WEBVTT
00:01.510 --> 00:04.810
If you've got a function
f(x), you know the value
00:04.806 --> 00:07.016
at some point,
and you want to go to a
00:07.023 --> 00:10.023
neighboring point,
a distance Δx away.
00:10.020 --> 00:12.690
You ask, "How much does the
function change?"
00:12.690 --> 00:16.880
And the answer is,
the change in the function
00:16.884 --> 00:22.514
Δf is the derivative of
the function at the starting
00:22.507 --> 00:26.127
point times the distance you
move.
00:26.130 --> 00:29.410
This is not an equality,
unless f happens to be a
00:29.406 --> 00:32.096
straight line;
it's an approximation and there
00:32.103 --> 00:33.513
are corrections to this.
00:33.510 --> 00:36.790
That's what all the dots mean;
corrections are proportional to
00:36.790 --> 00:38.750
Δx^(2) and
Δx^(3) and so on.
00:38.750 --> 00:41.560
But if Δx is tiny this
will do;
00:41.560 --> 00:45.120
that's it.
But today, we are going to move
00:45.124 --> 00:49.344
the whole Work Energy Theorem
and the Law of Conservation of
00:49.339 --> 00:51.339
Energy to two dimensions.
00:51.340 --> 00:54.730
So, when you go to two
dimensions, you've got to ask
00:54.728 --> 00:57.118
yourself, "What am I looking
for?"
00:57.120 --> 01:00.050
Well, in the end,
I'm hoping I will get some
01:00.046 --> 01:04.326
relation like K_1 +
U_1 = K_2 +
01:04.332 --> 01:07.852
U_2,
assuming there is no friction.
01:07.849 --> 01:10.499
U_2 is going
to be my new potential energy.
01:10.500 --> 01:14.440
Then, if it's the potential
energy and the particle is
01:14.440 --> 01:18.380
moving in two dimensions,
it's got to be a function of
01:18.381 --> 01:21.431
two variables,
x and y.
01:21.430 --> 01:24.290
So, I have to make sure that
you guys know enough about
01:24.294 --> 01:26.314
functions of more than one
variable.
01:26.310 --> 01:28.500
So, this is,
again, a crash course on
01:28.495 --> 01:30.615
reminding you of the main
points.
01:30.620 --> 01:31.990
There are not that many.
01:31.989 --> 01:34.559
By the time I get here I'll be
done.
01:34.560 --> 01:38.420
So, how do you visualize the
function of two variables?
01:38.420 --> 01:40.840
The function of one variable,
you know, you plot x
01:40.835 --> 01:42.685
this way and the function along
y.
01:42.690 --> 01:46.080
If it's two variables,
you plot x here and
01:46.075 --> 01:50.375
y here and the function
itself is shown by drawing some
01:50.377 --> 01:53.337
surface on top of the xy
plane,
01:53.340 --> 01:56.870
so that if you take a point and
you go right up till you hit the
01:56.872 --> 01:59.342
surface, that's the value of the
function,
01:59.340 --> 02:01.080
at that point xy.
02:01.079 --> 02:05.629
It's like a canopy on top of
the xy plane and how high
02:05.627 --> 02:08.277
you've got to go is the
function.
02:08.280 --> 02:10.140
For example,
starting with the floor,
02:10.144 --> 02:13.204
I can ask how high you have to
go till I hit the ceiling.
02:13.200 --> 02:16.910
That function varies;
there are dents and dimples and
02:16.911 --> 02:19.971
so on, so the function varies
with xy.
02:19.970 --> 02:21.800
Another example of a function
of x and
02:21.802 --> 02:24.222
y--x and y
could be coordinates in the
02:24.218 --> 02:26.798
United States and the function
could be the temperature at that
02:26.800 --> 02:29.040
point.
So, you plot it on top of that,
02:29.042 --> 02:31.912
each point you plot the
temperature at that point.
02:31.909 --> 02:35.389
So, once you've got the notion
of a function of two variables,
02:35.388 --> 02:38.918
if you're going to do calculus
the next thing is what about the
02:38.924 --> 02:40.754
derivatives of the function.
02:40.750 --> 02:42.060
How does it change?
02:42.060 --> 02:45.440
Well, in the old days,
it was dependent on x
02:45.440 --> 02:49.770
and I changed the x and I
found the change in the function
02:49.767 --> 02:53.487
divided by the change in
x and took the limit and
02:53.486 --> 02:55.646
that became my derivative.
02:55.650 --> 02:58.410
And now, I'm sitting in the
xy plane,
02:58.414 --> 03:01.184
so here is x and here is
y;
03:01.180 --> 03:04.160
I'm here, the function is
coming out of the blackboard.
03:04.159 --> 03:07.159
So, imagine something measured
out but I'm sitting here.
03:07.159 --> 03:10.849
Now, I want to move and ask how
the function changes.
03:10.850 --> 03:13.970
But now I have a lot of options;
in fact, an infinite number of
03:13.966 --> 03:15.446
options.
I can move along x,
03:15.445 --> 03:17.775
I can move along y,
I can move at some intermediate
03:17.781 --> 03:20.591
angle,
we have to ask what do you want
03:20.591 --> 03:24.451
me to do when it comes time to
take derivatives.
03:24.449 --> 03:26.799
So, it turns out,
and you will see it proven
03:26.802 --> 03:29.432
amply as we go along,
that you just have to think
03:29.428 --> 03:32.268
about derivatives on two
principle directions which I
03:32.273 --> 03:34.793
will choose to be x and
y.
03:34.789 --> 03:38.779
So, we're going to define one
derivative, which is defined as
03:38.784 --> 03:40.844
follows.
You start at the point
03:40.839 --> 03:44.249
xy, you go to the point
x + Δx,
03:44.250 --> 03:48.490
the same y and subtract
the function at the starting
03:48.486 --> 03:52.426
point, divide by Δx and
take all the limits,
03:52.430 --> 03:54.150
Δx goes to 0.
