magnitude.
Speaker A
There we go.
Speaker A
I already introduced, as you see there, length, time and mass.
Speaker A
And we call these the three fundamental quantities in physics.
Speaker A
I will give this the symbol capital L.
Speaker A
For length, capital T for time.
Speaker A
And capital M for mass.
Speaker A
All other quantities in physics can be derived from these fundamental quantities.
Speaker A
I give you an example.
Speaker A
I put a bracket around here.
Speaker A
I say speed.
Speaker A
And that means the dimensions of speed.
Speaker A
The dimensions of speed is the dimension of length divided by the dimension of time.
Speaker A
So I can write for that bracket L divided by bracket time.
Speaker A
Whether it's meters per second or inches per year, that's not what matters.
Speaker A
It has the dimension length per time.
Speaker A
Volume.
Speaker A
Would have the dimension of length to the power 3.
Speaker A
Density.
Speaker A
Would have the dimension of mass per unit volume.
Speaker A
So that means length to the power three.
Speaker A
All important in our course is acceleration.
Speaker A
We will deal a lot with acceleration.
Speaker A
Acceleration as you will see is length per time squared.
Speaker A
The unit is meters per second squared.
Speaker A
So you get length divided by time squared.
Speaker A
So all other quantities can be derived from these three fundamental.
Speaker A
So now that we have agreed on the units, we have the meter, the second and the kilogram.
Speaker A
We can start making measurements.
Speaker A
Now all important in making measurements, which is always ignored in every college book, is the uncertainty in your measurement.
Speaker A
Any measurement that you make without any knowledge of the uncertainty is meaningless.
Speaker A
I will repeat this.
Speaker A
I want you to hear it tonight at 3:00 when you wake up.
Speaker A
Any measurement that you make without a knowledge of its uncertainty is completely meaningless.
Speaker A
My grandmother used to tell me that, at least she believed it.
Speaker A
That someone who is lying in bed is longer than someone who stands up.
Speaker A
And in honor of my grandmother.
Speaker A
I'm going to bring this today to a test.
Speaker A
I have here a setup where I can measure a person standing up.
Speaker A
And a person lying down.
Speaker A
It's not the greatest bed, but lying down.
Speaker A
I have to convince you about the uncertainty in my measurements.
Speaker A
Because a measurement without knowledge of the uncertainty is meaningless.
Speaker A
And therefore what I will do is the following.
Speaker A
I have here an aluminum bar.
Speaker A
And I make the reasonable, plausible assumption that when this aluminum bar is sleeping, when it is horizontal.
Speaker A
That it is not longer than when it is standing up.
Speaker A
If you accept that, we can compare the length of this aluminum bar.
Speaker A
With this setup and with this setup.
Speaker A
At least we have some kind of calibration to start with.
Speaker A
I will measure it.
Speaker A
You have to trust me.
Speaker A
During these three months we have to trust each other.
Speaker A
So I measure here 149.9 centimeters.
Speaker A
However.
Speaker A
I would think that the so this is the aluminum bar, this is in vertical position.
Speaker A
149.9, but I would think that the uncertainty of my measurement is probably 1 millimeter.
Speaker A
I can't really guarantee you that I did it accurately any better.
Speaker A
So that's the vertical one.
Speaker A
Now we're going to measure the bar horizontally.
Speaker A
For which we have a setup here.
Speaker A
Oh, the scale is on your side.
Speaker A
So now I measure the length of this bar.
Speaker A
150.0.
Speaker A
Horizontally.
Speaker A
150.0, again plus or minus 0.1 cm.
Speaker A
So you would agree with me that I am capable of measuring plus or minus.
Speaker A
And standing up if that were one foot, we would all know it, wouldn't we?
Speaker A
You get out of bed in the morning, you lie down, you get up and you go.
Speaker A
And you're one foot shorter.
Speaker A
And we know that that's not the case.
Speaker A
If the difference were only one millimeter, we would never know.
Speaker A
Therefore, I suspect that if my grandmother was right.
Speaker A
That it's probably only a few centimeters.
Speaker A
Maybe an inch.
