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PROFESSOR: Let's get
on with some dynamics.

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00:00:26,490 --> 00:00:32,020
So the place I'm
going to begin is just

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a comment about mechanical
engineering courses.

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00:00:34,820 --> 00:00:39,010
The first, and you may have
heard this already in classes,

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00:00:39,010 --> 00:00:42,440
you'll be taking
subject 2001 if you're

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00:00:42,440 --> 00:00:45,770
Course 2 majors through
2009, and if you're

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00:00:45,770 --> 00:00:48,700
2-A, most of the odd ones.

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00:00:48,700 --> 00:00:57,920
But the subjects
2001 through 2005

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00:00:57,920 --> 00:01:00,510
are really basically engineering
science subjects that are all

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00:01:00,510 --> 00:01:02,650
foundational to
mechanical engineering,

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00:01:02,650 --> 00:01:05,780
and they all have a common
or property through them.

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And that is that we make
observations of the world,

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and we try to understand them.

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We pose problems.

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Why-- 400 years ago, is
the sun in the center

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of the solar system or not?

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00:01:23,620 --> 00:01:32,470
And we try to produce models
that explain the problem.

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So here's the problem,
the question of the day.

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We try to produce
models to describe it,

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and we make observations,
measurements, to see

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if our models are correct.

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And if we feed that information
back into the models,

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we try out the models, we test
it against more observations,

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and you go round and round.

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And this is kind of
the fundamental--

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this is the way all of these
basic first five subjects use,

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00:02:08,630 --> 00:02:14,140
basically, this
method of inquiry.

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00:02:14,140 --> 00:02:20,220
So in 2003, the way
this system works,

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00:02:20,220 --> 00:02:29,140
my kind of mental conception
of this modeling process,

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00:02:29,140 --> 00:02:30,201
is three things.

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And this applies to you.

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00:02:31,200 --> 00:02:32,580
You have a homework problem.

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00:02:32,580 --> 00:02:34,850
How do you attack
a homework problem?

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00:02:34,850 --> 00:02:38,690
You're going to need
to describe the motion.

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00:02:44,680 --> 00:02:48,840
You're going to need to choose
the physical laws-- pick,

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00:02:48,840 --> 00:02:51,960
I'll call it
because it's short--

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00:02:51,960 --> 00:02:59,180
the physical law that
you want to apply

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like f equals ma,
conservation of energy,

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00:03:02,210 --> 00:03:03,875
conservation of momentum.

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00:03:03,875 --> 00:03:06,590
You got to know which
physical laws to apply.

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00:03:06,590 --> 00:03:16,370
And then finally, third you
need to apply the correct math.

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00:03:16,370 --> 00:03:18,224
And that's really--
most dynamic problems

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00:03:18,224 --> 00:03:19,390
can be broken down this way.

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00:03:19,390 --> 00:03:21,530
That's the way I like to
conceptually break them down.

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You might have another
model, but this is

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the way I'm going to teach it.

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Can you describe the motion,
pick the correct physical laws

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00:03:28,210 --> 00:03:32,630
to apply to the problem, and
able to do the correct math,

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00:03:32,630 --> 00:03:35,260
solving the equation
of motion, for example.

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And all this is what
fits in our models box.

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And we test it against
observations and measurements

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and improve those
things over time.

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So I'm going to give you--
how many of you like history?

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I find history and
history technology

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00:03:52,884 --> 00:03:54,050
kind of fun and interesting.

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So I'm going to throw
a little bit of history

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into giving you a
little quick course

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00:03:58,320 --> 00:04:05,000
outline of what we're going to
do in this subject this term.

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Because the history
dynamics and what

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we're going to do in the course
actually track one another

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remarkably closely.

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So if I ever gave you a bunch
of names like Galileo, Kepler,

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Descartes, Newton, Copernicus,
Euler, Lagrange and Brahe,

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which one comes first?

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Take a guess.

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00:04:38,384 --> 00:04:40,764
AUDIENCE: Copernicus.

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PROFESSOR: Good.

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Copernicus.

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So Copernicus was
Polish, and the story

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starts long before then, but
in about 1,500 Copernicus

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said what?

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AUDIENCE: [INAUDIBLE]

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PROFESSOR: The sun's the center?

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AUDIENCE: [INAUDIBLE]

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PROFESSOR: Or the
Earth is the center?

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AUDIENCE: [INAUDIBLE]

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PROFESSOR: Which did say?

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Yes, so Ptolemy, back
around 130 AD said,

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well the Earth's the
center of the solar system.

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Copernicus came
along and said, nope

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00:05:19,720 --> 00:05:23,080
I think that, in fact, the sun's
the center of the solar system.

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And it for the next 100 years--
more than 100 years, couple

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hundred years-- there was
a really raging controversy

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00:05:29,250 --> 00:05:30,250
about that.

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00:05:30,250 --> 00:05:38,520
So Copernicus, Brahe
Kepler-- so I'm

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00:05:38,520 --> 00:05:40,520
putting them in rough
chronological order here.

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00:05:40,520 --> 00:05:41,995
Now, I'm going to
run out of board.

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00:05:41,995 --> 00:05:42,710
Oh well.

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00:05:45,370 --> 00:06:23,130
Galileo, Descartes-- I'm gonna
cheat-- OK, Descartes, Newton,

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00:06:23,130 --> 00:06:26,580
Euler, and Lagrange.

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So we're going to talk and say
a little bit about each of them.

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And now that I'm--
like I told you,

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00:06:34,955 --> 00:06:37,330
I haven't used this classroom
before so I gotta learn how

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00:06:37,330 --> 00:06:39,740
to play this game.

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00:06:39,740 --> 00:06:41,780
I need to be able to
reach this for a minute.

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So Brahe, he was
along about 1,600.

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Brahe was the
mathematician that wrote--

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00:06:47,450 --> 00:06:51,060
the imperial mathematician
to the emperor in Prague.

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00:06:51,060 --> 00:06:54,530
And he did 20 years
of observations.

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And he was out to prove
that the Earth was

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the center of the solar system.

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And then Kepler actually worked
with him as a mathematician,

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00:07:02,460 --> 00:07:07,250
and then took over as the
imperial mathematician.

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00:07:07,250 --> 00:07:12,160
And he took Brahe's data--
20 years of astronomical data

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00:07:12,160 --> 00:07:14,370
without the use
of the telescope--

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00:07:14,370 --> 00:07:20,720
and used it come up with the
three laws of planetary motion.

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00:07:20,720 --> 00:07:26,130
And so his first and second
laws were put out about 1609.

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00:07:26,130 --> 00:07:28,650
And one of the laws
is, like, equal area

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00:07:28,650 --> 00:07:30,160
swept out in equal time.

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00:07:30,160 --> 00:07:31,330
Have you hear that one?

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00:07:31,330 --> 00:07:34,620
That actually turns out to be
a statement of conservation

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00:07:34,620 --> 00:07:37,100
of angular momentum, which
we'll talk quite a bit

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about the course.

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Then came Galileo, and I'm not
putting their birth and death

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00:07:43,520 --> 00:07:44,020
dates here.

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00:07:44,020 --> 00:07:45,700
I'm kind of putting
in a period of time

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00:07:45,700 --> 00:07:49,070
in which kind of important
things happened around him.

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So 401 years ago a really
important thing happened.

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Galileo, in 1609, turned
the telescope on Jupiter,

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00:08:00,460 --> 00:08:02,192
and saw what?

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00:08:02,192 --> 00:08:03,104
AUDIENCE: [INAUDIBLE]

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00:08:03,104 --> 00:08:05,420
PROFESSOR: Four moons, right?

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00:08:05,420 --> 00:08:08,060
And then they really
started having some data

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00:08:08,060 --> 00:08:12,000
with which to really argue
against the Ptolymaic view

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00:08:12,000 --> 00:08:13,840
of the solar system.

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Descartes is an
important figure to us.

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And in the period of about
1630 to 1644-- in that period

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Descartes began what is today
known as analytic geometry.

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He was geometer, he
studied Euclid a lot.

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00:08:36,130 --> 00:08:40,110
But then he came up with a
Cartesian coordinate system,

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00:08:40,110 --> 00:08:43,370
xyz, and the beginnings
of analytic geometry,

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00:08:43,370 --> 00:08:47,790
which is essentially algebra,
coordinates, and geometry all

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00:08:47,790 --> 00:08:48,540
put together.

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And we are going to make
great use of analytic geometry

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00:08:52,170 --> 00:08:53,830
in this course.

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Then came Newton, kind of in
his actual lifespan, 1643.

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00:08:59,433 --> 00:09:04,070
It's kind of interesting
that he spans these people.