03:54.150 --> 03:57.700
That means you go from here to
here, you move a distance
03:57.695 --> 04:00.845
Δx, you'd find the
change in the function,
04:00.854 --> 04:02.664
you take the derivative.
04:02.659 --> 04:06.009
As you move horizontally,
you notice you don't do
04:06.013 --> 04:09.713
anything to y;
y will be left fixed at
04:09.709 --> 04:12.659
whatever y you had;
x will be changed by
04:12.656 --> 04:14.846
Δx;
you find the change;
04:14.849 --> 04:16.339
you take the derivative and
take the limit that is denoted
04:16.343 --> 04:17.393
by the symbol df/dx.
[Reader note:
04:17.391 --> 04:18.911
Partial derivatives will be
written as ordinary derivatives
04:18.911 --> 04:20.431
to avoid using fonts that may
not be universally available.
04:20.430 --> 04:21.710
It should be clear from the
context that a partial
04:21.714 --> 04:23.654
derivative is intended here,
since f depends on
04:23.647 --> 04:25.667
x and y.]
So, this curly d instead
04:25.672 --> 04:27.772
of the straight d tells
you it's called a "partial
04:27.772 --> 04:31.362
derivative."
Some people may want to make it
04:31.362 --> 04:37.082
very explicit by saying this is
the derivative with subscript
04:37.077 --> 04:39.547
y.
That means y is being
04:39.550 --> 04:42.310
held constant when x is
varied, but we don't have to
04:42.308 --> 04:44.968
write that because we know we've
got two coordinates.
04:44.970 --> 04:48.010
If I'm changing one,
the other guy is y so we
04:48.008 --> 04:52.178
won't write that;
that's the partial derivative.
04:52.180 --> 04:56.010
So, you can also move from here
to here, up and down,
04:56.011 --> 04:58.591
and see how the function
changes.
04:58.589 --> 05:02.389
I won't write the details,
you can define obviously a
05:02.385 --> 05:05.805
df/dy.
So, one tells you how the
05:05.810 --> 05:09.910
function changes,
with x I should move
05:09.912 --> 05:12.752
along x,
and the other tells you how it
05:12.751 --> 05:14.611
changes at y as you move
along y.
05:14.610 --> 05:16.580
Let's get some practice.
05:16.579 --> 05:20.609
So, I'm going to write some
function, f = x^(3)y^(2),
05:20.613 --> 05:23.693
that's a function of x
and y.
05:23.689 --> 05:26.899
You can write any--let's make
it a little more interesting,
05:26.902 --> 05:28.732
plus y,
or y^(2);
05:28.730 --> 05:32.810
that's some function of
x and y.
05:32.810 --> 05:37.700
So, when I say df/dx,
the rule is find out how it
05:37.701 --> 05:42.061
varies with x keeping
y constant.
05:42.060 --> 05:44.420
That means, really treat it
like a constant,
05:44.421 --> 05:47.771
because number five--What will
you do if y was equal to
05:47.770 --> 05:49.760
5?
This'll be 25 and it's not part
05:49.761 --> 05:52.281
of taking derivatives because
it's not changing,
05:52.281 --> 05:55.391
and here it'll be just standing
in front doing nothing;
05:55.389 --> 05:58.069
it will just take the x
derivative, treating y as
05:58.068 --> 06:00.018
a constant.
You're supposed to do that here;
06:00.019 --> 06:03.839
this is a derivative
3x^(2)y;
06:03.839 --> 06:05.769
so that's the x
derivative.
06:05.769 --> 06:10.389
Now, you can take the y
part, y^(2),
06:10.387 --> 06:14.897
yes, thank you.
Then, I can take df/dy,
06:14.904 --> 06:20.644
so then I look for y
changes, this is 2y here,
06:20.637 --> 06:24.327
so it's 2x^(3)y +
2y;
06:24.329 --> 06:27.459
so that's the x
derivative and the partial
06:27.463 --> 06:28.903
y derivative.
06:28.899 --> 06:33.929
Okay, so then you can now take
higher derivatives.
06:33.930 --> 06:35.960
We know that from calculus and
one variable,
06:35.961 --> 06:38.231
you can take the derivative of
the derivative.
06:38.230 --> 06:43.090
So, one thing you can think
about is d^(2)f/dx^(2),
06:43.093 --> 06:47.703
that really means take the
d by dx of the
06:47.701 --> 06:49.921
d by dx.
06:49.920 --> 06:51.140
That's what it means.
06:51.139 --> 06:53.639
First, take a derivative and
take the derivative of the
06:53.638 --> 06:56.048
derivative.
So, let's see what I get here.
06:56.050 --> 06:57.990
I already took df/dx.
06:57.990 --> 07:01.560
I want to take its derivative,
right, the derivative of
07:01.563 --> 07:05.073
x^(2) is 2x,
so I get 6xy^(2).
07:05.070 --> 07:09.400
07:09.399 --> 07:13.509
Then, I can take the y
derivative of the y
07:13.506 --> 07:16.116
derivative,
d^(2)f/dy^(2),
07:16.120 --> 07:17.850
that's d by dy of
df/dy,
07:17.850 --> 07:19.580
if you take the y
derivative there.
07:19.579 --> 07:22.919
You guys should keep an eye out
for me when I do this.
07:22.920 --> 07:28.530
I get this.
But now, we have an interesting
07:28.532 --> 07:31.342
possibility you didn't have in
one dimension,
07:31.339 --> 07:37.069
which is to take the x
derivative of the y
07:37.073 --> 07:42.603
derivative, so I want to take
d by dx of
07:42.597 --> 07:46.957
df/dy.
That is written as
07:46.956 --> 07:49.826
d^(2)f/dxdy.
07:49.830 --> 07:51.740
Let's see what I get.
07:51.740 --> 07:55.160
So, I want the x
derivative of the y
07:55.162 --> 07:58.232
derivative.
So, I go to this guy and take
07:58.227 --> 08:02.317
his x derivative,
I get 3x^(2) from there,
08:02.319 --> 08:05.369
so I get 6x^(2)y and
that's it.
