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PROFESSOR: --get started.
00:00:24.510 --> 00:00:27.880
What do you think are some
important new concepts
00:00:27.880 --> 00:00:32.189
that we've been talking
about in the last week?
00:00:32.189 --> 00:00:32.800
Make a list.
00:00:32.800 --> 00:00:33.526
Betsy.
00:00:33.526 --> 00:00:34.984
AUDIENCE: Various
ways to calculate
00:00:34.984 --> 00:00:37.377
kinetic energy [INAUDIBLE].
00:00:37.377 --> 00:00:37.960
PROFESSOR: OK.
00:00:45.896 --> 00:00:47.384
All right?
00:00:47.384 --> 00:00:48.561
AUDIENCE: Virtual work.
00:00:48.561 --> 00:00:49.560
PROFESSOR: Virtual work.
00:01:02.150 --> 00:01:04.309
Something else.
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[INAUDIBLE]
00:01:05.295 --> 00:01:07.267
AUDIENCE: The Lagrange equation.
00:01:07.267 --> 00:01:08.253
PROFESSOR: All right.
00:01:18.610 --> 00:01:19.364
[INAUDIBLE]
00:01:19.364 --> 00:01:21.280
AUDIENCE: [INAUDIBLE]
generalized coordinates.
00:01:21.280 --> 00:01:22.738
PROFESSOR: Generalized
coordinates.
00:01:34.670 --> 00:01:37.286
Anything else on your list?
00:01:37.286 --> 00:01:38.494
AUDIENCE: Generalized forces.
00:01:43.490 --> 00:01:45.050
PROFESSOR: OK.
00:01:45.050 --> 00:01:46.455
Lagrange equations.
00:01:46.455 --> 00:01:48.820
That's a pretty good list.
00:01:48.820 --> 00:01:51.340
Now, we've talked about quite
a few new things lately.
00:01:51.340 --> 00:01:53.650
So is there any-- did
you come in here today
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with any questions
about something
00:01:56.400 --> 00:02:00.645
that just isn't sitting right on
these topics or anything else?
00:02:00.645 --> 00:02:02.520
And I think write up a
couple extra questions
00:02:02.520 --> 00:02:05.540
that we may be able to
cover them as we go along.
00:02:05.540 --> 00:02:06.788
Steven?
00:02:06.788 --> 00:02:08.784
AUDIENCE: So when you
solve the entire thing,
00:02:08.784 --> 00:02:10.780
everything in general
is coordinates, right?
00:02:10.780 --> 00:02:12.776
But say you want to
find the actual numbers.
00:02:12.776 --> 00:02:18.196
Do you calculate them
as your [INAUDIBLE],
00:02:18.196 --> 00:02:19.320
or do you just [INAUDIBLE]?
00:02:22.030 --> 00:02:26.100
PROFESSOR: So this is really how
to use the equations you end up
00:02:26.100 --> 00:02:28.970
with in practical situations?
00:02:28.970 --> 00:02:31.015
OK.
00:02:31.015 --> 00:02:33.270
I'm just going to
answer this one.
00:02:33.270 --> 00:02:36.400
So when you choose your
generalized coordinates,
00:02:36.400 --> 00:02:39.620
you'll probably choose
about the same ones
00:02:39.620 --> 00:02:42.470
you'd choose if you
did the direct method.
00:02:42.470 --> 00:02:48.580
So if you did direct method
and got equations of motion,
00:02:48.580 --> 00:02:50.960
would you be asking
the same question?
00:02:50.960 --> 00:02:51.918
AUDIENCE: I don't know.
00:02:51.918 --> 00:02:52.700
PROFESSOR: All right?
00:02:52.700 --> 00:02:53.200
OK.
00:02:53.200 --> 00:02:57.050
So really, you want
to choose coordinates
00:02:57.050 --> 00:03:02.330
that are going to be
practically useful in the end.
00:03:02.330 --> 00:03:02.830
OK?
00:03:02.830 --> 00:03:04.538
And so it shouldn't
matter whether you're
00:03:04.538 --> 00:03:06.930
doing it by Lagrange or
doing it by the direct method
00:03:06.930 --> 00:03:08.336
to get to them, you're
going to use them
00:03:08.336 --> 00:03:09.480
in the same way in the end.
00:03:12.140 --> 00:03:15.580
Now, we have two methods to
get equations of motion now.
00:03:15.580 --> 00:03:17.910
When you're doing tough
problems, complicated problems,
00:03:17.910 --> 00:03:22.030
I would always do it one way and
use the other way to check it.
00:03:22.030 --> 00:03:23.280
I'd end up doing it both ways.
00:03:23.280 --> 00:03:26.410
If it was really important,
you do it both ways.
00:03:26.410 --> 00:03:28.720
And one will give you
insight about the other.
00:03:28.720 --> 00:03:30.920
Yesterday in lecture,
I was talking about,
00:03:30.920 --> 00:03:37.570
does it make sense to have a
Coriolis term in this equation?
00:03:37.570 --> 00:03:39.610
Would you expect it?
00:03:39.610 --> 00:03:41.820
And that kind of
common sense checking.
00:03:41.820 --> 00:03:42.320
All right.
00:03:42.320 --> 00:03:43.230
Any other questions?
00:03:43.230 --> 00:03:45.770
Good question.
00:03:45.770 --> 00:03:48.140
Generalized coordinates
or forces or anything?
00:03:48.140 --> 00:03:48.640
All right.
00:03:48.640 --> 00:03:49.830
Let's get on.
00:03:49.830 --> 00:03:53.180
Let's keep working here.
00:03:53.180 --> 00:04:02.390
So I have an assignment
for you, and it's here.
00:04:02.390 --> 00:04:04.040
This is a familiar
problem that you've
00:04:04.040 --> 00:04:06.180
worked before by other methods.