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00:09:04,070 --> 00:09:10,174
And in about 1666 is when he
first-- the first statement

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00:09:10,174 --> 00:09:11,340
of the three laws of motion.

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00:09:15,370 --> 00:09:30,430
Then Euler, and he's 1707 to
1783, and that's his lifespan.

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00:09:30,430 --> 00:09:34,780
Euler came up-- Newton never
talked about angular momentum.

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00:09:34,780 --> 00:09:36,260
He mostly talked
about particles.

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00:09:36,260 --> 00:09:40,950
Euler put Newton's three
laws into mathematics.

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00:09:40,950 --> 00:09:44,810
Euler taught us about
angular momentum,

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00:09:44,810 --> 00:09:53,140
and torque being dh
dt in most cases.

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He's the most prolific
mathematician all time,

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00:09:55,630 --> 00:09:58,230
solved all sorts of
important problems.

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And then finally, is Lagrange.

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00:10:00,410 --> 00:10:08,740
And Lagrange, in about 1788,
uses an energy method, energy

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00:10:08,740 --> 00:10:17,140
and the concept of work to
give us equations of motion.

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So the course, 203,
stands on the shoulders

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00:10:24,940 --> 00:10:26,140
of all these people.

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00:10:26,140 --> 00:10:29,810
But with Descartes, we start
with kinematics, really.

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00:10:29,810 --> 00:10:33,350
This is analytic geometry.

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00:10:33,350 --> 00:10:36,200
And that's where we're going to
start today is with kinematics.

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And very soon
thereafter, we're going

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00:10:37,840 --> 00:10:40,210
to review Newton, the
three laws, and what

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00:10:40,210 --> 00:10:45,400
we call the direct method for
finding equations of motion.

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00:10:45,400 --> 00:10:48,020
Conservation of
momentum, fact that

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00:10:48,020 --> 00:10:50,070
force-- some of the
forces on an object

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00:10:50,070 --> 00:10:53,390
equals mass times acceleration,
or it's a time derivative

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00:10:53,390 --> 00:10:55,730
of its linear momentum.

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00:10:55,730 --> 00:10:59,350
And we use that to derive
equations of motion.

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00:10:59,350 --> 00:11:05,490
So we're going to go kinematics
into doing the direct method

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00:11:05,490 --> 00:11:09,580
to getting equations of motion.

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00:11:09,580 --> 00:11:13,750
And we go from there
into angular momentum,

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00:11:13,750 --> 00:11:18,500
and what Euler gave us--
the same thing, torque.

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00:11:18,500 --> 00:11:21,840
We're going to do quite a
lot with angular momentum.

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00:11:21,840 --> 00:11:25,020
Because I know you know
a lot about f equals ma

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00:11:25,020 --> 00:11:28,380
and you've done lots of
problems 801 applying that.

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00:11:28,380 --> 00:11:30,960
You've done some problems
on rigid body rotations.

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00:11:30,960 --> 00:11:33,084
But I think there's
a lot more you

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00:11:33,084 --> 00:11:34,750
need to understand
about this, and we'll

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00:11:34,750 --> 00:11:36,083
spend quite a bit of time on it.

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And then near the
last third the course

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we shift, because Lagrange said
that if you just write down

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00:11:47,800 --> 00:11:53,030
expressions for energy,
kinetic and potential energy,

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00:11:53,030 --> 00:11:57,340
without any consideration
of Newton's laws

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00:11:57,340 --> 00:12:01,769
and the direct method, you can
derive the equations of motion.

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00:12:01,769 --> 00:12:02,810
That's pretty remarkable.

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00:12:02,810 --> 00:12:06,200
So there are actually two
independent roots to coming up

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00:12:06,200 --> 00:12:07,650
with equations of motion.

190
00:12:07,650 --> 00:12:10,400
And in this course, about
the last third of the course,

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00:12:10,400 --> 00:12:13,400
we're going to teach
you about Lagrange.

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00:12:13,400 --> 00:12:17,320
And then all these
things are going

193
00:12:17,320 --> 00:12:20,522
to be-- one of the applications
that are important engineers

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00:12:20,522 --> 00:12:21,605
is the study of vibration.

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00:12:25,320 --> 00:12:28,540
So we'll be looking
at vibration examples

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00:12:28,540 --> 00:12:32,250
as we go through the
course, and applying

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00:12:32,250 --> 00:12:36,090
these different methods
to first, modeling,

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00:12:36,090 --> 00:12:38,455
and then solving interesting
vibration problems.

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Which brings-- ah, I
have a question for you.

200
00:12:42,980 --> 00:12:46,270
So how many of you were
in this classroom last May

201
00:12:46,270 --> 00:12:49,060
with Professor Haynes Miller,
and I showed up one day

202
00:12:49,060 --> 00:12:51,420
and we talked about vibration?

203
00:12:51,420 --> 00:12:52,220
How many remember?

204
00:12:52,220 --> 00:12:54,480
I told you I was going to
ask this question, right?

205
00:12:54,480 --> 00:12:55,000
Great.

206
00:12:55,000 --> 00:12:57,670
OK, it's good to
see you here again,

207
00:12:57,670 --> 00:13:01,400
and we will talk about
vibration in this course.

208
00:13:01,400 --> 00:13:03,750
So there's kind of the
subject outline built

209
00:13:03,750 --> 00:13:07,350
on the shoulders of
these people in history

210
00:13:07,350 --> 00:13:09,400
that made important
contributions to dynamics.

211
00:13:16,020 --> 00:13:17,790
Any questions about the history?

212
00:13:17,790 --> 00:13:19,920
If you want to
know, one of my TAs

213
00:13:19,920 --> 00:13:22,590
compiled a pretty
neat little summary.

214
00:13:22,590 --> 00:13:24,880
Maybe I will see if I
go back and find this.

215
00:13:24,880 --> 00:13:26,810
I just printed out and
sent it-- how many of

216
00:13:26,810 --> 00:13:29,018
you like to know a little
bit more about the history?

217
00:13:29,018 --> 00:13:31,194
These are like two
liners on each person.

218
00:13:31,194 --> 00:13:31,860
Anybody want it?

219
00:13:31,860 --> 00:13:33,580
Is it worth my time
to send this out?

220
00:13:33,580 --> 00:13:36,450
OK, it's kind of fun.

221
00:13:36,450 --> 00:13:45,980
So let's do an example of this
modeling describing the motion,

222
00:13:45,980 --> 00:13:49,760
picking physical laws,
applying the math.

223
00:13:49,760 --> 00:13:55,150
And that'll get us
launched in the course.

224
00:13:55,150 --> 00:13:58,890
And we'll do it using Newton
and the direct method.

225
00:14:19,510 --> 00:14:24,760
So last May, Haynes Miller
and I talked about vibration.

226
00:14:24,760 --> 00:14:27,400
So I'm going to start
with a vibration problem.

227
00:14:30,180 --> 00:14:31,310
And I brought one.

228
00:14:31,310 --> 00:14:35,370
So here's my couple
of lead weights

229
00:14:35,370 --> 00:14:37,262
and a couple of springs.

230
00:14:37,262 --> 00:14:39,720
So really I just want to talk
about-- this is the problem I

231
00:14:39,720 --> 00:14:40,960
want to talk about.

232
00:14:40,960 --> 00:14:42,560
Now you've done
this problem before.

233
00:14:42,560 --> 00:14:46,480
Haynes Miller and
I did it last May.

234
00:14:46,480 --> 00:14:48,870
And you've no doubt
it in other classes.

235
00:14:48,870 --> 00:14:51,830
OK, it's a system which
has a spring, a mass,

236
00:14:51,830 --> 00:14:54,310
it exhibits something
called a natural frequency.

237
00:14:54,310 --> 00:14:58,220
But let's see what it takes
to just initially begin

238
00:14:58,220 --> 00:15:03,410
to follow this modeling
method to arrive

239
00:15:03,410 --> 00:15:05,160
at an equation of
motion for this problem.

240
00:15:14,770 --> 00:15:17,930
So what do I mean by when
I say, describe the motion?

241
00:15:17,930 --> 00:15:19,590
Really what that
boils down to if you

242
00:15:19,590 --> 00:15:23,240
have to assign a
coordinate system

243
00:15:23,240 --> 00:15:26,419
so that you can actually say
where the object's moving.

244
00:15:26,419 --> 00:15:27,710
And I'm going to pick one here.

245
00:15:27,710 --> 00:15:30,980
So here's-- coordinate system
going to be really important

246
00:15:30,980 --> 00:15:31,660
in this course.

247
00:15:34,480 --> 00:15:41,670
And I'll give us an xyz
Cartesian coordinate system.

