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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
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
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