00:04:06.180 --> 00:04:08.820
And you did, in fact, even
find the kinetic and potential
00:04:08.820 --> 00:04:11.200
energies, I think,
last week for this.
00:04:11.200 --> 00:04:16.490
So here's a cart with
a rod, uniform rod.
00:04:16.490 --> 00:04:21.329
No dashpot at the moment,
and no external forces.
00:04:21.329 --> 00:04:23.810
I'm going to give you
coordinates to use.
00:04:23.810 --> 00:04:26.930
So here's our inertial
system, deflection of the cart
00:04:26.930 --> 00:04:30.080
x, rotation of the rod theta.
00:04:30.080 --> 00:04:32.720
We're going to break you
into groups of about-- we
00:04:32.720 --> 00:04:36.625
got a big group here-- 5, 10.
00:04:39.190 --> 00:04:45.860
Well, I want you to break
into three groups, kind
00:04:45.860 --> 00:04:48.121
of like five or six
there, five or six there,
00:04:48.121 --> 00:04:49.120
and the same thing here.
00:04:49.120 --> 00:04:52.670
And one group each-- so
this group in the front
00:04:52.670 --> 00:04:57.030
left, compute the kinetic
energy of this system.
00:04:57.030 --> 00:04:59.910
And the group behind them,
compute the potential.
00:04:59.910 --> 00:05:01.890
And this group
over here, come up
00:05:01.890 --> 00:05:14.070
with the velocity of G and the
G dot VG, the velocity squared.
00:05:14.070 --> 00:05:14.919
OK?
00:05:14.919 --> 00:05:16.710
And this won't you take
long because you've
00:05:16.710 --> 00:05:17.880
done the stuff before.
00:05:17.880 --> 00:05:20.088
And then when you get done,
somebody from your group,
00:05:20.088 --> 00:05:22.430
just when you're done,
come up and write it down.
00:05:22.430 --> 00:05:23.892
All right.
00:05:23.892 --> 00:05:25.270
Let's sort these out.
00:05:28.850 --> 00:05:31.780
So let's start
with the velocity.
00:05:31.780 --> 00:05:35.700
We're going to need the velocity
to be able to finish out
00:05:35.700 --> 00:05:36.525
the kinetic energy.
00:05:36.525 --> 00:05:42.940
So the vector,
right, and they've
00:05:42.940 --> 00:05:44.165
broken it into two pieces.
00:05:46.860 --> 00:05:49.040
And since the others, if
you haven't worked on it,
00:05:49.040 --> 00:05:49.750
are you guys--
00:05:49.750 --> 00:05:52.205
STUDENT: Can you switch
the sine and cosine?
00:05:52.205 --> 00:05:56.560
PROFESSOR: I was about
to ask you about that.
00:05:56.560 --> 00:05:58.080
I think that that's better.
00:05:58.080 --> 00:05:59.699
So this is sine, right.
00:05:59.699 --> 00:06:01.136
STUDENT: No, that's cosine.
00:06:01.136 --> 00:06:02.573
PROFESSOR: I just erased it.
00:06:02.573 --> 00:06:05.447
That's cosine, sine.
00:06:10.240 --> 00:06:14.310
This is your l theta dot piece
and you want it this way.
00:06:14.310 --> 00:06:18.420
And that's theta, so it should
be sine theta j, cosine theta
00:06:18.420 --> 00:06:18.920
i.
00:06:21.920 --> 00:06:23.710
I think that's good
for the velocity.
00:06:23.710 --> 00:06:29.254
And then velocity squared, just
the square reach of pieces.
00:06:52.650 --> 00:06:54.559
So we have v dot b.
00:06:54.559 --> 00:06:56.850
We need to put this into the
kinetic energy expression,
00:06:56.850 --> 00:07:00.480
but one at a time here.
00:07:00.480 --> 00:07:03.210
The potential energy
expression, 1/2 kx squared.
00:07:03.210 --> 00:07:06.160
Everybody good with that?
00:07:06.160 --> 00:07:09.530
Looks like the spring
potential energy.
00:07:09.530 --> 00:07:16.530
And m2g, 1/2 l is a position
of the center of mass.
00:07:16.530 --> 00:07:19.760
1 minus cosine l that
times 1 minus cosine theta
00:07:19.760 --> 00:07:23.230
is the amount that it changes
height from the reference.
00:07:23.230 --> 00:07:25.785
What's the reference position?
00:07:25.785 --> 00:07:27.410
What have you assumed
for the reference
00:07:27.410 --> 00:07:29.520
position in this formulation?
00:07:29.520 --> 00:07:30.020
Yeah?
00:07:30.020 --> 00:07:31.619
STUDENT: [INAUDIBLE].
00:07:31.619 --> 00:07:33.160
PROFESSOR: So your
reference position
00:07:33.160 --> 00:07:37.330
is center of mass when
it's hanging straight down.
00:07:37.330 --> 00:07:37.990
Great.
00:07:37.990 --> 00:07:40.804
So there's your potential
energy expression.
00:07:40.804 --> 00:07:45.010
Kinetic energy
expression-- everybody
00:07:45.010 --> 00:07:45.990
comfortable with that?
00:07:57.370 --> 00:07:59.040
I'm not totally
comfortable with it.
00:08:02.460 --> 00:08:05.800
Let's talk about a minute
about kinetic energy and say,
00:08:05.800 --> 00:08:11.390
what can we use, what formulas
can we use here for t?
00:08:11.390 --> 00:08:15.230
There's totally general and
then we can narrow it down.
00:08:15.230 --> 00:08:16.390
What do you suggest?
00:08:16.390 --> 00:08:22.018
STUDENT: For the
[INAUDIBLE] you can--
00:08:22.018 --> 00:08:24.196
you have to take into account
that it's not spinning
00:08:24.196 --> 00:08:25.406
about its center of mass.