08:05.370 --> 08:09.070
08:09.069 --> 08:13.759
So, make sure that I got the
proper x derivative of
08:13.757 --> 08:16.057
this, I think that's fine.
08:16.060 --> 08:22.630
Then, I can also take
d^(2)f/dydx.
08:22.629 --> 08:25.979
That means, take the y
derivative of the x
08:25.982 --> 08:28.972
derivative, but here's the
x derivative,
08:28.970 --> 08:32.220
take the y derivative of
that, put a 2y there,
08:32.215 --> 08:33.695
and I get 6x^(2)y.
08:33.700 --> 08:38.740
08:38.740 --> 08:41.170
So, you're supposed to notice
something.
08:41.169 --> 08:43.729
If you already know this,
you're not surprised.
08:43.730 --> 08:48.340
If you've never seen this
before, you will notice that the
08:48.335 --> 08:51.445
cross derivative,
y followed by x
08:51.448 --> 08:54.278
and x followed by
y, will come up being
08:54.279 --> 08:57.779
equal.
That's a general property of
08:57.776 --> 09:00.456
any reasonable function.
09:00.460 --> 09:03.120
By reasonable,
I mean you cannot call the
09:03.115 --> 09:07.165
mathematicians to help because
they will always find something
09:07.165 --> 09:09.285
where this won't work,
okay?
09:09.289 --> 09:11.999
But if you write down any
function that you are capable of
09:11.998 --> 09:14.748
writing down with powers of
x and powers of y
09:14.754 --> 09:17.504
and sines and cosines,
it'll always be true that you
09:17.496 --> 09:20.816
could take the cross derivatives
in either order and get the same
09:20.820 --> 09:23.280
answer.
I'd like to give a little bit
09:23.283 --> 09:25.633
of a feeling for why that is
true;
09:25.629 --> 09:29.179
it's true but it's helpful to
know why it's true.
09:29.179 --> 09:31.789
So, let's ask the following
question.
09:31.789 --> 09:38.849
Let's take a function and let's
ask how much the function
09:38.853 --> 09:44.533
changes when I go from some
point x,
09:44.529 --> 09:48.569
y to another point
x + Δx,
09:48.574 --> 09:50.644
y + Δy.
09:50.639 --> 09:54.159
I want to find the change in
the function.
09:54.159 --> 09:59.529
So, I'm asking you what is
f(x + Δx,
09:59.530 --> 10:07.100
y + Δy) - f(x,y) for small
values of Δx and
10:07.104 --> 10:10.584
Δy.
For neighboring points,
10:10.582 --> 10:11.752
what's the change?
10:11.750 --> 10:14.670
So, we're going to do it in two
stages.
10:14.669 --> 10:19.529
We introduce an intermediate
point here, whose coordinate is
10:19.527 --> 10:24.447
x + Δ x and y,
and I'm going to add and
10:24.446 --> 10:28.486
subtract the value of the
function here.
10:28.490 --> 10:31.790
Adding and subtracting is free;
it doesn't cost anything,
10:31.786 --> 10:32.616
so let's do that.
10:32.620 --> 10:33.660
Then, what do I get?
10:33.659 --> 10:47.059
I get f(x + Δx,
y + Δy ) - f(x + Δx,
10:47.064 --> 10:58.184
y) + f(x + Δx,
y) - f(x,y).
10:58.180 --> 11:02.870
11:02.870 --> 11:06.040
I'm just saying,
the change of that guy minus
11:06.043 --> 11:10.013
this guy is the same as that
minus this, plus this minus
11:10.010 --> 11:12.960
that;
that's a trivial substitution.
11:12.960 --> 11:17.910
But I write it this way because
I look at the first entity here,
11:17.910 --> 11:21.290
it looks like I'm just changing
y.
11:21.289 --> 11:25.699
I'm not changing x,
you agree?
11:25.700 --> 11:27.460
So what will this be?
11:27.460 --> 11:32.250
This is going to be the rate of
change of the function with
11:32.249 --> 11:35.469
respect to x times
Δx,
11:35.470 --> 11:37.780
and this--I'm sorry,
I got it wrong,
11:37.778 --> 11:40.678
with respect to y times
Δy.
11:40.680 --> 11:47.740
11:47.740 --> 11:54.760
This one is df/dx times
Δx.
11:54.759 --> 12:02.689
Therefore, the change in the
function, if I add it all up,
12:02.691 --> 12:07.841
I get df/dx Δx + df/dy
Δy.
12:07.840 --> 12:11.800
But if you are pedantic,
you will notice there is
12:11.799 --> 12:15.759
something I have to be a little
careful about.
12:15.759 --> 12:20.439
What do you think I'm referring
to, in my notation,
12:20.438 --> 12:22.378
yes?
Student: [inaudible]
12:22.376 --> 12:24.666
Professor Ramamurti
Shankar: Meaning?
12:24.669 --> 12:27.459
Student: [inaudible]
Professor Ramamurti
12:27.456 --> 12:30.066
Shankar: Oh,
but this is nothing to do with
12:30.072 --> 12:31.382
dimension, you see.
12:31.379 --> 12:33.119
It's one number minus another
number.
12:33.120 --> 12:37.520
I put another number in between
and in this function;
12:37.520 --> 12:43.380
it's a function only of here.
12:43.379 --> 12:46.409
Let me see, x is not
changing at all and y is
12:46.412 --> 12:48.402
changing, so it is df/dy
Δy.
12:48.400 --> 12:50.240
I meant something else.
12:50.240 --> 12:52.820
That is, in fact,
a good approximation but there
12:52.815 --> 12:55.825
is one thing you should be
careful about which has to do
12:55.829 --> 12:58.459
with where the derivatives are
really taken.
12:58.460 --> 13:01.700
For the term here,
f(x + Δx,
13:01.700 --> 13:05.570
y) - f(xy),
I took the derivative,
13:05.567 --> 13:08.177
at my starting point.
