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visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Let’s get

on with some dynamics. So the place I’m

going to begin is just a comment about mechanical

engineering courses. The first, and you may have

heard this already in classes, you’ll be taking

subject 2001 if you’re Course 2 majors through

2009, and if you’re 2-A, most of the odd ones. But the subjects

2001 through 2005 are really basically engineering

science subjects that are all foundational to

mechanical engineering, and they all have a common

or property through them. And that is that we make

observations of the world, and we try to understand them. We pose problems. Why– 400 years ago, is

the sun in the center of the solar system or not? And we try to produce models

that explain the problem. So here’s the problem,

the question of the day. We try to produce

models to describe it, and we make observations,

measurements, to see if our models are correct. And if we feed that information

back into the models, we try out the models, we test

it against more observations, and you go round and round. And this is kind of

the fundamental– this is the way all of these

basic first five subjects use, basically, this

method of inquiry. So in 2003, the way

this system works, my kind of mental conception

of this modeling process, is three things. And this applies to you. You have a homework problem. How do you attack

a homework problem? You’re going to need

to describe the motion. You’re going to need to choose

the physical laws– pick, I’ll call it

because it’s short– the physical law that

you want to apply like f equals ma,

conservation of energy, conservation of momentum. You got to know which

physical laws to apply. And then finally, third you

need to apply the correct math. And that’s really–

most dynamic problems can be broken down this way. That’s the way I like to

conceptually break them down. You might have another

model, but this is the way I’m going to teach it. Can you describe the motion,

pick the correct physical laws to apply to the problem, and

able to do the correct math, solving the equation

of motion, for example. And all this is what

fits in our models box. And we test it against

observations and measurements and improve those

things over time. So I’m going to give you–

how many of you like history? I find history and

history technology kind of fun and interesting. So I’m going to throw

a little bit of history into giving you a

little quick course outline of what we’re going to

do in this subject this term. Because the history

dynamics and what we’re going to do in the course

actually track one another remarkably closely. So if I ever gave you a bunch

of names like Galileo, Kepler, Descartes, Newton, Copernicus,

Euler, Lagrange and Brahe, which one comes first? Take a guess. AUDIENCE: Copernicus. PROFESSOR: Good. Copernicus. So Copernicus was

Polish, and the story starts long before then, but

in about 1,500 Copernicus said what? AUDIENCE: [INAUDIBLE] PROFESSOR: The sun’s the center? AUDIENCE: [INAUDIBLE] PROFESSOR: Or the

Earth is the center? AUDIENCE: [INAUDIBLE] PROFESSOR: Which did say? Yes, so Ptolemy, back

around 130 AD said, well the Earth’s the

center of the solar system. Copernicus came

along and said, nope I think that, in fact, the sun’s

the center of the solar system. And it for the next 100 years–

more than 100 years, couple hundred years– there was

a really raging controversy about that. So Copernicus, Brahe

Kepler– so I’m putting them in rough

chronological order here. Now, I’m going to

run out of board. Oh well. Galileo, Descartes– I’m gonna

cheat– OK, Descartes, Newton, Euler, and Lagrange. So we’re going to talk and say

a little bit about each of them. And now that I’m–

like I told you, I haven’t used this classroom

before so I gotta learn how to play this game. I need to be able to

reach this for a minute. So Brahe, he was

along about 1,600. Brahe was the

mathematician that wrote– the imperial mathematician

to the emperor in Prague. And he did 20 years

of observations. And he was out to prove

that the Earth was the center of the solar system. And then Kepler actually worked

with him as a mathematician, and then took over as the

imperial mathematician. And he took Brahe’s data–

20 years of astronomical data without the use

of the telescope– and used it come up with the

three laws of planetary motion. And so his first and second

laws were put out about 1609. And one of the laws

is, like, equal area swept out in equal time. Have you hear that one? That actually turns out to be

a statement of conservation of angular momentum, which

we’ll talk quite a bit about the course. Then came Galileo, and I’m not

putting their birth and death dates here. I’m kind of putting

in a period of time in which kind of important

things happened around him. So 401 years ago a really

important thing happened. Galileo, in 1609, turned

the telescope on Jupiter, and saw what? AUDIENCE: [INAUDIBLE] PROFESSOR: Four moons, right? And then they really