00:08:25.406 --> 00:08:28.310
So you use the general
mass formula one half omega
00:08:28.310 --> 00:08:30.730
dot, its angular
momentum [INAUDIBLE].
00:08:30.730 --> 00:08:33.590
PROFESSOR: OK, so we have
some general formulas.
00:08:33.590 --> 00:08:37.510
And let's take a step by
step approach to this.
00:08:37.510 --> 00:08:41.250
We have how many rigid bodies?
00:08:41.250 --> 00:08:42.340
Two.
00:08:42.340 --> 00:08:45.110
And you compute
the kinetic energy
00:08:45.110 --> 00:08:48.545
for each one individually.
00:08:48.545 --> 00:08:50.280
That's the safest
way to go about this.
00:08:50.280 --> 00:08:55.290
So I would stay away
initially from doing this.
00:08:55.290 --> 00:08:58.319
You lumping the two together.
00:08:58.319 --> 00:08:59.860
So I would take the
two individually.
00:08:59.860 --> 00:09:05.110
So if you take the main
mass, the block on wheels,
00:09:05.110 --> 00:09:06.065
what is its velocity?
00:09:11.170 --> 00:09:13.030
Its velocity is
this x dot, right?
00:09:13.030 --> 00:09:14.897
And its kinetic energy is?
00:09:14.897 --> 00:09:15.914
STUDENT: [INAUDIBLE].
00:09:15.914 --> 00:09:17.330
PROFESSOR: So for
the first block,
00:09:17.330 --> 00:09:20.480
you have m1x dot squared.
00:09:20.480 --> 00:09:21.312
And you're done.
00:09:21.312 --> 00:09:22.270
That's the first block.
00:09:22.270 --> 00:09:24.050
Now you need the
second rigid body.
00:09:24.050 --> 00:09:30.407
So the second rigid
body, you could
00:09:30.407 --> 00:09:34.090
do-- the full,
general expression
00:09:34.090 --> 00:09:50.960
is one half m2 vg in o dot
vg in o plus a half omega H,
00:09:50.960 --> 00:09:55.200
with respect to g.
00:09:55.200 --> 00:09:56.200
Good with that, Vicente?
00:09:59.057 --> 00:10:00.015
You need it transposed?
00:10:02.580 --> 00:10:05.460
Good.
00:10:05.460 --> 00:10:10.750
And can we simplify that at all?
00:10:16.290 --> 00:10:19.870
For example, is it that rod
rotating about a fixed point?
00:10:23.272 --> 00:10:24.150
It's not.
00:10:24.150 --> 00:10:26.940
The point it rotates
about moves, right?
00:10:26.940 --> 00:10:30.180
So you can't say it's just--
you can't use, for example,
00:10:30.180 --> 00:10:32.370
parallel axis theorem
and just say it's 1/2
00:10:32.370 --> 00:10:35.540
I with respect to that
point, theta dot squared.
00:10:35.540 --> 00:10:37.780
Won't work.
00:10:37.780 --> 00:10:39.710
Can't use that one.
00:10:39.710 --> 00:10:41.690
You will find, that
if you work this out,
00:10:41.690 --> 00:10:47.226
you can say 1/2 I with
respect to g omega squared,
00:10:47.226 --> 00:10:48.600
in this problem
it will work out.
00:10:48.600 --> 00:10:52.340
It'll come out to that because
this is a planar motion problem
00:10:52.340 --> 00:10:56.340
and there's only one
component of rotation.
00:10:56.340 --> 00:11:06.840
So this will work out to be
a 1/2 I with respect zz of m2
00:11:06.840 --> 00:11:10.360
with respect to g
omega z squared,
00:11:10.360 --> 00:11:12.339
in this case theta dot squared.
00:11:12.339 --> 00:11:13.880
That's what this
term will reduce to.
00:11:13.880 --> 00:11:16.320
But don't assume it
just out of the box.
00:11:16.320 --> 00:11:17.730
And then you need this term.
00:11:17.730 --> 00:11:18.900
And that's why we need bg.
00:11:21.530 --> 00:11:24.630
So you need to do that.
00:11:24.630 --> 00:11:26.690
Put that piece in over there.
00:11:26.690 --> 00:11:40.370
So we need to-- so what
is Izz about g for m2?
00:11:45.320 --> 00:11:47.020
For a rod, slender rod.
00:11:51.410 --> 00:11:56.090
So m2 l squared over 12.
00:11:56.090 --> 00:11:58.340
That's what you
need to put in here.
00:11:58.340 --> 00:12:02.930
We have an expression
for v over there.
00:12:02.930 --> 00:12:04.470
We know everything now.
00:12:07.380 --> 00:12:10.485
So now let's apply our
Lagrange equations.
00:12:16.160 --> 00:12:19.179
And I'm going to need to
rearrange the board here
00:12:19.179 --> 00:12:19.720
a little bit.
00:12:22.680 --> 00:12:24.580
I'm going to need
that board space.
00:12:24.580 --> 00:12:33.760
So our T is 1/2 m1 x1 dot
squared, or x dot squared
00:12:33.760 --> 00:12:42.110
plus 1/2 m2 vg.vg.
00:12:57.160 --> 00:13:00.820
And now I can cover this up.
00:13:00.820 --> 00:13:02.020
All right.
00:13:02.020 --> 00:13:05.910
So the next task
is, let's work out,
00:13:05.910 --> 00:13:08.384
do our Lagrange equations work?
00:13:08.384 --> 00:13:10.300
So how many generalized
coordinates do we have
00:13:10.300 --> 00:13:13.130
and what are they?
00:13:13.130 --> 00:13:14.990
x and theta, that we've chosen.
00:13:14.990 --> 00:13:19.470
Two degrees of freedom--
they're complete, independent,
00:13:19.470 --> 00:13:21.000
polynomic.
00:13:21.000 --> 00:13:22.920
And we can use
Lagrange equations.