248
00:15:44,680 --> 00:15:48,250
And I'm going to try to adopt
the habit, for the most part,

249
00:15:48,250 --> 00:15:51,960
during the course that
this o marks this origin,

250
00:15:51,960 --> 00:15:54,340
but it also names the frame.

251
00:15:54,340 --> 00:15:56,650
So we're going to
talk about things

252
00:15:56,650 --> 00:15:58,860
in that are reference frames.

253
00:15:58,860 --> 00:16:01,500
And most important
one that we need

254
00:16:01,500 --> 00:16:04,080
to know about in the course is
an inertial reference frame,

255
00:16:04,080 --> 00:16:06,250
and when you can use
it, and when a system

256
00:16:06,250 --> 00:16:08,210
is inertial and is not.

257
00:16:08,210 --> 00:16:10,691
So I'm gonna say that
this is inertial.

258
00:16:10,691 --> 00:16:11,690
It's fixed to the Earth.

259
00:16:11,690 --> 00:16:12,960
It's not moving.

260
00:16:12,960 --> 00:16:17,250
And we're going to use this
coordinate x to describe

261
00:16:17,250 --> 00:16:19,110
the motion of this mass.

262
00:16:19,110 --> 00:16:31,560
And the motion is going to be--
this x is from the zero spring

263
00:16:31,560 --> 00:16:33,850
force position.

264
00:16:33,850 --> 00:16:35,440
It's actually quite
important that you

265
00:16:35,440 --> 00:16:38,230
pick-- that you
have to say what's

266
00:16:38,230 --> 00:16:42,580
the condition in the spring
of the system when x is 0

267
00:16:42,580 --> 00:16:44,300
So we're going to say
it's, when there's

268
00:16:44,300 --> 00:16:45,841
no force in the
spring means it's not

269
00:16:45,841 --> 00:16:48,782
stretch, that's where 0 is.

270
00:16:48,782 --> 00:16:50,490
So we've established
a coordinate system.

271
00:16:55,370 --> 00:16:57,630
Second, we need to
apply physical laws.

272
00:17:07,300 --> 00:17:11,800
Now, I'm going to do
this problem by f equals

273
00:17:11,800 --> 00:17:13,680
ma, Newton's second law.

274
00:17:13,680 --> 00:17:16,190
Sum of the external
forces is equal to mass

275
00:17:16,190 --> 00:17:17,740
times the acceleration.

276
00:17:17,740 --> 00:17:21,460
So that's the law
I'm going to apply.

277
00:17:21,460 --> 00:17:25,700
Sum of the external
forces, it's a vector

278
00:17:25,700 --> 00:17:28,640
but we're just doing the x
component only so we don't have

279
00:17:28,640 --> 00:17:32,930
to carry along vector notation,
is equal to, in this case,

280
00:17:32,930 --> 00:17:36,350
mass times acceleration.

281
00:17:36,350 --> 00:17:38,520
So that's the law
we're going to apply.

282
00:17:38,520 --> 00:17:44,975
And then finally the math to
solve the equation of motion

283
00:17:44,975 --> 00:17:46,970
that we find, that'll
be the third piece.

284
00:17:46,970 --> 00:17:49,090
But part of applying
the physics, in order

285
00:17:49,090 --> 00:17:57,230
to do this now, we need
what I call an FBD.

286
00:17:57,230 --> 00:17:58,602
What do you suppose that is?

287
00:17:58,602 --> 00:17:59,810
AUDIENCE: Free body diagrams.

288
00:17:59,810 --> 00:18:01,400
PROFESSOR: Free body diagrams.

289
00:18:01,400 --> 00:18:03,140
You've used these
many times before,

290
00:18:03,140 --> 00:18:04,950
so we're going to do those.

291
00:18:04,950 --> 00:18:10,210
And free body diagrams--

292
00:18:10,210 --> 00:18:12,780
And I'm going to
teach you, at least

293
00:18:12,780 --> 00:18:14,960
the way I go about doing
free body diagrams,

294
00:18:14,960 --> 00:18:17,150
as things get more
and more complicated,

295
00:18:17,150 --> 00:18:20,280
you're going to have to be
more sophisticated in the way

296
00:18:20,280 --> 00:18:22,350
that you do these things.

297
00:18:22,350 --> 00:18:25,750
So I just have some
simple little rules

298
00:18:25,750 --> 00:18:29,610
to do free body diagrams that
keep you from getting hung up

299
00:18:29,610 --> 00:18:31,480
on sign conventions.

300
00:18:31,480 --> 00:18:33,940
I think the thing people
make most mistakes about is

301
00:18:33,940 --> 00:18:36,030
they get confused about signs.

302
00:18:36,030 --> 00:18:39,390
So I'll try to show
you how I do it.

303
00:18:39,390 --> 00:18:54,540
So first you draw forces
that you know, basically

304
00:18:54,540 --> 00:18:56,720
in the direction
in which they act.

305
00:18:56,720 --> 00:18:59,150
Seems obvious.

306
00:18:59,150 --> 00:19:07,115
So when you know
the direction-- so

307
00:19:07,115 --> 00:19:08,490
this is a really
trivial problem,

308
00:19:08,490 --> 00:19:17,590
but the method here
is very specific.

309
00:19:17,590 --> 00:19:18,780
So what's an example?

310
00:19:18,780 --> 00:19:20,070
Well, gravity.

311
00:19:20,070 --> 00:19:23,960
So we'll start our
free body diagram.

312
00:19:23,960 --> 00:19:26,490
Gravity acts at
the center of mass.

313
00:19:26,490 --> 00:19:27,550
It's downward.

314
00:19:27,550 --> 00:19:29,970
This is what I mean by the
direction in which it acts.

315
00:19:29,970 --> 00:19:34,120
And it has magnitude, mg.

316
00:19:34,120 --> 00:19:35,740
OK.

317
00:19:35,740 --> 00:19:38,220
Now the other forces
aren't so obvious.

318
00:19:38,220 --> 00:19:41,050
The force that's put on by
the stiffness and this damper

319
00:19:41,050 --> 00:19:43,600
in the spring, which
way do you draw them?

320
00:19:43,600 --> 00:19:44,370
What's the sign?

321
00:19:44,370 --> 00:19:46,950
What's the sign convention?

322
00:19:46,950 --> 00:19:49,560
So the convention, the way I
go about doing these things,

323
00:19:49,560 --> 00:20:08,900
is I assume positive values for
the deflections and velocities.

324
00:20:08,900 --> 00:20:11,960
So in this case, x and x dot.

325
00:20:11,960 --> 00:20:16,530
You just require that the
deflections that you're going

326
00:20:16,530 --> 00:20:18,920
to work with are positive.

327
00:20:18,920 --> 00:20:20,650
And then from the
positive deflection,

328
00:20:20,650 --> 00:20:22,660
you say which way is
the resulting force?

329
00:20:22,660 --> 00:20:25,650
So if the deflection
in this is downwards,

330
00:20:25,650 --> 00:20:29,540
which direction is the force
that the spring applies

331
00:20:29,540 --> 00:20:32,020
to the mass?

332
00:20:32,020 --> 00:20:32,690
Up, right?

333
00:20:32,690 --> 00:20:35,660
What about if the
velocity is downwards,

334
00:20:35,660 --> 00:20:41,360
which direction is the force
is the damper puts on the mass?

335
00:20:41,360 --> 00:20:42,080
Also up, right?

336
00:20:42,080 --> 00:20:42,580
OK.

337
00:20:42,580 --> 00:20:48,720
So this allows-- this gives us--
so here's f spring and here's

338
00:20:48,720 --> 00:20:50,070
the f damper.

339
00:20:50,070 --> 00:20:53,610
And other any other
forces on this mass?

340
00:20:53,610 --> 00:20:57,190
So spring force, damper force,
and the gravitational force.

341
00:21:07,940 --> 00:21:18,385
And so third, you deduce
the signs basically

342
00:21:18,385 --> 00:21:19,760
from the direction
of the arrows.

343
00:21:25,880 --> 00:21:28,470
First we need what's called
your constitutive relationship.

344
00:21:28,470 --> 00:21:32,600
So the spring force,
fs, well you've

345
00:21:32,600 --> 00:21:35,370
made x positive so
it keeps things nice,

346
00:21:35,370 --> 00:21:41,340
the spring constant's a
positive number, so fs is kx.

347
00:21:41,340 --> 00:21:45,430
Fd is bx dot.

348
00:21:45,430 --> 00:21:47,440
And now we write the
statement that the sum

349
00:21:47,440 --> 00:21:49,755
of forces in the x direction.