started having some data with which to really argue

against the Ptolymaic view of the solar system. Descartes is an

important figure to us. And in the period of about

1630 to 1644– in that period Descartes began what is today

known as analytic geometry. He was geometer, he

studied Euclid a lot. But then he came up with a

Cartesian coordinate system, xyz, and the beginnings

of analytic geometry, which is essentially algebra,

coordinates, and geometry all put together. And we are going to make

great use of analytic geometry in this course. Then came Newton, kind of in

his actual lifespan, 1643. It’s kind of interesting

that he spans these people. And in about 1666 is when he

first– the first statement of the three laws of motion. Then Euler, and he’s 1707 to

1783, and that’s his lifespan. Euler came up– Newton never

talked about angular momentum. He mostly talked

about particles. Euler put Newton’s three

laws into mathematics. Euler taught us about

angular momentum, and torque being dh

dt in most cases. He’s the most prolific

mathematician all time, solved all sorts of

important problems. And then finally, is Lagrange. And Lagrange, in about 1788,

uses an energy method, energy and the concept of work to

give us equations of motion. So the course, 203,

stands on the shoulders of all these people. But with Descartes, we start

with kinematics, really. This is analytic geometry. And that’s where we’re going to

start today is with kinematics. And very soon

thereafter, we’re going to review Newton, the

three laws, and what we call the direct method for

finding equations of motion. Conservation of

momentum, fact that force– some of the

forces on an object equals mass times acceleration,

or it’s a time derivative of its linear momentum. And we use that to derive

equations of motion. So we’re going to go kinematics

into doing the direct method to getting equations of motion. And we go from there

into angular momentum, and what Euler gave us–

the same thing, torque. We’re going to do quite a

lot with angular momentum. Because I know you know

a lot about f equals ma and you’ve done lots of

problems 801 applying that. You’ve done some problems

on rigid body rotations. But I think there’s

a lot more you need to understand

about this, and we’ll spend quite a bit of time on it. And then near the

last third the course we shift, because Lagrange said

that if you just write down expressions for energy,

kinetic and potential energy, without any consideration

of Newton’s laws and the direct method, you can

derive the equations of motion. That’s pretty remarkable. So there are actually two

independent roots to coming up with equations of motion. And in this course, about

the last third of the course, we’re going to teach

you about Lagrange. And then all these

things are going to be– one of the applications

that are important engineers is the study of vibration. So we’ll be looking

at vibration examples as we go through the

course, and applying these different methods

to first, modeling, and then solving interesting

vibration problems. Which brings– ah, I

have a question for you. So how many of you were

in this classroom last May with Professor Haynes Miller,

and I showed up one day and we talked about vibration? How many remember? I told you I was going to

ask this question, right? Great. OK, it’s good to

see you here again, and we will talk about

vibration in this course. So there’s kind of the

subject outline built on the shoulders of

these people in history that made important

contributions to dynamics. Any questions about the history? If you want to

know, one of my TAs compiled a pretty

neat little summary. Maybe I will see if I

go back and find this. I just printed out and

sent it– how many of you like to know a little

bit more about the history? These are like two

liners on each person. Anybody want it? Is it worth my time

to send this out? OK, it’s kind of fun. So let’s do an example of this

modeling describing the motion, picking physical laws,

applying the math. And that’ll get us

launched in the course. And we’ll do it using Newton

and the direct method. So last May, Haynes Miller

and I talked about vibration. So I’m going to start

with a vibration problem. And I brought one. So here’s my couple

of lead weights and a couple of springs. So really I just want to talk

about– this is the problem I want to talk about. Now you’ve done

this problem before. Haynes Miller and

I did it last May. And you’ve no doubt

it in other classes. OK, it’s a system which

has a spring, a mass, it exhibits something

called a natural frequency. But let’s see what it takes

to just initially begin to follow this modeling

method to arrive at an equation of

motion for this problem. So what do I mean by when

I say, describe the motion? Really what that

boils down to if you have to assign a

coordinate system so that you can actually say

where the object’s moving. And I’m going to pick one here. So here’s– coordinate system