00:13:22.920 --> 00:13:26.820
We're going to come up with
two equations of motion.
00:13:26.820 --> 00:13:28.160
And we're going to apply this.
00:13:28.160 --> 00:13:29.420
That's the Lagrange equation.
00:13:29.420 --> 00:13:30.919
We're going to apply
it twice, where
00:13:30.919 --> 00:13:34.750
l is defined as T minus v.
00:13:34.750 --> 00:13:38.960
If you just plug in l into this
expression and just expand it,
00:13:38.960 --> 00:13:41.680
you get-- instead of two
terms, you get four terms,
00:13:41.680 --> 00:13:43.250
because you have these two guys.
00:13:43.250 --> 00:13:46.450
And this term-- for
mechanical systems,
00:13:46.450 --> 00:13:48.270
what can you say about
this term generally?
00:13:48.270 --> 00:13:52.110
That's the derivative
of v with respect
00:13:52.110 --> 00:13:55.941
to Q dots to velocities.
00:13:55.941 --> 00:13:56.440
Why?
00:13:59.940 --> 00:14:01.440
STUDENT: Conservative forces?
00:14:01.440 --> 00:14:04.460
PROFESSOR: No, just that you
find it for mechanical systems,
00:14:04.460 --> 00:14:09.030
springs and gravity, you will
never find that the potential
00:14:09.030 --> 00:14:12.877
energy as a function of
time or velocity just
00:14:12.877 --> 00:14:14.835
isn't-- and if it's not
a function of velocity,
00:14:14.835 --> 00:14:17.085
you take a derivative with
respect to velocity you get
00:14:17.085 --> 00:14:17.930
0's.
00:14:17.930 --> 00:14:20.115
So this goes to 0 for
mechanical systems.
00:14:23.160 --> 00:14:25.370
An exception for
non-mechanical would
00:14:25.370 --> 00:14:28.350
be like a charged particle
in a magnetic field.
00:14:28.350 --> 00:14:31.800
Then the forces get involved
with velocities and so forth.
00:14:31.800 --> 00:14:32.850
It gets messy.
00:14:32.850 --> 00:14:34.160
0 for mechanical systems.
00:14:34.160 --> 00:14:38.620
So we really don't have
to deal with three terms--
00:14:38.620 --> 00:14:41.650
that one, that one, that
one, and then on the right
00:14:41.650 --> 00:14:43.600
hand side are
generalized forces.
00:14:43.600 --> 00:14:48.430
So we can break into
four smaller groups.
00:14:48.430 --> 00:14:52.270
Two groups are going
to do the x equation.
00:14:52.270 --> 00:14:59.160
You have to take these Qj's
is this problem are Q1 is x
00:14:59.160 --> 00:15:02.500
and Q2 is theta.
00:15:02.500 --> 00:15:08.660
So for the x equations, we
need to do these derivatives.
00:15:08.660 --> 00:15:20.450
And for the theta equation, we
need to do these computations.
00:15:20.450 --> 00:15:27.380
So let's have one
group A do these.
00:15:27.380 --> 00:15:33.255
Group B do these and group C do
these and a D group do those.
00:15:33.255 --> 00:15:38.110
So break yourselves
into four groups
00:15:38.110 --> 00:15:45.360
and we'll do A, B, C
here in the center,
00:15:45.360 --> 00:15:47.400
group here, four or
five, four or five,
00:15:47.400 --> 00:15:50.822
or group here the C group,
and D group over here.
00:15:50.822 --> 00:15:53.030
Do these calculations and
let's get our two equations
00:15:53.030 --> 00:15:53.880
in motion.
00:15:53.880 --> 00:15:56.199
And when you get
your stuff done,
00:15:56.199 --> 00:15:58.740
so the A group, when you finish,
come up here and right here,
00:15:58.740 --> 00:16:00.110
write your stuff.
00:16:00.110 --> 00:16:02.834
And the B group, write your
answer here, and the C group
00:16:02.834 --> 00:16:03.500
and the D group.
00:16:03.500 --> 00:16:05.666
As soon as you get it done,
come up and put it down.
00:16:08.020 --> 00:16:16.120
We got this term, this term,
this term, and this term.
00:16:16.120 --> 00:16:17.560
Did you guys check?
00:16:17.560 --> 00:16:20.711
Did you guys get a
little time, B group
00:16:20.711 --> 00:16:21.710
to check on the A group?
00:16:26.700 --> 00:16:27.880
Which one was it?
00:16:27.880 --> 00:16:29.446
Who's checking on whom?
00:16:29.446 --> 00:16:31.070
You're checking on--
what do you think?
00:16:31.070 --> 00:16:32.798
STUDENT: We got the same answer.
00:16:32.798 --> 00:16:40.860
PROFESSOR: So main mass
acceleration, the second mass,
00:16:40.860 --> 00:16:45.390
its total acceleration,
these pieces, and there's
00:16:45.390 --> 00:16:47.450
an acceleration that's
Eulerian and then
00:16:47.450 --> 00:16:49.110
there's an
acceleration that is--
00:16:49.110 --> 00:16:52.790
what's this term related to?
00:16:52.790 --> 00:16:56.690
You expect it to come up?
00:16:56.690 --> 00:16:58.690
And the kx term,
and all these are
00:16:58.690 --> 00:17:00.510
going to equal to the
generalized forces
00:17:00.510 --> 00:17:02.950
of any non-conservative forces.
00:17:02.950 --> 00:17:04.569
So you're OK with this one.
00:17:04.569 --> 00:17:07.170
Let's move on to this one, then.
00:17:07.170 --> 00:17:10.490
Who's the check group here?
00:17:10.490 --> 00:17:11.240
What do you think?
00:17:11.240 --> 00:17:14.759
STUDENT: I think they
made [INAUDIBLE].
00:17:14.759 --> 00:17:17.353
PROFESSOR: Do you think
there's a problem here?