13:08.179 --> 13:10.309
If you want,
I will say at the starting
13:10.310 --> 13:12.150
point xy.
The second one,
13:12.147 --> 13:15.067
when I came here and I want to
move up, I'm taking the
13:15.070 --> 13:18.160
derivative with respect to
y at the new point.
13:18.159 --> 13:23.189
The new point is (x + Δx,
y), so derivatives are not
13:23.191 --> 13:25.921
quite taken at the same point.
13:25.920 --> 13:29.040
So, you've got to fix that.
13:29.040 --> 13:31.070
And how do you fix that part?
13:31.070 --> 13:34.920
You argue that the derivative
with respect to y is just
13:34.915 --> 13:38.005
another function of y;
f is a function of
13:38.007 --> 13:40.327
x and y,
its derivatives are functions
13:40.326 --> 13:41.596
of x and y.
13:41.600 --> 13:44.030
Everything is a function of
x and y,
13:44.025 --> 13:46.645
and we are saying this
derivative has been computed at
13:46.649 --> 13:50.669
x + Δx,
instead of x,
13:50.667 --> 13:58.357
so it is going to be
df/dy at xy +
13:58.360 --> 14:03.900
d^(2)f/dxdy times
Δx.
14:03.899 --> 14:06.029
In other words,
I am saying the derivative at
14:06.025 --> 14:08.485
this location is the derivative
at that location,
14:08.490 --> 14:11.520
plus the rate of change of the
derivative times the change in
14:11.515 --> 14:13.235
x.
In other words,
14:13.236 --> 14:15.956
the derivative itself is
changing.
14:15.960 --> 14:22.810
So, if you put that together
you find it is df/dx Δx +
14:22.806 --> 14:27.966
df/dy Δy,
where if I don't put any bars
14:27.971 --> 14:33.861
or anything it means at the
starting point xy +
14:33.856 --> 14:37.696
(d^(2)f/dxdy) Δx Δy.
14:37.700 --> 14:43.770
14:43.769 --> 14:47.909
Now, you've got to realize that
when you're doing these calculus
14:47.912 --> 14:51.332
problems, Δx is a tiny number,
Δy is a tiny number,
14:51.331 --> 14:53.371
Δx Δy is tiny times tiny.
14:53.370 --> 14:55.370
So normally,
we don't care about it,
14:55.368 --> 14:57.938
or if you want to be more
accurate, of course,
14:57.937 --> 14:59.647
you should keep that term.
14:59.649 --> 15:02.419
In the first approximation,
where you work to the first
15:02.421 --> 15:05.091
power of everything,
this will be your Δf.
15:05.090 --> 15:08.390
But if you're a little more
ambitious but you keep track of
15:08.393 --> 15:11.873
the fact the derivative itself
is changing, you will keep that
15:11.867 --> 15:15.007
term.
But another person comes along
15:15.012 --> 15:19.402
and says, "You know what,
I want to go like this.
15:19.399 --> 15:23.349
I want to introduce as my new
point, intermediate point,
15:23.352 --> 15:25.782
the one here.
It had a different value of
15:25.777 --> 15:28.617
y in the same x
and then I went horizontally
15:28.623 --> 15:30.473
when I keep track of the
changes."
15:30.470 --> 15:34.300
What do you think that person
will calculate for the change in
15:34.298 --> 15:37.638
the function?
That person will get exactly
15:37.641 --> 15:42.701
this part, but the extra term
that person will get will look
15:42.702 --> 15:46.822
like d^(2)f/dydx times
Δy Δx.
15:46.820 --> 15:49.990
15:49.990 --> 15:52.020
If you just do the whole thing
in your head,
15:52.015 --> 15:54.465
you can see that I'm just
exchanging the x and
15:54.465 --> 15:56.515
y roles.
So, everything that happened
15:56.524 --> 15:58.814
with x and y here
will come backwards with
15:58.810 --> 15:59.850
y and x.
15:59.850 --> 16:02.610
But then, the change between
these two points is the change
16:02.608 --> 16:05.228
between these two points and it
doesn't matter whether I
16:05.225 --> 16:08.075
introduce an intermediate point
here or an intermediate point
16:08.079 --> 16:10.089
there;
therefore, these changes have
16:10.086 --> 16:13.246
to be equal.
This part is of course equal;
16:13.250 --> 16:17.080
therefore, you want that part
to be equal, Δx Δy is
16:17.077 --> 16:21.267
clearly Δy Δx,
so the consequence of that is
16:21.273 --> 16:25.623
d^(2)f/dxdy =
d^(2)f/dydx and that's
16:25.623 --> 16:31.133
the reason it turns out when you
take cross derivatives you get
16:31.126 --> 16:34.196
the same answer.
It comes from the fact -- if
16:34.199 --> 16:36.829
you say, where is that result
coming from -- it comes from the
16:36.834 --> 16:38.974
fact,
if you start at some point and
16:38.973 --> 16:42.323
you go to another point and you
ask for the change in the
16:42.322 --> 16:44.592
function,
the change is accumulating as
16:44.586 --> 16:46.216
you move.
You can move horizontally and
16:46.216 --> 16:47.956
then vertically,
or you can move vertically and
16:47.955 --> 16:48.745
then horizontally.
16:48.750 --> 16:50.980
The change in the function is a
change in the function.
16:50.980 --> 16:53.010
You've got to get the same
answer both ways.
16:53.010 --> 16:55.790
That's the reason;
that's the requirement that
16:55.794 --> 16:57.384
leads to this requirement.
16:57.379 --> 17:01.919
Now, I will not be keeping
track of functions to this
17:01.919 --> 17:05.149
accuracy in everything we do
today.
17:05.150 --> 17:08.070
We'll be keeping the leading
powers in Δx and
17:08.069 --> 17:10.519
Δy.
So, you should bear in mind
17:10.515 --> 17:13.815
that if you make a movement in
the xy plane,
17:13.819 --> 17:17.689
which is Δx
horizontally and Δy
17:17.686 --> 17:22.726
vertically, then the change in
the function is this [df/dx
17:22.728 --> 17:24.828
Δx + df/dy Δy].