Speaker A
And so I would argue that if I can measure a length of a student to one millimeter accuracy.
Speaker A
That should settle the issue.
Speaker A
So I need a volunteer.
Speaker A
You want a volunteer.
Speaker A
Looks like you're very tall.
Speaker A
I hope that, yeah.
Speaker A
I hope we can, I hope that we don't run out of.
Speaker A
You're not taller than 178 or so.
Speaker B
No.
Speaker A
What is your name?
Speaker B
Rick Rider.
Speaker A
Rick, Rick Rider.
Speaker A
You're not nervous, right?
Speaker B
No.
Speaker A
Man.
Speaker A
Sit down.
Speaker A
I can't have tall guys here.
Speaker A
Come on.
Speaker A
We need someone more modest in size.
Speaker A
Don't take it personal, Rick.
Speaker A
Okay.
Speaker A
What is your name?
Speaker B
Zack.
Speaker A
Zack.
Speaker A
Nice day today, Zack.
Speaker A
Yeah.
Speaker A
You feel all right?
Speaker A
First lecture at MIT.
Speaker B
Yes.
Speaker A
No.
Speaker A
I don't.
Speaker A
Okay, man.
Speaker A
Stand there.
Speaker A
Yeah.
Speaker A
Okay, 183.2.
Speaker A
Stay there.
Speaker A
Stay there.
Speaker A
Don't move.
Speaker A
Zack.
Speaker A
And this is vertical.
Speaker A
What did I say?
Speaker A
180.
Speaker C
Three.
Speaker A
Only one person.
Speaker A
Three.
Speaker A
Come on.
Speaker A
0.2.
Speaker A
Okay.
Speaker A
183.2.
Speaker A
Yeah.
Speaker A
And an uncertainty of about 0.1 cm.
Speaker A
And now we're going to measure him horizontally.
Speaker A
Zack.
Speaker A
I don't want you to break your bones, so we have a little step for you here.
Speaker A
Put your feet there.
Speaker A
Oh, let me remove the aluminum bar.
Speaker A
Don't watch out for this scale, that you don't break that, because then it's all over.
Speaker A
Okay.
Speaker A
I'll come on your side.
Speaker A
I have to do that.
Speaker A
Yeah.
Speaker A
Yeah.
Speaker A
Relax.
Speaker A
Think of this as a small sacrifice for the sake of science, right?
Speaker A
Not.
Speaker A
Okay.
Speaker A
You good?
Speaker B
Yes.
Speaker A
You ready?
Speaker B
Yes.
Speaker A
Okay.
Speaker A
185.7.
Speaker A
Stay where you are.
Speaker A
185.7.
Speaker A
I'm sure I want to first make the subtraction, right?
Speaker A
185.7 plus or minus 0.1 cm.
Speaker A
Oh, that is 5, that is 2.5 plus or minus 0.2 cm.
Speaker A
You're about 1 inch taller when you sleep than when you stand up.
Speaker A
My grandmother was right.
Speaker A
She's always right.
Speaker A
Can you get off here?
Speaker A
I want you to appreciate that the accuracy, thank you very much, Zack.
Speaker A
That the accuracy of 1 millimeter was more than sufficient to make the case.
Speaker A
If the accuracy in my measurement would have been much less, this measurement would not have been convincing at all.
Speaker A
So whenever you make a measurement, you must know the uncertainty, otherwise it is meaningless.
Speaker A
Galileo Galilei.
Speaker A
Asked himself the question.
Speaker A
Why are mammals as large as they are and not much larger?
Speaker A
He had a very clever reasoning which I've never seen in print.
Speaker A
But it comes down to the fact that he argued that if the mammal becomes too massive.
Speaker A
That the bones will break.
Speaker A
And he thought that that was a limiting factor.
Speaker A
Even though I've never seen his reasoning in print, I will try to reconstruct it.
Speaker A
What could have gone through his head?
Speaker A
Here is a mammal.
Speaker A
And this is the one of the four legs of the mammal.
Speaker A
And this mammal has a size S.
Speaker A
And what I mean by that is a mouse is J big and a cat is J big.