00:17:17.353 --> 00:17:19.889
STUDENT: There might be.
00:17:19.889 --> 00:17:21.951
PROFESSOR: Can you
give me an alternative?
00:17:21.951 --> 00:17:26.720
STUDENT: It may be that
their dt d theta there should
00:17:26.720 --> 00:17:28.640
be in the time derivative.
00:17:33.220 --> 00:17:34.430
PROFESSOR: All right.
00:17:34.430 --> 00:17:38.775
So we need d, the derivative of
this with respect to theta dot.
00:17:41.960 --> 00:17:45.420
The first term doesn't give me
anything, the mx dot squared.
00:17:45.420 --> 00:17:49.730
The third term gives
you-- should give you
00:17:49.730 --> 00:17:56.200
an Izz g theta double
dot, eventually, right?
00:17:56.200 --> 00:18:05.370
So for sure, this d by
dt, the partial of T
00:18:05.370 --> 00:18:08.120
with respect to theta dot.
00:18:08.120 --> 00:18:09.570
So we just run through it.
00:18:09.570 --> 00:18:11.510
The first term gives us nothing.
00:18:11.510 --> 00:18:18.090
The third piece gives us,
with respect to theta dot Izz,
00:18:18.090 --> 00:18:19.030
theta dot.
00:18:19.030 --> 00:18:20.774
And no one wipes out the 1/2.
00:18:20.774 --> 00:18:22.690
The time derivative makes
it theta double dot.
00:18:22.690 --> 00:18:28.600
So the third term's going to
be an Izz g theta double dot.
00:18:28.600 --> 00:18:32.490
And it's the second term that
needs a derivative of this
00:18:32.490 --> 00:18:34.090
with respect to theta dot.
00:18:37.690 --> 00:18:45.470
So both terms are going to
yield some stuff, right?
00:18:45.470 --> 00:18:47.650
A lot of stuff.
00:18:47.650 --> 00:18:49.710
All right, I'm
going to write down
00:18:49.710 --> 00:18:54.070
how this should work it out,
rather than try to grind it out
00:18:54.070 --> 00:18:54.860
real time here.
00:19:06.750 --> 00:19:39.501
Izz
00:19:39.501 --> 00:19:40.000
All right.
00:19:40.000 --> 00:19:42.270
These are the terms
that should appear.
00:19:44.880 --> 00:19:47.260
This is the piece about g.
00:19:47.260 --> 00:19:52.470
This is the-- no,
that's not quite right.
00:19:55.470 --> 00:19:56.670
That's the piece about g.
00:19:56.670 --> 00:20:04.605
This should be about--
and then this term.
00:20:08.000 --> 00:20:10.080
That's how it actually
should check out.
00:20:10.080 --> 00:20:11.720
Do you agree with me?
00:20:11.720 --> 00:20:12.740
Looks OK?
00:20:12.740 --> 00:20:22.660
And then if we add to that
the m2 g l over 2 sine theta,
00:20:22.660 --> 00:20:25.350
which is our gravitational
potential energy, all of that
00:20:25.350 --> 00:20:30.510
added together ought
to be equal to q theta.
00:20:30.510 --> 00:20:34.400
So let's move on to looking
at the generalized forces
00:20:34.400 --> 00:20:36.040
for this problem.
00:20:36.040 --> 00:20:43.020
So don't know where you
guys went wrong on this.
00:20:43.020 --> 00:20:46.535
But if you have
any questions-- we
00:20:46.535 --> 00:20:47.910
can talk about
this for a minute.
00:20:52.370 --> 00:20:54.180
STUDENT: [INAUDIBLE].
00:20:54.180 --> 00:20:57.830
PROFESSOR: d t d theta?
00:20:57.830 --> 00:21:05.240
So you got to take the partial
derivative of this expression
00:21:05.240 --> 00:21:07.446
with respect to theta dot.
00:21:07.446 --> 00:21:10.180
This piece gives
you a contribution
00:21:10.180 --> 00:21:13.136
that will be Izz theta dot.
00:21:13.136 --> 00:21:14.510
And you take its
time derivative.
00:21:14.510 --> 00:21:19.840
So that's pretty obvious why
it gives you the first piece.
00:21:19.840 --> 00:21:21.714
STUDENT: I think the
problem is understanding
00:21:21.714 --> 00:21:25.286
in the first place that
they did [INAUDIBLE].
00:21:25.286 --> 00:21:26.760
PROFESSOR: Oh.
00:21:26.760 --> 00:21:27.672
STUDENT: [INAUDIBLE].
00:21:31.320 --> 00:21:33.180
PROFESSOR: Then the
second one, when
00:21:33.180 --> 00:21:36.740
you take the derivative of
this expression with respect
00:21:36.740 --> 00:21:41.180
to-- this is T with
respect to theta
00:21:41.180 --> 00:21:44.190
dot of this part,
this expression,
00:21:44.190 --> 00:21:49.370
you get 2 times
what's inside times
00:21:49.370 --> 00:21:56.160
the derivative of the inside
with respect to theta dot.
00:21:56.160 --> 00:22:03.270
And that'll give you another
L over 2 cosine theta.
00:22:03.270 --> 00:22:05.150
And I think you're done.
00:22:05.150 --> 00:22:08.690
This times this stuff, right?
00:22:08.690 --> 00:22:10.680
2 times this times
the derivative
00:22:10.680 --> 00:22:12.976
of the inside, which
is-- the derivative
00:22:12.976 --> 00:22:15.070
of the inside with
respect to theta dot
00:22:15.070 --> 00:22:17.790
should be L over 2 cosine theta.
00:22:17.790 --> 00:22:22.360
So you get an x dot
plus L over 2 theta
00:22:22.360 --> 00:22:29.490
dot cosine theta
2 times that times
00:22:29.490 --> 00:22:32.925
the derivative of the inside
with respect to theta dot.