going to be really important in this course. And I’ll give us an xyz

Cartesian coordinate system. And I’m going to try to adopt

the habit, for the most part, during the course that

this o marks this origin, but it also names the frame. So we’re going to

talk about things in that are reference frames. And most important

one that we need to know about in the course is

an inertial reference frame, and when you can use

it, and when a system is inertial and is not. So I’m gonna say that

this is inertial. It’s fixed to the Earth. It’s not moving. And we’re going to use this

coordinate x to describe the motion of this mass. And the motion is going to be–

this x is from the zero spring force position. It’s actually quite

important that you pick– that you

have to say what’s the condition in the spring

of the system when x is 0 So we’re going to say

it’s, when there’s no force in the

spring means it’s not stretch, that’s where 0 is. So we’ve established

a coordinate system. Second, we need to

apply physical laws. Now, I’m going to do

this problem by f equals ma, Newton’s second law. Sum of the external

forces is equal to mass times the acceleration. So that’s the law

I’m going to apply. Sum of the external

forces, it’s a vector but we’re just doing the x

component only so we don’t have to carry along vector notation,

is equal to, in this case, mass times acceleration. So that’s the law

we’re going to apply. And then finally the math to

solve the equation of motion that we find, that’ll

be the third piece. But part of applying

the physics, in order to do this now, we need

what I call an FBD. What do you suppose that is? AUDIENCE: Free body diagrams. PROFESSOR: Free body diagrams. You’ve used these

many times before, so we’re going to do those. And free body diagrams– And I’m going to

teach you, at least the way I go about doing

free body diagrams, as things get more

and more complicated, you’re going to have to be

more sophisticated in the way that you do these things. So I just have some

simple little rules to do free body diagrams that

keep you from getting hung up on sign conventions. I think the thing people

make most mistakes about is they get confused about signs. So I’ll try to show

you how I do it. So first you draw forces

that you know, basically in the direction

in which they act. Seems obvious. So when you know

the direction– so this is a really

trivial problem, but the method here

is very specific. So what’s an example? Well, gravity. So we’ll start our

free body diagram. Gravity acts at

the center of mass. It’s downward. This is what I mean by the

direction in which it acts. And it has magnitude, mg. OK. Now the other forces

aren’t so obvious. The force that’s put on by

the stiffness and this damper in the spring, which

way do you draw them? What’s the sign? What’s the sign convention? So the convention, the way I

go about doing these things, is I assume positive values for

the deflections and velocities. So in this case, x and x dot. You just require that the

deflections that you’re going to work with are positive. And then from the

positive deflection, you say which way is

the resulting force? So if the deflection

in this is downwards, which direction is the force

that the spring applies to the mass? Up, right? What about if the

velocity is downwards, which direction is the force

is the damper puts on the mass? Also up, right? OK. So this allows– this gives us–

so here’s f spring and here’s the f damper. And other any other

forces on this mass? So spring force, damper force,

and the gravitational force. And so third, you deduce

the signs basically from the direction

of the arrows. First we need what’s called

your constitutive relationship. So the spring force,

fs, well you’ve made x positive so

it keeps things nice, the spring constant’s a

positive number, so fs is kx. Fd is bx dot. And now we write the

statement that the sum of forces in the x direction. We look at up here, we

say well that’s going to fs plus fd minus mg. So that’s– whoops, I wrote

it the wrong way around. Minus, minus, plus. Because I’m plus

downwards, right? Well, spring minus fs

is minus kx minus bx dot plus mg equals mx double dot. And I rearranged this

to put all the motion variables on one side. mx double dot plus bx

dot plus kx equals mg. So there’s my equation of

motion, but with a method for doing the free

body diagrams, which will work with multiple bodies. So you have two bodies with

springs in between them. This is when the

confusion really comes up. Two bodies with a spring

trapped between them. What’s the sign convention? You do the same thing. Both bodies exhibit

positive motions, the force that results is

proportional to the difference, and you work it out. And you’ll get the signs right. OK, so here’s our