Speaker A
That's what I mean by size.
Speaker A
Very crudely defined.
Speaker A
The mass of the mammal is M.
Speaker A
And this mammal has a thigh bone, which we call the femur.
Speaker A
Which is here.
Speaker A
And the femur, of course, carries the body to a large extent.
Speaker A
And let's assume that the femur has a length L and has a thickness D.
Speaker A
Here is a femur.
Speaker A
This is what the femur approximately looks like.
Speaker A
So this would be the length of the femur.
Speaker A
And this would be the thickness D.
Speaker A
And this would be the cross-sectional area A.
Speaker A
I'm now going to take you through.
Speaker A
What we call in physics a scaling argument.
Speaker A
I would argue that the length of the femur must be proportional to the size of the animal.
Speaker A
That's completely plausible.
Speaker A
If an animal is four times larger than another, you would need four times longer legs.
Speaker A
And that's all this is saying.
Speaker A
It's very reasonable.
Speaker A
It is also very reasonable that the mass of an animal is proportional to the third power of the size.
Speaker A
Because that's related to its volume.
Speaker A
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.
Speaker A
Because of this relationship.
Speaker A
Okay, that's one.
Speaker A
Now comes the argument.
Speaker A
Pressure.
Speaker A
On the femur is proportional to the weight of the animal.
Speaker A
Divided by the cross section A of the femur.
Speaker A
That's what pressure is.
Speaker A
And that is the mass of the animal, that's proportional to the mass of the animal divided by D squared.
Speaker A
Because we want the area here.
Speaker A
It's proportional to D squared.
Speaker A
Now follow me closely.
Speaker A
If the pressure is higher than a certain level, the bones will break.
Speaker A
Therefore, for an animal not to break its bones when the mass goes up by a certain factor.
Speaker A
Say a factor of four.
Speaker A
In order for the bones not to break, D squared must also go up by a factor of four.
Speaker A
That's a key argument in the scaling here.
Speaker A
You really have to think that through carefully.
Speaker A
Therefore, I would argue that the mass must be proportional to D squared.
Speaker A
This is the breaking argument.
Speaker A
Now compare these two.
Speaker A
The mass is proportional to the length of the femur to the power three.
Speaker A
And to the thickness of the femur to the power two.
Speaker A
Therefore, the thickness of the femur to the power two must be proportional to the length L.
Speaker A
And therefore the thickness of the femur must be proportional to L to the power 3/2.
Speaker A
A very interesting result.
Speaker A
What is this result telling you?
Speaker A
It tells you that if I have two animals and one is 10 times larger than the other.
Speaker A
That S is 10 times larger.
Speaker A
That the lengths of the legs are 10 times larger.
Speaker A
But that the thickness of the femur is 30 times larger because it is L to the power 3/2.
Speaker A
If I were to compare a mouse with an elephant.
Speaker A
An elephant is about 100 times larger in size.
Speaker A
So the length of the femur of the elephant would be 100 times larger than that of a mouse.
Speaker A
But the thickness of the femur would have to be 1,000 times larger.
Speaker A
And that.
Speaker A
May have convinced Galileo Galilei that that's the reason why.
Speaker A
The largest animals are as large as they are.
Speaker A
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.
Speaker A
You're all made of bones and that is biologically not feasible.
Speaker A
And so there is a limit somewhere.
Speaker A
Set by this scaling law.
Speaker A
Well, I I wanted to bring this to a test.
Speaker A
After all, I brought my grandmother's statement to a test.
Speaker A
So why not bringing Galileo Galilei's statement to a test?
Speaker A
And so I went to Harvard where they have a beautiful collection of femurs.
Speaker A
And I asked them for the femur of a raccoon and a horse.
Speaker A
A raccoon is this big.
Speaker A
A horse is about four times bigger.
Speaker A
So the length of the femur of a horse must be about four times the length of the raccoon.
Speaker A
Close.
Speaker A
So I was not surprised.
Speaker A
Then I measured the thickness.
Speaker A
And I said to myself, ah.