00:22:32.925 --> 00:22:35.840
This is the only term
that contributes is that.
00:22:35.840 --> 00:22:39.210
And then we've already done the
derivative of this with respect
00:22:39.210 --> 00:22:41.787
to theta dot.
00:22:41.787 --> 00:22:42.370
Wait a minute.
00:22:42.370 --> 00:22:42.869
We haven't.
00:22:42.869 --> 00:22:45.330
This one, now we got
another term here.
00:22:45.330 --> 00:22:52.030
So this one gives you 2
times the expression times
00:22:52.030 --> 00:23:04.954
the-- this would give you
L theta dot sine theta.
00:23:04.954 --> 00:23:07.120
But now you have to take
the derivative with respect
00:23:07.120 --> 00:23:12.300
to theta dot, which
gives you what?
00:23:12.300 --> 00:23:14.940
STUDENT: [INAUDIBLE].
00:23:14.940 --> 00:23:16.920
PROFESSOR: Another
L over 2 sine theta?
00:23:22.100 --> 00:23:24.540
Something like that.
00:23:24.540 --> 00:23:30.015
So you end up with the theta
dot L, L squared over 2,
00:23:30.015 --> 00:23:32.440
theta dot L squared
sine squared.
00:23:32.440 --> 00:23:35.220
And you probably get
a theta dot L squared
00:23:35.220 --> 00:23:36.520
cosine squared over here.
00:23:36.520 --> 00:23:38.250
And those two add
together to give you
00:23:38.250 --> 00:23:47.060
a theta dot L squared over--
theta dot L squared, I guess.
00:23:47.060 --> 00:23:50.170
Those collapse together.
00:23:50.170 --> 00:23:55.310
Those come together to give
you the other piece of this.
00:23:55.310 --> 00:23:59.094
STUDENT: [INAUDIBLE]
one of the coefficients.
00:23:59.094 --> 00:24:02.775
But where is the
derivative [INAUDIBLE].
00:24:02.775 --> 00:24:04.900
PROFESSOR: No, the derivative
with respect to theta
00:24:04.900 --> 00:24:08.254
only comes in in the
potential energy term.
00:24:08.254 --> 00:24:12.238
STUDENT: So what's number two?
00:24:12.238 --> 00:24:14.360
PROFESSOR: OK.
00:24:14.360 --> 00:24:15.267
All right, yep.
00:24:15.267 --> 00:24:15.850
You need that.
00:24:15.850 --> 00:24:19.130
And so T with respect
to theta, and you
00:24:19.130 --> 00:24:20.690
do have theta pieces in there.
00:24:20.690 --> 00:24:22.140
And it does kick
out more pieces.
00:24:22.140 --> 00:24:23.701
STUDENT: [INAUDIBLE].
00:24:23.701 --> 00:24:24.700
PROFESSOR: No, I didn't.
00:24:24.700 --> 00:24:27.270
Haven't even done
that piece yet.
00:24:27.270 --> 00:24:31.740
So you do that piece,
a couple things cancel.
00:24:31.740 --> 00:24:37.030
And you end up with-- so I
don't have time to work it out,
00:24:37.030 --> 00:24:38.440
to write it all up on the board.
00:24:38.440 --> 00:24:43.940
But the complete solution
for this is posted.
00:24:43.940 --> 00:24:47.600
So Professor Gossard, who
teaches the other three
00:24:47.600 --> 00:24:52.010
recitation sections, writes
these up and posts the answers.
00:24:52.010 --> 00:24:53.160
And so they're on Stellar.
00:24:53.160 --> 00:24:57.940
So you get the gory details
of each of these pieces.
00:24:57.940 --> 00:25:04.444
Let's go on to talk
about generalized forces,
00:25:04.444 --> 00:25:05.610
while we have a few minutes.
00:25:08.410 --> 00:25:10.200
The way it was set
up, were there--
00:25:10.200 --> 00:25:11.450
what are the right hand sides?
00:25:11.450 --> 00:25:14.000
Are there any
generalized external
00:25:14.000 --> 00:25:18.500
non-conservative forces, the
way the problem was first posed?
00:25:18.500 --> 00:25:19.140
None.
00:25:19.140 --> 00:25:20.190
So let's put in a couple.
00:25:20.190 --> 00:25:26.535
Let's add dashpot, b here,
and an external force here.
00:25:26.535 --> 00:25:29.240
Call it F1 of T.
00:25:29.240 --> 00:25:33.560
So now, what's Qx and Q theta?
00:25:39.830 --> 00:25:43.260
That's an exercise I think you
can all go through, but just
00:25:43.260 --> 00:25:46.300
check with your groups.
00:25:46.300 --> 00:25:50.060
Figure out the
generalized forces.
00:25:50.060 --> 00:25:54.420
And do it by imagining-- for
this one, for the x equation,
00:25:54.420 --> 00:25:58.100
say OK, you have a small,
virtual displacement delta x.
00:25:58.100 --> 00:26:00.255
What's the virtual
work that's done?
00:26:00.255 --> 00:26:05.330
Your delta w, then, will be
thing you're looking for,
00:26:05.330 --> 00:26:06.150
delta x.
00:26:06.150 --> 00:26:07.250
And the same thing.
00:26:07.250 --> 00:26:09.005
This is the x1.
00:26:09.005 --> 00:26:12.480
And you get a similar x thing
when you do they one for theta.
00:26:12.480 --> 00:26:14.370
It would be Q
theta, delta theta.
00:26:14.370 --> 00:26:16.790
Figure out the work done
and that'll tell you
00:26:16.790 --> 00:26:20.650
what Qx and Q theta are.
00:26:20.650 --> 00:26:24.960
So the total work of the
non-conservative forces
00:26:24.960 --> 00:26:26.460
through these virtual
displacements,
00:26:26.460 --> 00:26:28.880
you can just add them up.