350
00:21:52,490 --> 00:21:55,000
We look at up here, we
say well that's going

351
00:21:55,000 --> 00:22:01,680
to fs plus fd minus mg.

352
00:22:01,680 --> 00:22:06,360
So that's-- whoops, I wrote
it the wrong way around.

353
00:22:06,360 --> 00:22:10,170
Minus, minus, plus.

354
00:22:10,170 --> 00:22:12,170
Because I'm plus
downwards, right?

355
00:22:12,170 --> 00:22:18,430
Well, spring minus fs
is minus kx minus bx dot

356
00:22:18,430 --> 00:22:24,809
plus mg equals mx double dot.

357
00:22:24,809 --> 00:22:26,600
And I rearranged this
to put all the motion

358
00:22:26,600 --> 00:22:29,720
variables on one side.

359
00:22:29,720 --> 00:22:37,700
mx double dot plus bx
dot plus kx equals mg.

360
00:22:40,350 --> 00:22:43,670
So there's my equation of
motion, but with a method

361
00:22:43,670 --> 00:22:45,710
for doing the free
body diagrams, which

362
00:22:45,710 --> 00:22:47,684
will work with multiple bodies.

363
00:22:47,684 --> 00:22:49,850
So you have two bodies with
springs in between them.

364
00:22:49,850 --> 00:22:52,740
This is when the
confusion really comes up.

365
00:22:52,740 --> 00:22:55,940
Two bodies with a spring
trapped between them.

366
00:22:55,940 --> 00:22:58,490
What's the sign convention?

367
00:22:58,490 --> 00:22:59,540
You do the same thing.

368
00:22:59,540 --> 00:23:02,870
Both bodies exhibit
positive motions,

369
00:23:02,870 --> 00:23:05,747
the force that results is
proportional to the difference,

370
00:23:05,747 --> 00:23:06,580
and you work it out.

371
00:23:06,580 --> 00:23:08,100
And you'll get the signs right.

372
00:23:08,100 --> 00:23:09,940
OK, so here's our
equation of motion arrived

373
00:23:09,940 --> 00:23:12,720
at by doing the direct method.

374
00:23:12,720 --> 00:23:21,130
And if we went on to the
third step, which we're not

375
00:23:21,130 --> 00:23:24,570
going to do today, and
that is apply the math,

376
00:23:24,570 --> 00:23:27,430
it might because I want you now
to describe the motion for me,

377
00:23:27,430 --> 00:23:28,520
solve for the motion.

378
00:23:28,520 --> 00:23:30,570
That means solving the
differential equation.

379
00:23:30,570 --> 00:23:33,600
And that's what we did last
may in Haynes Miller's class.

380
00:23:33,600 --> 00:23:35,560
We'll come back
to this later on.

381
00:23:35,560 --> 00:23:40,547
But for today's purposes,
we don't need to go there.

382
00:23:40,547 --> 00:23:42,130
Got something else
much more important

383
00:23:42,130 --> 00:23:45,270
to get to about kinematics.

384
00:23:45,270 --> 00:23:47,610
But I want to show
you one thing,

385
00:23:47,610 --> 00:23:51,700
and that is just a little
tiny introductory taste

386
00:23:51,700 --> 00:23:54,480
to this point.

387
00:23:54,480 --> 00:23:57,390
So I've derived the
equation of motion of this

388
00:23:57,390 --> 00:24:00,854
by Newton's laws.

389
00:24:00,854 --> 00:24:02,270
But I'm going to
ignore Newton now

390
00:24:02,270 --> 00:24:04,144
and saw I'm going to
drive equation of motion

391
00:24:04,144 --> 00:24:04,950
by another way.

392
00:24:04,950 --> 00:24:07,470
And it's an energy
technique, and that is-- well

393
00:24:07,470 --> 00:24:11,020
let's talk about the total
energy of the system.

394
00:24:11,020 --> 00:24:13,750
It's going to be the
sum of a kinetic energy

395
00:24:13,750 --> 00:24:14,920
and a potential energy.

396
00:24:18,560 --> 00:24:21,290
And we'll find that even
with Lagrange, there's

397
00:24:21,290 --> 00:24:24,400
a problem with forces
on systems that

398
00:24:24,400 --> 00:24:26,430
are what we call
non-conservative,

399
00:24:26,430 --> 00:24:28,840
things that either take
energy out of, or put energy

400
00:24:28,840 --> 00:24:29,710
into the system.

401
00:24:29,710 --> 00:24:31,850
And the dashpot does that.

402
00:24:31,850 --> 00:24:35,469
Dashpot generates heat and
takes energy out of the system.

403
00:24:35,469 --> 00:24:37,510
So I'm going to have to
ignore it for the moment.

404
00:24:37,510 --> 00:24:39,750
So the sum of the kinetic
and the potential energies

405
00:24:39,750 --> 00:24:45,680
in this problem is
a 1/2 kx squared

406
00:24:45,680 --> 00:24:53,040
for the potential of the spring,
plus a 1/2 mx dot squared

407
00:24:53,040 --> 00:25:00,600
for the kinetic energy of
the mass, and minus mgx

408
00:25:00,600 --> 00:25:04,880
for the potential energy that
is due to the object moving

409
00:25:04,880 --> 00:25:07,230
in the gravitational field.

410
00:25:07,230 --> 00:25:11,460
And that's the total
energy of the system.

411
00:25:11,460 --> 00:25:14,490
Now my problem, I've
allowed no forces.

412
00:25:14,490 --> 00:25:15,740
There's no excitation on here.

413
00:25:15,740 --> 00:25:18,085
This is just free
vibration only.

414
00:25:18,085 --> 00:25:20,460
That's all we're talking about,
make initial displacement

415
00:25:20,460 --> 00:25:22,530
and it vibrates.

416
00:25:22,530 --> 00:25:24,410
If there's no
damping, what can you

417
00:25:24,410 --> 00:25:28,884
say about the total
energy of the system?

418
00:25:28,884 --> 00:25:30,198
AUDIENCE: [INAUDIBLE]

419
00:25:30,198 --> 00:25:31,464
PROFESSOR: Say it again.

420
00:25:31,464 --> 00:25:32,380
I heard it over there.

421
00:25:32,380 --> 00:25:33,800
It's got to be constant, right?

422
00:25:33,800 --> 00:25:37,480
All right, well, so
this must be constant.

423
00:25:37,480 --> 00:25:55,200
Therefore, the time derivative
of my system, it better be 0.

424
00:25:55,200 --> 00:25:56,890
The energy is constant.

425
00:25:56,890 --> 00:25:58,860
Take it's time derivative,
it's got to be 0.

426
00:25:58,860 --> 00:26:00,610
Apply that to the
right-hand side of this,

427
00:26:00,610 --> 00:26:13,460
I get kxx dot plus mx dot
x double dot minus mgx dot

428
00:26:13,460 --> 00:26:15,530
equals zero.

429
00:26:15,530 --> 00:26:23,690
And I now cancel out the
common x dot terms go away.

430
00:26:23,690 --> 00:26:37,580
And I'm left with--
and I've essentially

431
00:26:37,580 --> 00:26:40,600
solved for the equation
of motion of this system

432
00:26:40,600 --> 00:26:44,010
without ever looking at
conservational momentum,

433
00:26:44,010 --> 00:26:46,695
Newton's laws, only by
energy considerations.

434
00:26:50,300 --> 00:26:52,570
OK, so that's a
very simple example

435
00:26:52,570 --> 00:26:55,350
of that you can use energy to
derive equations of motions.

436
00:26:55,350 --> 00:27:00,840
But you then have to go back and
fix it to account for the loss

437
00:27:00,840 --> 00:27:01,977
term, the damping term.

438
00:27:01,977 --> 00:27:04,060
And that you still have
to consider it as a force,

439
00:27:04,060 --> 00:27:04,800
we'll find out.

440
00:27:04,800 --> 00:27:09,920
Even was Lagrange you have to go
back and consider the work done

441
00:27:09,920 --> 00:27:13,050
by external forces.

442
00:27:13,050 --> 00:27:15,320
OK.

443
00:27:15,320 --> 00:27:18,010
So you've just kind of
seen the whole course.

444
00:27:18,010 --> 00:27:19,390
We've described
the motion, we've

445
00:27:19,390 --> 00:27:24,090
applied to Newton's laws, the
physics to the direct method

446
00:27:24,090 --> 00:27:26,360
to derive the
equations of motion,

447
00:27:26,360 --> 00:27:31,540
we have gone to a direct method,
and have derived the equations

448
00:27:31,540 --> 00:27:32,660
of motion that way.