Speaker A
If the length is four times higher, then the thickness has to be eight times higher if this holds.
Speaker A
And what I'm going to plot for you.
Speaker A
You will see that shortly.
Speaker A
Is D divided by L versus L.
Speaker A
And that, of course, must be proportional to L to the power 1/2.
Speaker A
I bring one L here.
Speaker A
So if I compare the horse and I compare the raccoon.
Speaker A
I would argue that the thickness divided by the length of the femur for the horse must be the square root of four.
Speaker A
Twice as much as that of the raccoon.
Speaker A
And so I was very anxious to plot that.
Speaker A
And I did that.
Speaker A
And I show you the result.
Speaker A
Here is my first result.
Speaker A
So, we see there D over L.
Speaker A
I explained to you why I prefer that to plot it.
Speaker A
And here you see the length.
Speaker A
You see here the raccoon.
Speaker A
And you see the horse.
Speaker A
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.
Speaker A
Well, I wasn't too disappointed.
Speaker A
One and a half is not two.
Speaker A
But it is in the right direction.
Speaker A
The horse clearly have a larger value for D over L than the raccoon.
Speaker A
I realized I needed more data.
Speaker A
So I went back to Harvard.
Speaker A
I said, look, I need a smaller animal.
Speaker A
An opossum, maybe, maybe a rat, maybe a mouse.
Speaker A
And they said, okay.
Speaker A
They gave me three more bones.
Speaker A
They gave me an antelope, which is actually a little larger than the raccoon.
Speaker A
And they gave me an opossum.
Speaker A
And they gave me a mouse.
Speaker A
Here is the bone of the antelope.
Speaker A
Here is the one of the raccoon.
Speaker A
Here is the one of the opossum.
Speaker A
And now you won't believe this.
Speaker A
This is so wonderful.
Speaker A
So romantic.
Speaker A
There is the mouse.
Speaker A
Isn't that beautiful?
Speaker A
Teeny, weeny, little mouse, it's only a teeny, weeny, little femur.
Speaker A
And there it is.
Speaker A
And I, um.
Speaker A
I made the plot.
Speaker A
I was very curious what that plot would look like.
Speaker A
And.
Speaker A
Here it is.
Speaker A
I was shocked.
Speaker A
I was really shocked.
Speaker A
Because look.
Speaker A
The horse is 50 times larger in size than the mouse.
Speaker A
The difference in D over L is only a factor of two.
Speaker A
And I expected something more like a factor of seven.
Speaker A
And so in D over L where I expect a factor of seven, I only see a factor of two.
Speaker A
So I said to myself, oh my goodness.
Speaker A
Why didn't I ask them for an elephant?
Speaker A
The real clincher would be the elephant.
Speaker A
Because if that goes way off scale, maybe we can still rescue the statement by Galileo Galilei.
Speaker A
And so I went back.
Speaker A
And they said, okay, we'll give you the femur of an elephant.
Speaker A
They also gave me one of a moose, believe it or not.
Speaker A
I think they wanted to get rid of me by that time, to be frank of you.
Speaker A
And here is the femur of an elephant.
Speaker A
And I measured it.
Speaker A
The length and the thickness.
Speaker A
And it is very heavy.
Speaker A
It weighs a ton.
Speaker A
I plotted it.
Speaker A
I was full of expectation.
Speaker A
I couldn't sleep all night.
Speaker A
And there's the elephant.
Speaker A
There is no evidence whatsoever that D over L is really larger for the elephant than for the mouse.
Speaker A
These vertical bars indicate my uncertainty.
Speaker A
In measurements of thickness and the horizontal scale, which is a logarithmic scale, the uncertainty of the length measurements.
Speaker A
Is in the thickness of the red pen, so there's no need for me to indicate that any further.
Speaker A
And here you have your measurements.
Speaker A
In case you want to check them.
Speaker A
And look again at the mouse.
Speaker A
And look at the elephant.
Speaker A
The mouse has indeed only 1 cm length of the femur and the elephant is indeed 100 times longer.
Speaker A
So the first scaling argument that S is proportional to L, that is certainly what you expect.