00:26:28.880 --> 00:26:31.090
So there's a contribution
that comes from a delta x.
00:26:31.090 --> 00:26:32.460
And there will be
another contribution
00:26:32.460 --> 00:26:33.350
from the delta theta.
00:26:33.350 --> 00:26:35.110
And we can figure
out each piece.
00:26:35.110 --> 00:26:38.580
And you assign each piece to
the equation it goes with.
00:26:38.580 --> 00:26:44.830
So if you do a small virtual
deflection in the x direction,
00:26:44.830 --> 00:26:47.662
how much work is done?
00:26:47.662 --> 00:26:48.870
Somebody give me a term here.
00:26:52.710 --> 00:26:58.390
So work, remember,
is F dot d dot dr.
00:26:58.390 --> 00:27:00.550
And this dr is a
function of our delta
00:27:00.550 --> 00:27:04.792
x's delta theta's and so forth.
00:27:04.792 --> 00:27:07.780
STUDENT: F of T dx?
00:27:07.780 --> 00:27:09.140
PROFESSOR: Delta x?
00:27:09.140 --> 00:27:10.830
That'll be some work done.
00:27:10.830 --> 00:27:15.970
So that force, external force
moves through the full delta x.
00:27:15.970 --> 00:27:17.372
And what else?
00:27:17.372 --> 00:27:20.144
STUDENT: Minus dx [INAUDIBLE].
00:27:20.144 --> 00:27:22.865
PROFESSOR: So that
suggests then we have here,
00:27:22.865 --> 00:27:29.305
this is F1 of T in positive
I direction minus dx
00:27:29.305 --> 00:27:32.590
dot in the opposite
direction times delta
00:27:32.590 --> 00:27:36.780
x is a virtual work done
by-- as you do that.
00:27:36.780 --> 00:27:38.030
Now how about the delta theta?
00:27:43.460 --> 00:27:46.030
Somebody else-- how
much virtual work
00:27:46.030 --> 00:27:49.540
is done by these
forces F and minus bx?
00:27:49.540 --> 00:27:52.740
They're the only
non-conservative
00:27:52.740 --> 00:27:56.570
external forces in the problem
are the dashpot and the F,
00:27:56.570 --> 00:27:58.300
whatever it is.
00:27:58.300 --> 00:28:00.750
How much work is
done by those forces
00:28:00.750 --> 00:28:05.980
in a virtual
displacement delta theta?
00:28:05.980 --> 00:28:08.030
Hand up back there?
00:28:08.030 --> 00:28:10.740
I hear a bid for 0.
00:28:10.740 --> 00:28:13.120
What do other people think?
00:28:13.120 --> 00:28:17.310
Do those forces move at all
because you make motion delta
00:28:17.310 --> 00:28:18.800
theta?
00:28:18.800 --> 00:28:22.560
Is there any dr
here that results
00:28:22.560 --> 00:28:24.830
because of delta
theta in the direction
00:28:24.830 --> 00:28:26.570
of any of these applied forces?
00:28:26.570 --> 00:28:30.280
At the point of application
of these forces, do they move?
00:28:30.280 --> 00:28:33.040
So this force is right here.
00:28:33.040 --> 00:28:34.910
Does it move because
you do a delta theta?
00:28:34.910 --> 00:28:35.630
Nope.
00:28:35.630 --> 00:28:38.390
And this force,
applied right here,
00:28:38.390 --> 00:28:40.940
does that point move
because of delta theta?
00:28:40.940 --> 00:28:46.790
No, so in this problem then this
piece here is 0 delta theta.
00:28:46.790 --> 00:28:48.430
And total virtual
work done is that.
00:28:48.430 --> 00:28:51.410
And you assign each piece
to its appropriate equation.
00:28:51.410 --> 00:28:56.720
So Qx, this is Qx
right here, delta x.
00:28:56.720 --> 00:29:00.450
Qx belongs up here.
00:29:00.450 --> 00:29:03.080
The sum of these
things equals Qx.
00:29:03.080 --> 00:29:08.690
And in the second case, for the
beta equation, it's equal to 0.
00:29:08.690 --> 00:29:12.060
So let's make the problems a
little bit harder for a second.
00:29:12.060 --> 00:29:19.550
Let's put a force, apply
a horizontal force here.
00:29:19.550 --> 00:29:20.366
We'll call this F2.
00:29:24.130 --> 00:29:33.340
So now, what is the work
done in this system?
00:29:33.340 --> 00:29:36.320
We now have an additional force.
00:29:36.320 --> 00:29:39.170
Is there any work done
because of delta x?
00:29:42.700 --> 00:29:48.489
This is the real-- you
understand this piece,
00:29:48.489 --> 00:29:50.280
then you really begin
to understand how you
00:29:50.280 --> 00:29:53.000
do these generalized forces.
00:29:53.000 --> 00:30:00.440
Qx delta x-- is the generalized
force associated with delta x,
00:30:00.440 --> 00:30:02.880
is it affected by
this new force?
00:30:02.880 --> 00:30:04.940
Is any work done?
00:30:04.940 --> 00:30:09.190
So I now cause this
little delta x to happen.
00:30:09.190 --> 00:30:15.840
Is that force doing work,
assuming no other deflections
00:30:15.840 --> 00:30:18.501
aren't happening right now?
00:30:18.501 --> 00:30:19.000
Why?
00:30:23.230 --> 00:30:26.110
So is there-- it's
not allowed to move.
00:30:26.110 --> 00:30:28.080
Whatever instantaneous
position it's in,
00:30:28.080 --> 00:30:31.500
it's frozen there
in its coordinate.
00:30:31.500 --> 00:30:35.110
But if there's an x
component, it moves in x.
00:30:35.110 --> 00:30:38.485
So it's angled like
this off the cart.
00:30:38.485 --> 00:30:40.630
But now delta x does this to it?