449
00:27:32,660 --> 00:27:35,290
And that's basically what you're
going to do in the course.

450
00:27:35,290 --> 00:27:37,630
But now you're going
to do it with much more

451
00:27:37,630 --> 00:27:39,880
sophisticated tools.

452
00:27:39,880 --> 00:27:42,160
You'll have multiple
degree of freedom systems.

453
00:27:42,160 --> 00:27:44,520
The description
describing the motion,

454
00:27:44,520 --> 00:27:46,230
is maybe going to
be for some of you,

455
00:27:46,230 --> 00:27:48,410
the most challenging
part of the course.

456
00:27:48,410 --> 00:27:50,615
And this is a topic
we call kinematics.

457
00:27:53,960 --> 00:27:59,380
And that's what
we'll turn to next.

458
00:28:13,330 --> 00:28:15,420
So reference frames and vectors.

459
00:28:15,420 --> 00:28:16,840
That's the topic.

460
00:28:16,840 --> 00:28:20,110
This is now that we're
talking about kinematics,

461
00:28:20,110 --> 00:28:24,610
and this is all about
describing the motion.

462
00:28:24,610 --> 00:28:29,330
So Descartes gave us the
Cartesian coordinate system,

463
00:28:29,330 --> 00:28:30,340
and we'll start there.

464
00:28:30,340 --> 00:28:34,080
So imagine this
is a fixed frame--

465
00:28:34,080 --> 00:28:36,600
we'll talk about what
makes an inertial frame

466
00:28:36,600 --> 00:28:38,300
the next lecture.

467
00:28:38,300 --> 00:28:39,840
But here we have
an inertial frame.

468
00:28:42,860 --> 00:28:51,620
And it's the frame we'll
call O-xyz or O for short.

469
00:28:51,620 --> 00:29:05,720
And in this frame, maybe this
is me, and up here is a dog,

470
00:29:05,720 --> 00:29:12,840
and I'm going to call this
point A and this point B.

471
00:29:12,840 --> 00:29:18,190
And I'm going to describe the
positions of these two points

472
00:29:18,190 --> 00:29:19,890
by vectors.

473
00:29:19,890 --> 00:29:24,050
This one will be R, and the
notation that I'm going to use

474
00:29:24,050 --> 00:29:30,330
is point and it's measurement
with respect something.

475
00:29:30,330 --> 00:29:34,980
Well, it's with respect to this
point O in this inertial frame.

476
00:29:34,980 --> 00:29:38,910
So this is A with respect
O is the way to read this.

477
00:29:38,910 --> 00:29:41,310
There's another vector here.

478
00:29:41,310 --> 00:29:47,440
This is RB respect
to A And finally,

479
00:29:47,440 --> 00:29:53,190
R of B with respect to O They're
all vectors on the board.

480
00:29:53,190 --> 00:29:55,660
I'll try to remember
to underline them

481
00:29:55,660 --> 00:29:56,860
in the textbooks and things.

482
00:29:56,860 --> 00:30:00,420
They're usually-- vectors
are noted with bold letters.

483
00:30:03,970 --> 00:30:07,930
And vectors allow us
to say the following.

484
00:30:07,930 --> 00:30:13,180
That R, the position of
the dog and the reference

485
00:30:13,180 --> 00:30:16,690
with respect to O, is the sum
of these other two vectors.

486
00:30:16,690 --> 00:30:26,390
R of A with respect to O plus
R R of B with respect to A.

487
00:30:26,390 --> 00:30:29,200
And mostly to do
dynamics we're really

488
00:30:29,200 --> 00:30:32,040
interested in things like
velocities and accelerations.

489
00:30:32,040 --> 00:30:34,010
So to get the velocities
and accelerations,

490
00:30:34,010 --> 00:30:39,095
we have to take a time
derivative of our RBO dt.

491
00:30:41,730 --> 00:30:45,840
And that's going to give us
what we'll call the velocity,

492
00:30:45,840 --> 00:30:48,040
obviously you write
it as V. And it would

493
00:30:48,040 --> 00:30:53,060
be the velocity of point B with
respect to O. And no surprise,

494
00:30:53,060 --> 00:30:58,970
it'll be the velocity of
point A plus the velocity of B

495
00:30:58,970 --> 00:31:04,330
with respect to A.

496
00:31:04,330 --> 00:31:13,200
And finally, if we take two
derivatives, dt squared,

497
00:31:13,200 --> 00:31:17,690
we'll get the acceleration
of B with respect to O.

498
00:31:17,690 --> 00:31:22,730
And that'll be the sum of
A-- the acceleration of A

499
00:31:22,730 --> 00:31:28,880
with respect to O plus
the acceleration of B

500
00:31:28,880 --> 00:31:32,420
with respect to A.
All, again, vectors.

501
00:31:38,490 --> 00:31:41,502
Now, just to look ahead--
this seems all really trivial.

502
00:31:41,502 --> 00:31:43,210
You guys are going to
sleep on me, right?

503
00:31:46,620 --> 00:31:52,460
If these are rigid bodies, this
is a rigid body that is moving

504
00:31:52,460 --> 00:31:55,360
and maybe rotating.

505
00:31:55,360 --> 00:32:01,370
And B is on it, and A is
on it, and O isn't on it.

506
00:32:01,370 --> 00:32:05,000
It starts getting
a little tricky.

507
00:32:05,000 --> 00:32:12,000
And this, the derivative
of a vector that's attached

508
00:32:12,000 --> 00:32:19,220
to the body somehow has to
account for the fact that

509
00:32:19,220 --> 00:32:23,530
if I'm-- the
observer's on the body,

510
00:32:23,530 --> 00:32:25,010
this other point's on the body.

511
00:32:25,010 --> 00:32:28,550
Say it's, I'm on this asteroid,
and I've got a dog out there,

512
00:32:28,550 --> 00:32:31,000
and the dog's run away from me.

513
00:32:31,000 --> 00:32:36,460
The speed of the dog with
respect to me, I can measure.

514
00:32:36,460 --> 00:32:37,949
But if I'm down
here looking at it,

515
00:32:37,949 --> 00:32:39,740
it'll look different
because it's rotating.

516
00:32:39,740 --> 00:32:41,198
So how do you
account for all that?

517
00:32:41,198 --> 00:32:44,810
So taking these derivatives
of vectors in moving frames

518
00:32:44,810 --> 00:32:48,590
is where the devil's
in the details.

519
00:32:48,590 --> 00:32:54,190
And that's part of what I'm
going to be teaching you.

520
00:32:54,190 --> 00:32:55,620
OK.

521
00:32:55,620 --> 00:32:59,065
I'm still learning how
to optimize my board use.

522
00:32:59,065 --> 00:33:00,940
I haven't got it perfect
yet, but because I'm

523
00:33:00,940 --> 00:33:02,898
having to move around a
lot here and improvise.

524
00:33:02,898 --> 00:33:04,670
But we'll persevere.

525
00:33:04,670 --> 00:33:09,160
You need to remember a
couple things about vectors,

526
00:33:09,160 --> 00:33:13,916
how to add them, dot products.

527
00:33:13,916 --> 00:33:15,290
If you've forgotten
these things,

528
00:33:15,290 --> 00:33:18,469
you need to go back and
review them really quickly.

529
00:33:18,469 --> 00:33:20,510
There's usually a little
review section the book,

530
00:33:20,510 --> 00:33:22,301
so you need to practice
that sort of thing.

531
00:33:24,700 --> 00:33:26,710
Couple other little facts
you need to remember.

532
00:33:26,710 --> 00:33:32,687
So the derivative of
the sum of two vectors

533
00:33:32,687 --> 00:33:34,145
is just the sum of
the derivatives.

534
00:33:42,130 --> 00:33:45,180
And quite importantly,
we're going

535
00:33:45,180 --> 00:33:48,170
to make use of this one
a lot, is the derivative

536
00:33:48,170 --> 00:33:49,760
of a product of two things.

537
00:33:49,760 --> 00:33:53,340
One of them be in a vector,
some function maybe of time

538
00:33:53,340 --> 00:34:01,840
and a here is derivative of
f with respect to t times a,

539
00:34:01,840 --> 00:34:11,139
plus the derivative of a
with respect to t times f.

540
00:34:11,139 --> 00:34:12,750
That we'll make a lot use of.

541
00:34:12,750 --> 00:34:16,350
So just your basic calculus.

542
00:34:16,350 --> 00:34:20,070
So now, I want to take up--
let's talk about the simplest

543
00:34:20,070 --> 00:34:22,690
form of being able to
do these derivatives

544
00:34:22,690 --> 00:34:27,429
and calculate these
velocities, when everything's

545
00:34:27,429 --> 00:34:33,199
described in terms of
Cartesian coordinates.