Speaker A
Because elephant is about 100 times larger in size.
Speaker A
But when you go to D over L, you see it's all over.
Speaker A
The D over L for the mouth is really not all that different from the elephant.
Speaker A
And you would have expected that number to be with the square root of.
Speaker A
100, so you expect it to be 10 times larger instead of about the same.
Speaker A
I now want to discuss with you.
Speaker A
What we call in physics dimensional analysis.
Speaker A
I want to ask myself the question.
Speaker A
If I drop an apple from a certain height.
Speaker A
And I change that height.
Speaker A
What will happen with the time for the apple to fall?
Speaker A
Well.
Speaker A
I drop the apple from a height H.
Speaker A
And I want to know what happens with the time when it falls.
Speaker A
And I change H.
Speaker A
So I said to myself, well.
Speaker A
The time that it takes must be proportional to the height to some power alpha.
Speaker A
Completely reasonable.
Speaker A
If I make the height larger, we all know that it takes longer for the apple to fall.
Speaker A
But that's a safe thing.
Speaker A
I said to myself, well.
Speaker A
If the apple has a mass M, it probably is also proportional to the mass of that apple to the power beta.
Speaker A
Turns out to be not so.
Speaker A
But you could think that.
Speaker A
But you could have said, well, let's not take the acceleration of the Earth.
Speaker A
But let's take the mass of the Earth itself.
Speaker A
Very reasonable, right?
Speaker A
I would think if I increase the mass of the Earth, that the apple will fall faster.
Speaker A
So now I would put in the mass of the Earth here.
Speaker A
And I start my dimensional analysis and I end up dead in the waters.
Speaker A
Because you see, there is no mass here.
Speaker A
There is a mass to the power beta here and one to the power gamma.
Speaker A
So what you would have found is beta plus gamma equals zero.
Speaker A
And that would be end of story.
Speaker A
Now you can ask yourself the question.
Speaker A
Well, is there something wrong with the analysis that we did?
Speaker A
Is ours perhaps better than this one?
Speaker A
Well, it's a different one.
Speaker A
We came to the conclusion that the time that it takes for the apple to fall is independent of the mass.
Speaker A
Do we believe that?
Speaker A
Yes, we do.
Speaker A
On the other hand, there are very prestigious physicists.
Speaker A
Who even nowadays do very fancy experiments and they try to demonstrate.
Speaker A
That the time for an apple to fall does depend on its mass.
Speaker A
Even though it probably is only very small if it's true, but they try to prove that.
Speaker A
If any of them succeeds or any one of you succeeds, that's certainly worth a Nobel Prize.
Speaker A
So we do believe that it's independent of the mass.
Speaker A
However.
Speaker A
This what I did with you was not a proof.
Speaker A
Because if you do it this way.
Speaker A
You get stuck.
Speaker A
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.
Speaker A
I think that's very useful.
Speaker A
We confirm that with experiment.
Speaker A
And indeed it came out that way.
Speaker A
So it was not a complete waste of time.
Speaker A
But when you do a dimensional analysis.
Speaker A
You better be careful.
Speaker A
I like you to think this over.
Speaker A
The comparison between the two.
Speaker A
At dinner and maybe at breakfast and maybe even while you're taking a shower.
Speaker A
Whether it's needed or not.
Speaker A
It is important that you digest and appreciate the difference between these two approaches.
Speaker A
It will give you an insight in the power.
Speaker A
And also into the limitations of dimensional analysis.
Speaker A
This goes to the very heart of our understanding and appreciation of physics.
Speaker A
It's important that you get a feel for this.
Speaker A
You're now at MIT.
Speaker A
This is the time.
Speaker A
Thank you, see you Friday.
Topics:unitsdimensionsmeasurement uncertaintyfundamental quantitiesscaling argumentsphysicsWalter Lewinclassical mechanicsPowers of 10Galileo Galilei
![Conceptos del LUGAR GEOMÉTRICO de las RAÍCES [LGR] – Si… — Transcript](https://i.ytimg.com/vi/uTMLBQ85wCE/hqdefault.jpg)