00:30:40.630 --> 00:30:42.590
Is that force F doing work?
00:30:42.590 --> 00:30:43.100
How much?
00:30:47.560 --> 00:30:52.600
So then, when you add that
one to it, we end up with a--
00:30:52.600 --> 00:30:54.905
and is it in plus
direction, and it's exactly
00:30:54.905 --> 00:30:56.820
in the same
direction as delta x.
00:30:56.820 --> 00:30:59.470
There's no components.
00:30:59.470 --> 00:31:04.050
The dot product of F2 is in
the I direction dot delta
00:31:04.050 --> 00:31:06.060
x, which is also in the I.
00:31:06.060 --> 00:31:09.940
So you get an additional
contribution of F2 delta x.
00:31:09.940 --> 00:31:14.540
And so now this generalized
force has an F2 in it.
00:31:14.540 --> 00:31:16.520
How about the other
direction, though?
00:31:16.520 --> 00:31:22.110
How about this now
Q theta delta theta?
00:31:22.110 --> 00:31:23.280
What does it give you?
00:31:28.370 --> 00:31:36.090
Now, if you now freeze x,
and allow a slight angular
00:31:36.090 --> 00:31:41.570
variation delta theta,
does this guy do any work?
00:31:41.570 --> 00:31:44.250
How much, and is
it F2 delta theta?
00:31:47.310 --> 00:31:49.010
That's wrong in two ways.
00:31:49.010 --> 00:31:51.610
Dimensions are wrong.
00:31:51.610 --> 00:31:52.610
Something else is wrong.
00:31:52.610 --> 00:31:54.320
So let's draw it.
00:31:54.320 --> 00:31:56.080
Here is this thing.
00:31:56.080 --> 00:31:58.150
Here's F2.
00:31:58.150 --> 00:32:03.352
Delta theta causes the motion
of this point in what direction?
00:32:03.352 --> 00:32:04.560
Perpendicular to this, right?
00:32:04.560 --> 00:32:05.934
So you can think
of it like this.
00:32:05.934 --> 00:32:10.190
And this would be L delta
theta is the actual distance
00:32:10.190 --> 00:32:10.865
that it moves.
00:32:13.440 --> 00:32:16.050
And so can either
say the component
00:32:16.050 --> 00:32:19.240
of that motion in the
direction of the force,
00:32:19.240 --> 00:32:21.160
or you can say the
component of the force
00:32:21.160 --> 00:32:24.980
in the direction of that
motion, dot product between this
00:32:24.980 --> 00:32:25.540
and that.
00:32:25.540 --> 00:32:27.900
And this is theta.
00:32:27.900 --> 00:32:30.390
So what is the component
of L delta theta
00:32:30.390 --> 00:32:33.510
in the direction of the force?
00:32:33.510 --> 00:32:34.570
That's this right here.
00:32:44.390 --> 00:32:51.360
And that's now in
the I hat dot F2 I.
00:32:51.360 --> 00:32:56.510
So our delta w in
the theta direction
00:32:56.510 --> 00:33:09.900
is Q theta delta theta equals my
F2 L cosine theta delta theta.
00:33:09.900 --> 00:33:14.070
And to solve for Q theta,
now these can disappear.
00:33:14.070 --> 00:33:16.860
And Q theta is F2
L cosine theta.
00:33:16.860 --> 00:33:18.230
Are the units right?
00:33:18.230 --> 00:33:22.370
What are the units of this
generalized force, meaning you
00:33:22.370 --> 00:33:24.705
got to think about
that equation.
00:33:24.705 --> 00:33:26.440
The theta equation--
we talked about this
00:33:26.440 --> 00:33:27.840
before-- has units of what?
00:33:41.610 --> 00:33:46.379
What are the dimensions of
term that look like that?
00:33:46.379 --> 00:33:48.920
This comes from, when you do it
the direct way, the summation
00:33:48.920 --> 00:33:50.990
of external torques.
00:33:50.990 --> 00:33:52.830
This had better have
units of torque, right,
00:33:52.830 --> 00:33:55.570
which is units of
force times distance.
00:33:55.570 --> 00:33:58.610
So force times
distance had better
00:33:58.610 --> 00:34:01.270
be the units of this equation.
00:34:01.270 --> 00:34:04.470
And therefore that had
better be a torque.
00:34:04.470 --> 00:34:08.360
Is force times length a
torque, of units a torque?
00:34:08.360 --> 00:34:08.880
Sure.
00:34:08.880 --> 00:34:11.949
So that's the correct unit.
00:34:11.949 --> 00:34:14.920
And over here, for
the x equation,
00:34:14.920 --> 00:34:17.480
do we come out with
the correct units?
00:34:17.480 --> 00:34:19.760
Yeah, come out force.
00:34:19.760 --> 00:34:21.989
This naturally works,
because the delta theta
00:34:21.989 --> 00:34:23.000
is dimensionless.
00:34:23.000 --> 00:34:27.190
So the length dr, the length
piece that comes in here
00:34:27.190 --> 00:34:29.179
stays with this to
give you a torque.
00:34:29.179 --> 00:34:31.560
Over here, delta x
has length in it.
00:34:31.560 --> 00:34:33.199
And you're just
left with the force.
00:34:33.199 --> 00:34:36.739
So you get a force--
this generalized force
00:34:36.739 --> 00:34:39.090
for the x direction is a force.
00:34:39.090 --> 00:34:41.889
The generalized force
in the theta direction
00:34:41.889 --> 00:34:44.630
is a moment, a torque.
00:34:44.630 --> 00:34:45.610
All right?
00:34:45.610 --> 00:34:47.909
Good.
00:34:47.909 --> 00:34:50.610
So we're officially done.
00:34:50.610 --> 00:34:53.389
But if you have any
last questions on this,
00:34:53.389 --> 00:34:56.439
I'll stick around and
we can chat about it.