equation of motion arrived at by doing the direct method. And if we went on to the

third step, which we’re not going to do today, and

that is apply the math, it might because I want you now

to describe the motion for me, solve for the motion. That means solving the

differential equation. And that’s what we did last

may in Haynes Miller’s class. We’ll come back

to this later on. But for today’s purposes,

we don’t need to go there. Got something else

much more important to get to about kinematics. But I want to show

you one thing, and that is just a little

tiny introductory taste to this point. So I’ve derived the

equation of motion of this by Newton’s laws. But I’m going to

ignore Newton now and saw I’m going to

drive equation of motion by another way. And it’s an energy

technique, and that is– well let’s talk about the total

energy of the system. It’s going to be the

sum of a kinetic energy and a potential energy. And we’ll find that even

with Lagrange, there’s a problem with forces

on systems that are what we call

non-conservative, things that either take

energy out of, or put energy into the system. And the dashpot does that. Dashpot generates heat and

takes energy out of the system. So I’m going to have to

ignore it for the moment. So the sum of the kinetic

and the potential energies in this problem is

a 1/2 kx squared for the potential of the spring,

plus a 1/2 mx dot squared for the kinetic energy of

the mass, and minus mgx for the potential energy that

is due to the object moving in the gravitational field. And that’s the total

energy of the system. Now my problem, I’ve

allowed no forces. There’s no excitation on here. This is just free

vibration only. That’s all we’re talking about,

make initial displacement and it vibrates. If there’s no

damping, what can you say about the total

energy of the system? AUDIENCE: [INAUDIBLE] PROFESSOR: Say it again. I heard it over there. It’s got to be constant, right? All right, well, so

this must be constant. Therefore, the time derivative

of my system, it better be 0. The energy is constant. Take it’s time derivative,

it’s got to be 0. Apply that to the

right-hand side of this, I get kxx dot plus mx dot

x double dot minus mgx dot equals zero. And I now cancel out the

common x dot terms go away. And I’m left with–

and I’ve essentially solved for the equation

of motion of this system without ever looking at

conservational momentum, Newton’s laws, only by

energy considerations. OK, so that’s a

very simple example of that you can use energy to

derive equations of motions. But you then have to go back and

fix it to account for the loss term, the damping term. And that you still have

to consider it as a force, we’ll find out. Even was Lagrange you have to go

back and consider the work done by external forces. OK. So you’ve just kind of

seen the whole course. We’ve described

the motion, we’ve applied to Newton’s laws, the

physics to the direct method to derive the

equations of motion, we have gone to a direct method,

and have derived the equations of motion that way. And that’s basically what you’re

going to do in the course. But now you’re going

to do it with much more sophisticated tools. You’ll have multiple

degree of freedom systems. The description

describing the motion, is maybe going to

be for some of you, the most challenging

part of the course. And this is a topic

we call kinematics. And that’s what

we’ll turn to next. So reference frames and vectors. That’s the topic. This is now that we’re

talking about kinematics, and this is all about

describing the motion. So Descartes gave us the

Cartesian coordinate system, and we’ll start there. So imagine this

is a fixed frame– we’ll talk about what

makes an inertial frame the next lecture. But here we have

an inertial frame. And it’s the frame we’ll

call O-xyz or O for short. And in this frame, maybe this

is me, and up here is a dog, and I’m going to call this

point A and this point B. And I’m going to describe the

positions of these two points by vectors. This one will be R, and the

notation that I’m going to use is point and it’s measurement

with respect something. Well, it’s with respect to this

point O in this inertial frame. So this is A with respect

O is the way to read this. There’s another vector here. This is RB respect

to A And finally, R of B with respect to O They’re

all vectors on the board. I’ll try to remember

to underline them in the textbooks and things. They’re usually– vectors

are noted with bold letters. And vectors allow us

to say the following. That R, the position of

the dog and the reference with respect to O, is the sum

of these other two vectors. R of A with respect to O plus

R R of B with respect to A. And mostly to do

dynamics we’re really interested in things like

velocities and accelerations. So to get the velocities

and accelerations, we have to take a time

derivative of our RBO dt. And that’s going to give us

what we’ll call the velocity, obviously you write

it as V. And it would be the velocity of point B with

respect to O. And no surprise, it’ll be the velocity of

point A plus the velocity of B with respect to A. And finally, if we take two

derivatives, dt squared, we’ll get the acceleration

of B with respect to O. And that’ll be the sum of

A– the acceleration of A with respect to O plus

the acceleration of B with respect to A.