546
00:34:33,199 --> 00:34:36,946
Now I'm going to give you a
little look ahead because I'm

547
00:34:36,946 --> 00:34:42,620
going to try to avoid confusion
as much as possible here.

548
00:34:42,620 --> 00:34:47,204
The hardest problem is
when you have a rigid body,

549
00:34:47,204 --> 00:34:49,570
you got the dog on it, you've
got the observer on it,

550
00:34:49,570 --> 00:34:52,300
it's rotating, and translating.

551
00:34:52,300 --> 00:34:56,260
And to take this derivative, you
end up with a number of terms.

552
00:34:56,260 --> 00:34:58,470
The simplest problem
is just something

553
00:34:58,470 --> 00:35:00,410
in a fixed Cartesian
coordinate system.

554
00:35:00,410 --> 00:35:02,240
So we're going to start
with a simple one,

555
00:35:02,240 --> 00:35:04,239
and build our way up to
the complicated one, OK?

556
00:35:07,030 --> 00:35:10,650
But let's now, we're going to
do the really, the simplest one.

557
00:35:10,650 --> 00:35:15,050
We're going to do
velocity and acceleration

558
00:35:15,050 --> 00:35:16,980
in Cartesian coordinates.

559
00:35:23,750 --> 00:35:29,750
And basically I should say
fixed Cartesian coordinates,

560
00:35:29,750 --> 00:35:30,410
not moving.

561
00:35:33,160 --> 00:35:37,900
All right, so now let's
consider the dog out here,

562
00:35:37,900 --> 00:35:42,840
and his position in the
Cartesian coordinate system.

563
00:35:42,840 --> 00:35:45,620
And I could write
that and you'll,

564
00:35:45,620 --> 00:35:48,250
without any loss of
generality here, you'll

565
00:35:48,250 --> 00:35:52,427
know what I mean if
I say RBx component.

566
00:35:52,427 --> 00:35:54,260
And I'm going to stop
writing the slash O's,

567
00:35:54,260 --> 00:35:57,320
because this is now all in
this fixed reference frame.

568
00:35:57,320 --> 00:36:01,080
And it's in I-hat direction.

569
00:36:01,080 --> 00:36:10,590
And I've got another component,
RBy in the J-hat, and an RBz

570
00:36:10,590 --> 00:36:13,344
in the K-hat.

571
00:36:13,344 --> 00:36:15,010
And I want to take
the time derivative--

572
00:36:15,010 --> 00:36:16,301
I was looking for the velocity.

573
00:36:16,301 --> 00:36:18,680
I want to calculate
the velocity.

574
00:36:18,680 --> 00:36:25,175
So the velocity here of
BNO is d by dt of RBO.

575
00:36:25,175 --> 00:36:28,070
.

576
00:36:28,070 --> 00:36:31,200
And now this is now the
product of two things,

577
00:36:31,200 --> 00:36:33,950
so I've got to use
that formula over here.

578
00:36:33,950 --> 00:36:35,960
Product one turn times
the other, and so forth.

579
00:36:35,960 --> 00:36:39,330
So I go to these,
and I say OK, so this

580
00:36:39,330 --> 00:36:53,690
is R dot Bx times I plus R
dot By times J plus R dot

581
00:36:53,690 --> 00:37:00,610
Bz times K. And then the other--
the flip side of that is I

582
00:37:00,610 --> 00:37:06,650
have to take the derivatives of
I times RBx, the derivative J

583
00:37:06,650 --> 00:37:07,810
and so forth.

584
00:37:07,810 --> 00:37:10,940
But what's the derivative
of, let's say, I?

585
00:37:10,940 --> 00:37:15,380
Capital I is my unit vector
in the fixed reference

586
00:37:15,380 --> 00:37:17,795
frame, my O-xyz frame.

587
00:37:17,795 --> 00:37:19,045
0 So it's a constant.

588
00:37:19,045 --> 00:37:23,390
It is unit length, and it points
in a direction that it's fixed.

589
00:37:23,390 --> 00:37:25,240
So what's its derivative?

590
00:37:25,240 --> 00:37:26,990
It's going to have
a 0 derivative.

591
00:37:26,990 --> 00:37:30,770
So the second part of this--
second bits of that is zero.

592
00:37:30,770 --> 00:37:36,480
So that's the velocity in
Cartesian coordinates of my dog

593
00:37:36,480 --> 00:37:40,750
out there running around.

594
00:37:40,750 --> 00:37:43,477
And the acceleration,
in a similar way,

595
00:37:43,477 --> 00:37:45,810
now to get the acceleration,
you take another derivative

596
00:37:45,810 --> 00:37:46,309
of this.

597
00:37:46,309 --> 00:37:48,920
And again, you'll have to take
derivatives of I, J, and K,

598
00:37:48,920 --> 00:37:50,620
and again they're going to be 0.

599
00:37:50,620 --> 00:37:57,300
So you will find that the
acceleration then, is just R

600
00:37:57,300 --> 00:38:02,910
double dot x term
in the plus R double

601
00:38:02,910 --> 00:38:10,485
dot By in the J plus r
double dot Bz in the K.

602
00:38:10,485 --> 00:38:12,610
That would be our acceleration
term, and it's easy.

603
00:38:20,219 --> 00:38:22,760
Now imagine that we are doing
this in polar coordinates, unit

604
00:38:22,760 --> 00:38:25,390
vectors in polar coordinates.

605
00:38:25,390 --> 00:38:27,350
Let me check, last year
the students told me

606
00:38:27,350 --> 00:38:30,110
that in your
physics courses, you

607
00:38:30,110 --> 00:38:34,380
use unit vectors R-hat,
theta-hat, and K. Is

608
00:38:34,380 --> 00:38:35,450
that right?

609
00:38:35,450 --> 00:38:38,580
So I'll use those unit
vectors so they look familiar,

610
00:38:38,580 --> 00:38:40,090
because in polar
coordinates people

611
00:38:40,090 --> 00:38:41,760
use lots of different things.

612
00:38:41,760 --> 00:38:45,710
But think about it, in
polar coordinates, theta--

613
00:38:45,710 --> 00:38:48,460
it's a fixed, maybe,
coordinate system,

614
00:38:48,460 --> 00:38:54,020
but now theta goes like this
and R moves with theta, right?

615
00:38:54,020 --> 00:38:55,550
So the unit vector
is pointing here,

616
00:38:55,550 --> 00:38:58,070
but over time it might
move down to here.

617
00:38:58,070 --> 00:39:01,710
And unit vector has
changed direction,

618
00:39:01,710 --> 00:39:04,210
and its derivative in
time is no longer 0.

619
00:39:04,210 --> 00:39:06,810
So it starts getting messy
as soon as the unit vectors

620
00:39:06,810 --> 00:39:07,680
change in time.

621
00:39:07,680 --> 00:39:10,690
And so that's one of
our objectives here

622
00:39:10,690 --> 00:39:12,420
is to get to that
point and describe

623
00:39:12,420 --> 00:39:14,130
how you handle those cases.

624
00:39:46,450 --> 00:39:51,460
So a quick point about velocity.

625
00:39:51,460 --> 00:39:54,250
You need to really understand
what we mean by velocity.

626
00:39:54,250 --> 00:40:00,010
So here's our Cartesian system.

627
00:40:00,010 --> 00:40:05,300
Here's this point out
here B. And now, this

628
00:40:05,300 --> 00:40:10,070
is the dog running around,
and the path of the dog

629
00:40:10,070 --> 00:40:14,440
might have been like this.

630
00:40:14,440 --> 00:40:16,885
And right in here he's
going this direction.

631
00:40:19,460 --> 00:40:30,800
And in a little
time, in delta t,

632
00:40:30,800 --> 00:40:38,610
he moves by an amount
delta RB with respect to O.

633
00:40:38,610 --> 00:40:41,100
And that's what this is.

634
00:40:41,100 --> 00:40:44,015
He's moved this little
bit in time delta t.

635
00:40:44,015 --> 00:40:46,320
And he happens to be going
off in that direction.

636
00:40:46,320 --> 00:40:49,450
So this then is R
prime, I'll call it,

637
00:40:49,450 --> 00:40:53,790
of B with respect to O, and this
is our original RB with respect

638
00:40:53,790 --> 00:40:59,780
to O. So we can say that
his new position, RB

639
00:40:59,780 --> 00:41:09,030
with respect to prime
is RBO plus delta R.

640
00:41:09,030 --> 00:41:10,550
And these are all vectors.