17:24.829 --> 17:27.429
Draw a box around that,
because that's going to
17:27.434 --> 17:30.724
be--This is just the naive
generalization to two dimensions
17:30.718 --> 17:32.868
of what you know in one
dimension.
17:32.869 --> 17:34.779
We were saying look,
the function is changing
17:34.783 --> 17:37.133
because the independent
variables x and y
17:37.130 --> 17:39.490
are changing,
and the change in the function
17:39.489 --> 17:42.349
is one part, which I blame on
the changing x,
17:42.349 --> 17:44.489
and a second part,
which I blame on the changing
17:44.487 --> 17:45.667
y and I add them.
17:45.670 --> 17:48.740
So, you're worried about the
fact that we're moving in the
17:48.736 --> 17:50.346
plane and there are vectors.
17:50.349 --> 17:53.319
That's all correct,
but f is not a vector,
17:53.321 --> 17:56.971
f is just a number and
this change has got two parts,
17:56.972 --> 18:00.422
okay.
So, this is basically all the
18:00.418 --> 18:05.398
math we will need to do,
what I want to do today.
18:05.400 --> 18:09.410
So, let's now go back to our
original goal,
18:09.407 --> 18:15.127
which was to derive something
like the Law of Conservation of
18:15.131 --> 18:19.331
Energy in two dimensions instead
of one.
18:19.329 --> 18:22.769
You remember what I did last
time so I will remind you one
18:22.769 --> 18:24.639
more time what the trick was.
18:24.640 --> 18:29.270
We found out that the change in
the kinetic energy of some
18:29.271 --> 18:33.011
object is equal to the work done
by a force,
18:33.009 --> 18:38.469
which was some F times
Δx and then if you add
18:38.468 --> 18:43.648
all the changes over a not
infinitesimal displacement,
18:43.650 --> 18:46.240
but a macroscopic displacement,
that was given by integral of
18:46.237 --> 18:48.737
F times dx and the
integral of F times
18:48.739 --> 18:50.339
dx from
x_1 to
18:50.335 --> 18:51.495
x_2.
18:51.500 --> 18:54.360
If F is a function only
of x, from the rules of
18:54.358 --> 18:57.068
calculus, the integral can be
written as a difference of a
18:57.073 --> 18:59.173
function at this limit minus
that limit,
18:59.170 --> 19:02.140
and that was
U(x_1) -
19:02.140 --> 19:06.820
U(x_2),
where U is that function
19:06.819 --> 19:11.409
whose derivative with a minus
sign is F.
19:11.410 --> 19:14.650
That's what we did.
19:14.650 --> 19:17.900
Then, it's very simple now to
take the U_1 to
19:17.900 --> 19:19.020
the left-hand side.
19:19.019 --> 19:20.629
Let me see,
K_1 to the
19:20.633 --> 19:22.713
right-hand side and
U(x_2) to the
19:22.713 --> 19:24.583
left-hand side,
to get K_2 +
19:24.577 --> 19:27.237
U_2 = K_1 +
U_1 and that's the
19:27.240 --> 19:28.340
conservation of energy.
19:28.339 --> 19:32.399
You want to try the same thing
in two dimensions;
19:32.400 --> 19:34.010
that's your goal.
19:34.009 --> 19:39.539
So, the first question is,
"What should I use for the work
19:39.543 --> 19:42.823
done?"
What expression should I use
19:42.817 --> 19:47.547
for the work done in two
dimensions, because the force
19:47.545 --> 19:50.425
now is a vector;
force is not one number,
19:50.433 --> 19:52.773
it's got an x part and a
y part.
19:52.769 --> 19:56.429
My displacement has also got an
x part and a y
19:56.426 --> 19:59.336
part and I can worry about what
I should use.
19:59.339 --> 20:02.629
So, I'm going to deduce the
quantity I want to use for
20:02.631 --> 20:05.671
ΔW, namely,
the tiny work done which is an
20:05.673 --> 20:07.043
extension of this.
20:07.039 --> 20:12.219
I'm going to demand that since
I'm looking for a Work Energy
20:12.224 --> 20:17.414
Theorem, I'm going to demand
that, remember in one dimension
20:17.408 --> 20:19.428
ΔK = F Δx.
20:19.430 --> 20:23.200
If we divide it by the time
over which it happens,
20:23.201 --> 20:26.281
I find dK/dt is equal to
force [F]
20:26.279 --> 20:28.049
times velocity [v].
20:28.049 --> 20:33.219
I'm going to demand that that's
the power, force times velocity,
20:33.222 --> 20:35.442
is what I call the power.
20:35.440 --> 20:40.770
And I'm going to look for
dK/dt in two dimensions,
20:40.766 --> 20:43.236
of a body that's moving.
20:43.240 --> 20:45.400
What's the rate at which
kinetic energy is changing when
20:45.402 --> 20:46.152
a body is moving?
20:46.150 --> 20:48.830
For that, I need a formula for
kinetic energy.
20:48.829 --> 20:53.049
A formula for kinetic energy is
going to be again ½
20:53.054 --> 20:56.144
mv^(2).
I want that to be the same
20:56.140 --> 21:00.130
entity, so I will choose it to
be that, but v^(2) has
21:00.134 --> 21:04.404
got now v_x^(2)
+ v_y^(2),
21:04.400 --> 21:07.620
because you know whenever you
take a vector V,
21:07.617 --> 21:10.957
then its length is the square
root of the x part
21:10.959 --> 21:13.619
squared plus the y part
squared.
21:13.620 --> 21:18.590
21:18.590 --> 21:25.020
Any questions?
Okay, so let's take the rate of
21:25.020 --> 21:27.760
change of this,
dK/dt.
21:27.759 --> 21:30.079
Again, you have to know your
calculus.
21:30.079 --> 21:34.619
What's the time derivative of
v_x^(2)?