All, again, vectors. Now, just to look ahead–

this seems all really trivial. You guys are going to

sleep on me, right? If these are rigid bodies, this

is a rigid body that is moving and maybe rotating. And B is on it, and A is

on it, and O isn’t on it. It starts getting

a little tricky. And this, the derivative

of a vector that’s attached to the body somehow has to

account for the fact that if I’m– the

observer’s on the body, this other point’s on the body. Say it’s, I’m on this asteroid,

and I’ve got a dog out there, and the dog’s run away from me. The speed of the dog with

respect to me, I can measure. But if I’m down

here looking at it, it’ll look different

because it’s rotating. So how do you

account for all that? So taking these derivatives

of vectors in moving frames is where the devil’s

in the details. And that’s part of what I’m

going to be teaching you. OK. I’m still learning how

to optimize my board use. I haven’t got it perfect

yet, but because I’m having to move around a

lot here and improvise. But we’ll persevere. You need to remember a

couple things about vectors, how to add them, dot products. If you’ve forgotten

these things, you need to go back and

review them really quickly. There’s usually a little

review section the book, so you need to practice

that sort of thing. Couple other little facts

you need to remember. So the derivative of

the sum of two vectors is just the sum of

the derivatives. And quite importantly,

we’re going to make use of this one

a lot, is the derivative of a product of two things. One of them be in a vector,

some function maybe of time and a here is derivative of

f with respect to t times a, plus the derivative of a

with respect to t times f. That we’ll make a lot use of. So just your basic calculus. So now, I want to take up–

let’s talk about the simplest form of being able to

do these derivatives and calculate these

velocities, when everything’s described in terms of

Cartesian coordinates. Now I’m going to give you a

little look ahead because I’m going to try to avoid confusion

as much as possible here. The hardest problem is

when you have a rigid body, you got the dog on it, you’ve

got the observer on it, it’s rotating, and translating. And to take this derivative, you

end up with a number of terms. The simplest problem

is just something in a fixed Cartesian

coordinate system. So we’re going to start

with a simple one, and build our way up to

the complicated one, OK? But let’s now, we’re going to

do the really, the simplest one. We’re going to do

velocity and acceleration in Cartesian coordinates. And basically I should say

fixed Cartesian coordinates, not moving. All right, so now let’s

consider the dog out here, and his position in the

Cartesian coordinate system. And I could write

that and you’ll, without any loss of

generality here, you’ll know what I mean if

I say RBx component. And I’m going to stop

writing the slash O’s, because this is now all in

this fixed reference frame. And it’s in I-hat direction. And I’ve got another component,

RBy in the J-hat, and an RBz in the K-hat. And I want to take

the time derivative– I was looking for the velocity. I want to calculate

the velocity. So the velocity here of

BNO is d by dt of RBO. . And now this is now the

product of two things, so I’ve got to use

that formula over here. Product one turn times

the other, and so forth. So I go to these,

and I say OK, so this is R dot Bx times I plus R

dot By times J plus R dot Bz times K. And then the other–

the flip side of that is I have to take the derivatives of

I times RBx, the derivative J and so forth. But what’s the derivative

of, let’s say, I? Capital I is my unit vector

in the fixed reference frame, my O-xyz frame. 0 So it’s a constant. It is unit length, and it points

in a direction that it’s fixed. So what’s its derivative? It’s going to have

a 0 derivative. So the second part of this–

second bits of that is zero. So that’s the velocity in

Cartesian coordinates of my dog out there running around. And the acceleration,

in a similar way, now to get the acceleration,

you take another derivative of this. And again, you’ll have to take

derivatives of I, J, and K, and again they’re going to be 0. So you will find that the