641
00:41:14,800 --> 00:41:24,332
And the velocity of
B with respect to O

642
00:41:24,332 --> 00:41:32,800
is just equal to this limit
of delta RBO over delta t

643
00:41:32,800 --> 00:41:37,740
as t goes to 0.

644
00:41:37,740 --> 00:41:39,520
So what direction
is the velocity?

645
00:41:43,858 --> 00:41:47,410
The velocity is in the
direction of the change, not

646
00:41:47,410 --> 00:41:50,600
the original vector, it was in
the direction of the change.

647
00:41:50,600 --> 00:41:52,580
And in fact, if
the path of the dog

648
00:41:52,580 --> 00:41:58,310
is like this, at the instant
you compute the velocity,

649
00:41:58,310 --> 00:42:02,710
you're computing the tangent
to the path of the dog.

650
00:42:02,710 --> 00:42:04,950
So that's what velocity
is at any instant time

651
00:42:04,950 --> 00:42:06,990
is a tangent to the path.

652
00:42:06,990 --> 00:42:08,650
And that's a good
concept to remember.

653
00:42:49,660 --> 00:42:53,400
So we're still in this
fixed Cartesian space,

654
00:42:53,400 --> 00:42:55,190
and I have of couple of points.

655
00:42:55,190 --> 00:42:57,250
I'll make it really
trivial here.

656
00:42:57,250 --> 00:43:04,820
Here's B, and here's A,
and the velocity of B--

657
00:43:04,820 --> 00:43:05,590
where's my number?

658
00:43:09,950 --> 00:43:13,920
We'll make this 10
feet per second.

659
00:43:13,920 --> 00:43:19,300
And it's in the J-hat direction.

660
00:43:19,300 --> 00:43:25,610
And A, this is the
velocity of BNO.

661
00:43:25,610 --> 00:43:33,190
The velocity of ANO, we'll
say is 4 feet per second,

662
00:43:33,190 --> 00:43:35,270
also in the J direction.

663
00:43:35,270 --> 00:43:38,590
And I want to know what's the
velocity of B with respect

664
00:43:38,590 --> 00:43:44,005
to A. So now I'm chasing
the dog, he's running at 10,

665
00:43:44,005 --> 00:43:47,460
I'm running at 4.

666
00:43:47,460 --> 00:43:50,060
How do I perceive
the speed of the dog?

667
00:43:50,060 --> 00:43:52,170
Well, to do this
in vectors, which

668
00:43:52,170 --> 00:43:54,420
is the point of
the exercise here,

669
00:43:54,420 --> 00:43:58,000
is we have the expressions
we started with over there.

670
00:43:58,000 --> 00:44:00,370
And we're going to use
these a lot in the course.

671
00:44:00,370 --> 00:44:04,120
So the velocity of
B with respect to O

672
00:44:04,120 --> 00:44:05,730
is the velocity
of A with respect

673
00:44:05,730 --> 00:44:09,520
to O plus the velocity
of B with respect to A.

674
00:44:09,520 --> 00:44:12,140
And if I want to know velocity
of B with respect to A,

675
00:44:12,140 --> 00:44:12,990
I just solve this.

676
00:44:18,140 --> 00:44:22,540
So velocity of B with respect
to O minus the velocity of A

677
00:44:22,540 --> 00:44:26,340
with respect to O, and in
this case that's 10 minus 4

678
00:44:26,340 --> 00:44:32,380
is 6 in the J.

679
00:44:32,380 --> 00:44:35,470
Point of the exercise is
to manipulate the vector

680
00:44:35,470 --> 00:44:36,930
expressions like this.

681
00:44:36,930 --> 00:44:39,220
So take whatever known
quantities you have

682
00:44:39,220 --> 00:44:40,470
and solve for the unknown one.

683
00:44:40,470 --> 00:44:42,553
In this case, I want to
know the relative velocity

684
00:44:42,553 --> 00:44:44,490
between the two, and it's this.

685
00:44:49,970 --> 00:44:53,620
If I'm here, and I'm
watching the dog,

686
00:44:53,620 --> 00:44:57,110
that's how I perceive the speed
of the dog relative to me,

687
00:44:57,110 --> 00:44:58,200
right?

688
00:44:58,200 --> 00:45:00,180
6 feet per second
in the J direction.

689
00:45:00,180 --> 00:45:05,510
What's the speed of the dog from
the point of view of over here?

690
00:45:05,510 --> 00:45:09,570
The speed of the
dog relative to me.

691
00:45:13,450 --> 00:45:17,140
So it's again the velocity
of B with respect to A,

692
00:45:17,140 --> 00:45:22,300
but from a different position
in this fixed reference frame.

693
00:45:29,930 --> 00:45:31,740
Really important
point, actually.

694
00:45:31,740 --> 00:45:34,085
This is a really important
conceptual point.

695
00:45:37,300 --> 00:45:38,570
Somebody be bold.

696
00:45:38,570 --> 00:45:40,490
What's the speed
with respect to O?

697
00:45:40,490 --> 00:45:44,570
The velocity of B with
respect to A seen from O,

698
00:45:44,570 --> 00:45:48,074
as computed from O, measured
from O. Got radar down there,

699
00:45:48,074 --> 00:45:49,115
and you're tracking them.

700
00:45:54,065 --> 00:45:56,045
AUDIENCE: [INAUDIBLE]

701
00:45:56,045 --> 00:45:59,470
PROFESSOR: In what direction?

702
00:45:59,470 --> 00:46:00,430
AUDIENCE: [INAUDIBLE]

703
00:46:03,310 --> 00:46:04,760
PROFESSOR: Yeah.

704
00:46:04,760 --> 00:46:06,110
It's the same.

705
00:46:06,110 --> 00:46:09,160
The point is it's the same.

706
00:46:09,160 --> 00:46:15,740
If you're in a fixed reference
frame, a vector of velocity

707
00:46:15,740 --> 00:46:20,760
is the same as seen from
any point in the frame.

708
00:46:20,760 --> 00:46:23,680
Any fixed point in the frame
of velocity is always the same.

709
00:46:23,680 --> 00:46:27,140
And in fact, in this case, the
velocity-- this is a moving

710
00:46:27,140 --> 00:46:30,820
point and the velocity
of him with respect to me

711
00:46:30,820 --> 00:46:33,590
this is different
six feet per second.

712
00:46:33,590 --> 00:46:35,570
And I, from here,
say the velocity

713
00:46:35,570 --> 00:46:39,140
of that guy with respect to this
guy is still 6 feet per second.

714
00:46:39,140 --> 00:46:43,930
Any place in that
frame or even any point

715
00:46:43,930 --> 00:46:45,470
moving at constant
velocity, you're

716
00:46:45,470 --> 00:46:47,955
going to see the same answer.

717
00:46:47,955 --> 00:46:50,740
So it doesn't
matter where you are

718
00:46:50,740 --> 00:46:54,510
to compute the velocity
of B with respect to A.

719
00:46:54,510 --> 00:46:57,011
That's the important point.

720
00:46:57,011 --> 00:46:57,510
OK.

721
00:47:12,010 --> 00:47:15,550
OK, we got to pick up with,
and I may not quite finish,

722
00:47:15,550 --> 00:47:22,795
but I am going to introduce
the next complexity.

723
00:47:40,960 --> 00:47:41,460
OK.

724
00:48:03,340 --> 00:48:06,120
So what we just
arrived at a minute ago

725
00:48:06,120 --> 00:48:08,590
is that the velocity
as seen from O

726
00:48:08,590 --> 00:48:13,220
is the same as the velocity
as seen from A. And A is me,

727
00:48:13,220 --> 00:48:15,140
and I'm moving, and
I'm chasing the dog.

728
00:48:15,140 --> 00:48:18,295
So I'm a moving
reference frame, I'm

729
00:48:18,295 --> 00:48:20,590
what's called a translating
reference frame.

730
00:48:20,590 --> 00:48:22,490
So now we're going to
take the next step.

731
00:48:22,490 --> 00:48:24,890
We had a fixed reference
frame before purely,

732
00:48:24,890 --> 00:48:26,390
and now I want to
talk about having

733
00:48:26,390 --> 00:48:28,680
the idea, the concept of
having a moving reference

734
00:48:28,680 --> 00:48:32,200
frame within a fixed one.

735
00:48:32,200 --> 00:48:35,571
So this is the reference
frame O capital XYZ.

736
00:48:35,571 --> 00:48:37,820
And this little reference
frame now is attached to me,

737
00:48:37,820 --> 00:48:41,050
and it's A, and I call
it x-prime y-prime.

738
00:48:41,050 --> 00:48:43,360
So just so you can--
it's going to be

739
00:48:43,360 --> 00:48:46,910
hard to tell this X from this
X if I don't do something

740
00:48:46,910 --> 00:48:49,860
like a prime.