21:34.619 --> 21:36.929
The rule from calculus says,
first take the derivative of
21:36.934 --> 21:39.584
v_x^(2) with
respect to v_x,
21:39.579 --> 21:42.829
which is 2v_x,
then take the derivative of
21:42.832 --> 21:45.302
v_x with
respect to time.
21:45.299 --> 21:49.399
So, you do the same thing for
the second term,
21:49.396 --> 21:53.306
2 v_y
dv_y/dt,
21:53.309 --> 21:58.909
and that gives me--the 2s
cancel, gives me
21:58.905 --> 22:04.085
mdv_x
/dtv_x + m
22:04.091 --> 22:09.551
dv_y
/dtv_y.
22:09.550 --> 22:13.330
22:13.329 --> 22:16.609
This is the rate at which the
kinetic energy of a body is
22:16.614 --> 22:20.054
changing.
So, what's my next step, yes?
22:20.049 --> 22:24.579
Student: So what
happened to the half?
22:24.579 --> 22:26.599
Professor Ramamurti
Shankar: The half got
22:26.599 --> 22:27.759
canceled by the two here.
22:27.760 --> 22:32.280
Yes?
Student: [inaudible]
22:32.279 --> 22:38.249
Professor Ramamurti
Shankar: That's correct.
22:38.250 --> 22:41.530
So, what we want to do is to
recognize this as ma in
22:41.534 --> 22:44.204
the x direction,
because m times
22:44.202 --> 22:46.942
dv_x/dt is
a_x,
22:46.935 --> 22:49.835
and this is m times
a_y.
22:49.839 --> 22:53.749
Therefore, they are the forces
in the x and y
22:53.745 --> 22:57.305
directions, so I write
F_xv_x +
22:57.313 --> 22:59.673
F_yv_y.
22:59.670 --> 23:02.970
So the power,
when you go to two dimensions,
23:02.970 --> 23:06.270
is not very different from one
dimension.
23:06.269 --> 23:08.509
In one dimension,
you had only one force and you
23:08.514 --> 23:09.664
had only one velocity.
23:09.660 --> 23:12.150
In two dimensions,
you got an x and
23:12.154 --> 23:15.504
y component for each one,
and it becomes this.
23:15.500 --> 23:22.090
So, this is what I will define
to be the power.
23:22.089 --> 23:24.649
When a body is moving and a
force is acting on it,
23:24.652 --> 23:27.222
the force has two components,
the velocity has two
23:27.215 --> 23:30.405
components;
this combination shall be
23:30.414 --> 23:33.784
called "Power."
But now, let's multiply both
23:33.777 --> 23:37.667
sides by Δt and write
the change in kinetic energy is
23:37.671 --> 23:41.441
equal to F_x
and you guys think about what
23:41.436 --> 23:44.416
happens when I multiply
v_x by
23:44.421 --> 23:46.921
Δt;
v_x =
23:46.916 --> 23:49.766
dx/dt.
Multiplying by Δt just
23:49.770 --> 23:53.790
gives me the distance traveled
in the x direction +
23:53.790 --> 23:56.270
F_y times
dy.
23:56.269 --> 24:02.079
So, this is the tiny amount of
work done by a force and it
24:02.084 --> 24:07.494
generalizes what we had here,
work done is force times
24:07.490 --> 24:10.130
Δx.
In two dimensions,
24:10.127 --> 24:13.017
it's F_xdx +
F_ydy.
24:13.019 --> 24:15.579
It's not hard to guess that but
the beauty is--this
24:15.581 --> 24:17.771
combination--Now you can say,
you know what,
24:17.769 --> 24:20.169
I didn't have to do all this,
I could have always guessed
24:20.166 --> 24:22.666
that in two dimensions,
when you had an x and a
24:22.674 --> 24:24.584
y you obviously have to
add them.
24:24.579 --> 24:27.249
There's nothing very obvious
about it because this
24:27.246 --> 24:30.726
combination is now guaranteed to
have the property that if I call
24:30.728 --> 24:31.978
this the work done.
24:31.980 --> 24:35.530
Then, it has the advantage that
the work done is in fact the
24:35.526 --> 24:37.146
change in kinetic energy.
24:37.150 --> 24:40.720
I want to define work so that
its effect on kinetic energy is
24:40.724 --> 24:43.414
the same as in 1D,
namely, work done should be
24:43.405 --> 24:45.665
equal to change in kinetic
energy.
24:45.670 --> 24:49.310
And I engineered that by taking
the change in kinetic energy,
24:49.311 --> 24:53.071
seeing whatever it came out to,
and calling that the work done,
24:53.074 --> 24:54.414
I cannot go wrong.
24:54.410 --> 24:58.200
But now, if you notice
something repeating itself all
24:58.202 --> 25:01.852
the time, which is that I had a
vector F,
25:01.849 --> 25:05.429
which you can write as I
times F_x + J
25:05.428 --> 25:08.558
times F_y,
I had a vector velocity,
25:08.560 --> 25:11.720
which is I times
v_x + J times
25:11.719 --> 25:13.149
v_y.
25:13.150 --> 25:17.080
Then, I had a tiny distance
moved by the particle,
25:17.084 --> 25:21.584
which is I times dx +
J times dy.
25:21.579 --> 25:24.989
So, the particle moves from one
point to another point.
25:24.990 --> 25:28.710
The vector describing its
location changes by this tiny
25:28.710 --> 25:30.460
vector.
It's just the step in the
25:30.457 --> 25:32.707
x direction times
I, plus step in the
25:32.711 --> 25:34.381
y direction times
J.
25:34.380 --> 25:38.370
So, what you're finding is the
following combination.
25:38.369 --> 25:40.789
The x component of
F times the x
25:40.792 --> 25:43.222
component of v,
plus the y component of
25:43.223 --> 25:45.503
F, times the y
component of v.
25:45.500 --> 25:48.060
Or F component of
x times the distance
25:48.060 --> 25:51.170
moved in x plus y
component of F times the
25:51.173 --> 25:52.633
displacement in y.