acceleration then, is just R double dot x term

in the plus R double dot By in the J plus r

double dot Bz in the K. That would be our acceleration

term, and it’s easy. Now imagine that we are doing

this in polar coordinates, unit vectors in polar coordinates. Let me check, last year

the students told me that in your

physics courses, you use unit vectors R-hat,

theta-hat, and K. Is that right? So I’ll use those unit

vectors so they look familiar, because in polar

coordinates people use lots of different things. But think about it, in

polar coordinates, theta– it’s a fixed, maybe,

coordinate system, but now theta goes like this

and R moves with theta, right? So the unit vector

is pointing here, but over time it might

move down to here. And unit vector has

changed direction, and its derivative in

time is no longer 0. So it starts getting messy

as soon as the unit vectors change in time. And so that’s one of

our objectives here is to get to that

point and describe how you handle those cases. So a quick point about velocity. You need to really understand

what we mean by velocity. So here’s our Cartesian system. Here’s this point out

here B. And now, this is the dog running around,

and the path of the dog might have been like this. And right in here he’s

going this direction. And in a little

time, in delta t, he moves by an amount

delta RB with respect to O. And that’s what this is. He’s moved this little

bit in time delta t. And he happens to be going

off in that direction. So this then is R

prime, I’ll call it, of B with respect to O, and this

is our original RB with respect to O. So we can say that

his new position, RB with respect to prime

is RBO plus delta R. And these are all vectors. And the velocity of

B with respect to O is just equal to this limit

of delta RBO over delta t as t goes to 0. So what direction

is the velocity? The velocity is in the

direction of the change, not the original vector, it was in

the direction of the change. And in fact, if

the path of the dog is like this, at the instant

you compute the velocity, you’re computing the tangent

to the path of the dog. So that’s what velocity

is at any instant time is a tangent to the path. And that’s a good

concept to remember. So we’re still in this

fixed Cartesian space, and I have of couple of points. I’ll make it really

trivial here. Here’s B, and here’s A,

and the velocity of B– where’s my number? We’ll make this 10

feet per second. And it’s in the J-hat direction. And A, this is the

velocity of BNO. The velocity of ANO, we’ll

say is 4 feet per second, also in the J direction. And I want to know what’s the

velocity of B with respect to A. So now I’m chasing

the dog, he’s running at 10, I’m running at 4. How do I perceive

the speed of the dog? Well, to do this

in vectors, which is the point of

the exercise here, is we have the expressions

we started with over there. And we’re going to use

these a lot in the course. So the velocity of

B with respect to O is the velocity

of A with respect to O plus the velocity

of B with respect to A. And if I want to know velocity

of B with respect to A, I just solve this. So velocity of B with respect

to O minus the velocity of A with respect to O, and in

this case that’s 10 minus 4 is 6 in the J. Point of the exercise is

to manipulate the vector expressions like this. So take whatever known

quantities you have and solve for the unknown one. In this case, I want to

know the relative velocity between the two, and it’s this. If I’m here, and I’m

watching the dog, that’s how I perceive the speed

of the dog relative to me, right? 6 feet per second

in the J direction. What’s the speed of the dog from

the point of view of over here? The speed of the

dog relative to me. So it’s again the velocity

of B with respect to A, but from a different position

in this fixed reference frame. Really important

point, actually. This is a really important

conceptual point. Somebody be bold. What’s the speed

with respect to O? The velocity of B with

respect to A seen from O, as computed from O, measured

from O. Got radar down there, and you’re tracking them. AUDIENCE: [INAUDIBLE] PROFESSOR: In what direction? AUDIENCE: [INAUDIBLE] PROFESSOR: Yeah. It’s the same. The point is it’s the same. If you’re in a fixed reference

frame, a vector of velocity is the same as seen from

any point in the frame. Any fixed point in the frame

of velocity is always the same. And in fact, in this case, the

velocity– this is a moving point and the velocity

of him with respect to me this is different

six feet per second. And I, from here,

say the velocity of that guy with respect to this

guy is still 6 feet per second. Any place in that

frame or even any point moving at constant

velocity, you’re going to see the same answer. So it doesn’t

matter where you are to compute the velocity

of B with respect to A. That’s the important point. OK. OK, we got to pick up with,