741
00:48:49,860 --> 00:48:52,990
So that this is the concept of
a translating coordinate system

742
00:48:52,990 --> 00:48:56,410
attached to a body, like
a rigid body, for example.

743
00:48:56,410 --> 00:48:59,180
We're going to do lots of
rigid body dynamics here.

744
00:48:59,180 --> 00:49:01,720
And within this
coordinate system,

745
00:49:01,720 --> 00:49:05,759
I can compute the velocity
of B with respect to A,

746
00:49:05,759 --> 00:49:07,300
and I'll get exactly
the same answer.

747
00:49:07,300 --> 00:49:10,930
I'll get that 6 feet per
second in the J direction.

748
00:49:10,930 --> 00:49:14,340
So it's as if-- so
this concept of being

749
00:49:14,340 --> 00:49:17,110
able to have a reference
frame attached to a body

750
00:49:17,110 --> 00:49:20,220
and translating with it, you
can measure things within it,

751
00:49:20,220 --> 00:49:24,160
get the answer, and then
convert that answer to here

752
00:49:24,160 --> 00:49:27,780
if you're using a
different coordinate.

753
00:49:27,780 --> 00:49:31,070
You could use polar coordinates
here and rectangular here,

754
00:49:31,070 --> 00:49:33,610
but they still can be
related to one another.

755
00:49:33,610 --> 00:49:34,830
We'll do problems like that.

756
00:49:43,410 --> 00:49:45,820
OK.

757
00:49:45,820 --> 00:49:51,340
So now what I'm doing is I
told you like in the readings,

758
00:49:51,340 --> 00:49:56,740
the end game is to be able
to talk about translating

759
00:49:56,740 --> 00:49:59,510
and rotating bodies,
and do dynamics

760
00:49:59,510 --> 00:50:01,950
in three dimensions with
translating and rotating

761
00:50:01,950 --> 00:50:03,090
objects.

762
00:50:03,090 --> 00:50:07,740
And we're going to get
there somewhat step by step.

763
00:50:07,740 --> 00:50:10,390
But I want you to
understand the end game

764
00:50:10,390 --> 00:50:11,950
so you know where we're going.

765
00:50:11,950 --> 00:50:14,110
And you need to have a
couple of concepts in mind.

766
00:50:17,820 --> 00:50:23,950
So the first concept is that
this is a rigid body now.

767
00:50:23,950 --> 00:50:27,800
And you can describe the
motion of rigid bodies

768
00:50:27,800 --> 00:50:34,360
by the summation, the
combination of a translation

769
00:50:34,360 --> 00:50:36,410
and a rotation.

770
00:50:36,410 --> 00:50:39,680
And of the rigid body, if you
can describe its translation,

771
00:50:39,680 --> 00:50:41,510
and you can describe
its rotation,

772
00:50:41,510 --> 00:50:43,430
you have the complete motion.

773
00:50:43,430 --> 00:50:47,470
So you got to understand what
do we mean by what's really

774
00:50:47,470 --> 00:50:49,700
the definition of translation.

775
00:50:49,700 --> 00:50:53,096
So translation--
so I've got this--

776
00:50:53,096 --> 00:50:55,450
I'll call it a merry-go-round.

777
00:50:55,450 --> 00:50:58,510
We'll use a merry-go-round
example in a minute.

778
00:50:58,510 --> 00:51:01,845
And you're observers in
a fixed inertial frame

779
00:51:01,845 --> 00:51:05,140
up above this
merry-go-round looking down.

780
00:51:05,140 --> 00:51:06,640
OK.

781
00:51:06,640 --> 00:51:09,890
But so you can see it, I
got to turn it on its side.

782
00:51:09,890 --> 00:51:11,680
So here's my merry-go-round.

783
00:51:11,680 --> 00:51:13,840
And if it's not
rotating, but let's

784
00:51:13,840 --> 00:51:20,320
say it's sitting on a train,
on a flat bed and moving along.

785
00:51:20,320 --> 00:51:22,360
It's translating.

786
00:51:22,360 --> 00:51:24,620
And when you say
a body translates,

787
00:51:24,620 --> 00:51:31,510
any two points on the body
move in parallel paths.

788
00:51:31,510 --> 00:51:33,844
So two points, my
thumb and my finger--

789
00:51:33,844 --> 00:51:35,260
if I'm just going
along with this,

790
00:51:35,260 --> 00:51:37,510
those two paths are traveling
parallel to one another.

791
00:51:40,780 --> 00:51:50,180
If I got Y pointing
up, the body does this,

792
00:51:50,180 --> 00:51:52,605
is it rotating and translating?

793
00:51:52,605 --> 00:51:53,480
AUDIENCE: [INAUDIBLE]

794
00:51:53,480 --> 00:51:59,095
PROFESSOR: Are any two points
on a moving in parallel paths?

795
00:51:59,095 --> 00:52:00,050
Right?

796
00:52:00,050 --> 00:52:01,200
OK.

797
00:52:01,200 --> 00:52:02,800
When it goes through
curved things,

798
00:52:02,800 --> 00:52:05,090
it's called curvilinear
translation.

799
00:52:05,090 --> 00:52:07,220
But it's still just translation.

800
00:52:07,220 --> 00:52:09,200
OK, so I'll stop
and hold steady.

801
00:52:09,200 --> 00:52:13,620
The train stopped, and
the thing-- let it rotate.

802
00:52:13,620 --> 00:52:17,200
So that's pure rotation.

803
00:52:17,200 --> 00:52:20,340
And the thing to remember
about pure rotation

804
00:52:20,340 --> 00:52:26,372
is that anywhere on the body
rotates at the same rate.

805
00:52:26,372 --> 00:52:29,800
If this is going
around once a second,

806
00:52:29,800 --> 00:52:31,880
the rotation rate
is one rotation

807
00:52:31,880 --> 00:52:35,450
per second, 360 degrees,
2 pi radians per second

808
00:52:35,450 --> 00:52:37,180
is its rotation rate.

809
00:52:37,180 --> 00:52:39,710
Every point on the body
experiences the same rotation

810
00:52:39,710 --> 00:52:40,950
rate.

811
00:52:40,950 --> 00:52:42,760
That's a really important
one to remember.

812
00:52:46,240 --> 00:52:48,640
If I'm holding still,
merry-go-round's

813
00:52:48,640 --> 00:52:53,300
going round and round, it has a
fixed axis of rotation, right?

814
00:52:53,300 --> 00:52:58,590
But do rotating bodies have to
have fixed axes of rotation?

815
00:52:58,590 --> 00:53:07,605
So if I throw that up in the
air, not hanging onto it,

816
00:53:07,605 --> 00:53:11,200
it's got gravity acting
on it, it's rotating.

817
00:53:11,200 --> 00:53:12,675
What's a rotate about?

818
00:53:12,675 --> 00:53:13,550
AUDIENCE: [INAUDIBLE]

819
00:53:13,550 --> 00:53:15,180
PROFESSOR: Center of mass, OK.

820
00:53:15,180 --> 00:53:18,570
Is the center of mass moving?

821
00:53:18,570 --> 00:53:21,700
So this is clearly-- this
is an example of rotation

822
00:53:21,700 --> 00:53:22,580
plus translation.

823
00:53:25,430 --> 00:53:28,660
It rotates about an axis
but the axis can move.

824
00:53:28,660 --> 00:53:31,099
That's another important
concept that we

825
00:53:31,099 --> 00:53:33,390
have to allow in order to be
able to do these problems.

826
00:53:33,390 --> 00:53:35,410
But this is now
general motion, it's

827
00:53:35,410 --> 00:53:38,050
a combination of
translation and rotation,

828
00:53:38,050 --> 00:53:40,329
and we figure out each
of those two pieces,

829
00:53:40,329 --> 00:53:42,620
then we can describe the
complete motion of the system.

830
00:53:47,140 --> 00:53:57,660
All right, where we'll pick up
next time is then doing that.

831
00:53:57,660 --> 00:53:59,076
And it would help
actually, if you

832
00:53:59,076 --> 00:54:03,570
go read that reading,
especially up to chapter 16,

833
00:54:03,570 --> 00:54:07,700
we have to get into to taking
derivatives of vectors which

834
00:54:07,700 --> 00:54:10,700
are rotating, and come
up with a general formula

835
00:54:10,700 --> 00:54:13,680
allows us to do velocities
and accelerations

836
00:54:13,680 --> 00:54:14,660
under those conditions.

837
00:54:14,660 --> 00:54:18,940
See you on Tuesday next.