25:52.630 --> 25:57.470
So, we are running into the
following combination.
25:57.470 --> 26:00.590
We are saying,
there seem to be in all these
26:00.592 --> 26:04.372
problems two vectors,
I times A_x
26:04.368 --> 26:07.198
+ J times
A_y.
26:07.200 --> 26:09.390
A, for example,
could be the force that I'm
26:09.386 --> 26:11.266
talking about.
There's another vector
26:11.266 --> 26:14.286
B, which is I
times B_x + J
26:14.289 --> 26:15.879
times B_y.
26:15.880 --> 26:20.990
26:20.990 --> 26:23.270
Right?
For example,
26:23.268 --> 26:25.988
this guy could be standing in
for F,
26:25.994 --> 26:28.724
this could be standing in for
v.
26:28.720 --> 26:33.950
The combination that seems to
appear very naturally is the
26:33.947 --> 26:37.797
combination
A_xB_x +
26:37.799 --> 26:41.009
A_yB_y.
26:41.009 --> 26:46.019
It appears too many times so I
take it seriously,
26:46.017 --> 26:47.997
give that a name.
26:48.000 --> 26:51.780
That name will be called a dot
product of A with
26:51.779 --> 26:54.299
B and is written like
this.
26:54.300 --> 26:58.210
26:58.210 --> 27:01.250
Whenever something appears all
the time you give it a name;
27:01.250 --> 27:04.670
this is A.B. So,
for any two vectors A
27:04.665 --> 27:07.545
and B,
that will be the definition of
27:07.545 --> 27:11.065
A.B.
Then, the work done by a force
27:11.067 --> 27:15.347
F, that displaces a
particle by a tiny vector
27:15.346 --> 27:17.776
dr, is F.dr.
27:17.779 --> 27:21.469
The particle's moving in the
xy plane.
27:21.470 --> 27:26.920
From one instant to the next,
it can move from here to there.
27:26.920 --> 27:30.760
That little guy is dr;
it's got a little bit
27:30.757 --> 27:33.447
horizontal and it's got a little
bit vertical,
27:33.453 --> 27:35.493
that's how you build dr.
27:35.490 --> 27:38.710
The force itself is some force
which at that point need not
27:38.707 --> 27:41.367
point at the direction in which
you're moving.
27:41.370 --> 27:42.540
It's some direction.
27:42.539 --> 27:44.869
At each point,
the force could have whatever
27:44.869 --> 27:46.829
value it likes.
So, the dot product,
27:46.829 --> 27:49.269
you know, sometimes you learn
the dot product as
27:49.270 --> 27:52.700
A_xB_x +
A_yB_y,
27:52.700 --> 27:54.260
you can ask,
"Who thought about it?"
27:54.260 --> 27:56.460
Why is it a natural quantity?
27:56.460 --> 27:59.330
And here is one way you can
understand why somebody would
27:59.330 --> 28:01.330
think of this particular
combination.
28:01.329 --> 28:04.579
So, once you've got the dot
product, you've got to get a
28:04.581 --> 28:06.061
feeling for what it is.
28:06.059 --> 28:09.579
The first thing we realize is
that if you take a dot product
28:09.579 --> 28:11.519
of A with itself,
then it's
28:11.516 --> 28:14.176
A_xA_x +
A_yA_y,
28:14.175 --> 28:16.615
which is A_x^(2) +
A_y^(2),
28:16.619 --> 28:19.169
which is the length of the
vector A,
28:19.165 --> 28:22.315
which you can either denote
this way or just write it
28:22.317 --> 28:23.587
without an arrow.
28:23.589 --> 28:26.839
So A.A is a positive
number that measures the length
28:26.842 --> 28:29.872
squared of the vector A,
likewise B.B.
28:29.869 --> 28:32.799
Now, we have to ask ourselves,
"What is A.B?"
28:32.800 --> 28:40.130
28:40.130 --> 28:43.270
So, somebody know what
A.B is?
28:43.270 --> 28:46.280
Yep?
Student: [inaudible]
28:46.278 --> 28:48.288
Professor Ramamurti
Shankar: Okay,
28:48.291 --> 28:49.601
so how do we know that?
28:49.599 --> 28:52.639
How do we know it's length of
A times length of
28:52.644 --> 28:54.774
B times cosine of the
angle?
28:54.769 --> 28:57.749
You should follow from--Is it
an independent definition or is
28:57.749 --> 28:59.089
it a consequence of this?
28:59.090 --> 29:00.630
Yep?
Student: Well,
29:00.632 --> 29:02.142
you can derive using the Law of
Cosines.
29:02.140 --> 29:02.770
Professor Ramamurti
Shankar: Yes.
29:02.769 --> 29:06.619
He said we can derive it using
the Law of Cosines and that's
29:06.619 --> 29:07.989
what I will do now.
29:07.990 --> 29:10.280
In other words,
that definition which you may
29:10.279 --> 29:13.609
have learnt about first is not
independent of this definition;
29:13.609 --> 29:15.979
it's a consequence of this
definition.
29:15.980 --> 29:17.690
So, let's see why that's true.
29:17.690 --> 29:19.870
Let's draw two vectors here.
29:19.869 --> 29:24.699
Here is A and here is
B, it's got a length
29:24.702 --> 29:28.132
A, it's got a length
B,
29:28.130 --> 29:31.050
this makes an angle,
θ_A with the
29:31.048 --> 29:33.208
x axis,
and that makes an angle
29:33.208 --> 29:36.008
θ_B for the
x axis.
29:36.009 --> 29:41.409
Now, do you guys agree that the
x component of A
29:41.405 --> 29:46.535
with the horizontal part of
A is the A cos
29:46.535 --> 29:48.565
θ_A?
29:48.569 --> 29:51.139
You must've seen a lot of
examples of that when you did
29:51.144 --> 29:52.484
all the force calculation.
29:52.480 --> 29:55.870
And A_y is
equal to the length of A
29:55.871 --> 29:59.381
times sin θ_A
and likewise for B;
29:59.380 --> 30:01.010
I don't feel like writing it.