and I may not quite finish, but I am going to introduce

the next complexity. OK. So what we just

arrived at a minute ago is that the velocity

as seen from O is the same as the velocity

as seen from A. And A is me, and I’m moving, and

I’m chasing the dog. So I’m a moving

reference frame, I’m what’s called a translating

reference frame. So now we’re going to

take the next step. We had a fixed reference

frame before purely, and now I want to

talk about having the idea, the concept of

having a moving reference frame within a fixed one. So this is the reference

frame O capital XYZ. And this little reference

frame now is attached to me, and it’s A, and I call

it x-prime y-prime. So just so you can–

it’s going to be hard to tell this X from this

X if I don’t do something like a prime. So that this is the concept of

a translating coordinate system attached to a body, like

a rigid body, for example. We’re going to do lots of

rigid body dynamics here. And within this

coordinate system, I can compute the velocity

of B with respect to A, and I’ll get exactly

the same answer. I’ll get that 6 feet per

second in the J direction. So it’s as if– so

this concept of being able to have a reference

frame attached to a body and translating with it, you

can measure things within it, get the answer, and then

convert that answer to here if you’re using a

different coordinate. You could use polar coordinates

here and rectangular here, but they still can be

related to one another. We’ll do problems like that. OK. So now what I’m doing is I

told you like in the readings, the end game is to be able

to talk about translating and rotating bodies,

and do dynamics in three dimensions with

translating and rotating objects. And we’re going to get

there somewhat step by step. But I want you to

understand the end game so you know where we’re going. And you need to have a

couple of concepts in mind. So the first concept is that

this is a rigid body now. And you can describe the

motion of rigid bodies by the summation, the

combination of a translation and a rotation. And of the rigid body, if you

can describe its translation, and you can describe

its rotation, you have the complete motion. So you got to understand what

do we mean by what’s really the definition of translation. So translation–

so I’ve got this– I’ll call it a merry-go-round. We’ll use a merry-go-round

example in a minute. And you’re observers in

a fixed inertial frame up above this

merry-go-round looking down. OK. But so you can see it, I

got to turn it on its side. So here’s my merry-go-round. And if it’s not

rotating, but let’s say it’s sitting on a train,

on a flat bed and moving along. It’s translating. And when you say

a body translates, any two points on the body

move in parallel paths. So two points, my

thumb and my finger– if I’m just going

along with this, those two paths are traveling

parallel to one another. If I got Y pointing

up, the body does this, is it rotating and translating? AUDIENCE: [INAUDIBLE] PROFESSOR: Are any two points

on a moving in parallel paths? Right? OK. When it goes through

curved things, it’s called curvilinear

translation. But it’s still just translation. OK, so I’ll stop

and hold steady. The train stopped, and

the thing– let it rotate. So that’s pure rotation. And the thing to remember

about pure rotation is that anywhere on the body

rotates at the same rate. If this is going

around once a second, the rotation rate

is one rotation per second, 360 degrees,

2 pi radians per second is its rotation rate. Every point on the body

experiences the same rotation rate. That’s a really important

one to remember. If I’m holding still,

merry-go-round’s going round and round, it has a

fixed axis of rotation, right? But do rotating bodies have to

have fixed axes of rotation? So if I throw that up in the

air, not hanging onto it, it’s got gravity acting

on it, it’s rotating. What’s a rotate about? AUDIENCE: [INAUDIBLE] PROFESSOR: Center of mass, OK. Is the center of mass moving? So this is clearly– this

is an example of rotation plus translation. It rotates about an axis

but the axis can move. That’s another important

concept that we have to allow in order to be

able to do these problems. But this is now

general motion, it’s a combination of

translation and rotation, and we figure out each

of those two pieces, then we can describe the

complete motion of the system. All right, where we’ll pick up

next time is then doing that. And it would help

actually, if you go read that reading,

especially up to chapter 16, we have to get into to taking

derivatives of vectors which are rotating, and come

up with a general formula allows us to do velocities

and accelerations under those conditions. See you on Tuesday next.