30:01.009 --> 30:06.449
If you do that,
then A.B will be length
30:06.454 --> 30:14.084
of A, length of B
times (cos θ_A cos
30:14.076 --> 30:18.666
θ_B + sin
θ_A sin
30:18.674 --> 30:21.824
θ_B).
30:21.819 --> 30:25.759
You've got to go back to your
good old trig and it'll tell you
30:25.758 --> 30:29.698
this cos cos plus sin
sin is cos (θ_A –
30:29.696 --> 30:32.876
θ_B).So,
you find it is length of
30:32.884 --> 30:36.204
A, length of B,
cos (θ_A –
30:36.195 --> 30:37.575
θ_B).
30:37.579 --> 30:40.899
Often, people simply say that
is AB cos θ,
30:40.904 --> 30:44.704
where it's understood that
θ is the angle between
30:44.703 --> 30:48.363
the two vectors.
So, the dot product that you
30:48.355 --> 30:53.515
learnt--I don't know which way
it was introduced to you first,
30:53.519 --> 30:57.399
but these are two equivalent
definitions of the dot product.
30:57.400 --> 30:59.600
In one of them,
if you're thinking more in
30:59.600 --> 31:01.640
terms of the components of
A,
31:01.640 --> 31:04.260
a pair of numbers for A
and a pair of numbers for
31:04.256 --> 31:07.106
B, this definition of the
dot product is very nice.
31:07.109 --> 31:10.299
If you're thinking of them as
two little arrows pointing in
31:10.303 --> 31:12.953
different directions,
then the other definition,
31:12.952 --> 31:15.932
in which the lengths and the
angle between them appear,
31:15.930 --> 31:17.010
that's more natural.
31:17.010 --> 31:18.670
But numerically they're equal.
31:18.670 --> 31:23.790
31:23.789 --> 31:27.239
An important property of the
dot product, which you can check
31:27.238 --> 31:30.858
either way, is the dot product
of A with B + C,
31:30.859 --> 31:34.179
is dot product of A with
B plus dot product of
31:34.182 --> 31:35.592
A with C.
31:35.589 --> 31:38.339
That just means you can open
all the brackets with dot
31:38.337 --> 31:40.667
products as you can with
ordinary products.
31:40.670 --> 31:44.510
31:44.509 --> 31:49.369
Now, that's a very important
property of the dot product.
31:49.369 --> 31:53.569
So, maybe I can ask somebody,
do you know one property,
31:53.566 --> 31:56.826
significant property of the dot
product?
31:56.829 --> 32:00.379
Student: When two
vectors are perpendicular the
32:00.379 --> 32:01.919
dot product equals 0.
32:01.920 --> 32:02.820
Professor Ramamurti
Shankar: Oh that's
32:02.815 --> 32:03.605
interesting, I didn't think
about it.
32:03.610 --> 32:06.580
Yes.
One thing is if two vectors are
32:06.578 --> 32:11.938
perpendicular the dot product is
0 because the cosine of 90 is 0.
32:11.940 --> 32:14.570
But what I had in mind--Of
course it's hard for me,
32:14.574 --> 32:17.744
it's not fair I ask you some
question without saying what I'm
32:17.736 --> 32:21.066
looking for.
What can you say if you use a
32:21.067 --> 32:23.037
different set of axis?
32:23.040 --> 32:26.160
Yep?
Student: In this case it
32:26.161 --> 32:27.861
doesn't matter.
Professor Ramamurti
32:27.862 --> 32:29.382
Shankar: If you go to the
rotated axis,
32:29.375 --> 32:31.135
the components of vector
A will change.
32:31.140 --> 32:32.860
We have done that in the
homework;
32:32.860 --> 32:34.040
we've done that in the class.
32:34.039 --> 32:35.739
The components of B will
also change.
32:35.740 --> 32:37.010
Everything will get a prime.
32:37.009 --> 32:39.779
But the combination,
A_xB_x +
32:39.778 --> 32:43.598
A_yB_y,
when you evaluate it before or
32:43.604 --> 32:47.944
after, will give the same answer
because it's not so obvious when
32:47.944 --> 32:49.644
you write it this way.
32:49.640 --> 32:52.960
But it's very obvious when you
write it this way because it's
32:52.956 --> 32:56.266
clear to us if you stand on your
head or you rotate the whole
32:56.272 --> 32:58.052
axis.
What you're looking for is the
32:58.052 --> 33:00.152
length of A,
which certainly doesn't change
33:00.146 --> 33:02.536
on your orientation,
or the length of B,
33:02.542 --> 33:04.012
or the angle between them.
33:04.009 --> 33:07.579
The angle θ_A
will change but the angle with
33:07.575 --> 33:10.025
the new x axis won't be
the same.
33:10.029 --> 33:12.789
The angle with the new y
axis won't be the same.
33:12.789 --> 33:15.009
Likewise for B,
but the angle between the
33:15.012 --> 33:16.812
vectors, it's an invariant
property,
33:16.809 --> 33:19.429
something intrinsic to the two
vectors, doesn't change,
33:19.432 --> 33:21.232
so the dot product is an
invariant.
33:21.230 --> 33:23.310
This is a very important notion.
33:23.309 --> 33:26.079
When you learn relativity,
you will find you have one
33:26.077 --> 33:28.677
observer saying something,
another observer saying
33:28.684 --> 33:30.854
something.
They will disagree on a lot of
33:30.847 --> 33:33.537
things, but there are few things
they will all agree on.
33:33.539 --> 33:36.499
Those few things will be analog
of A.B.
33:36.500 --> 33:41.040
So, it's very good to have this
part of it very clear in your
33:41.040 --> 33:43.410
head.
This part of elementary vector
33:43.413 --> 33:45.813
analysis should be clear in your
head.
33:45.809 --> 33:55.949
Okay, so any questions about
this?
33:55.950 --> 33:58.170
So, if you want,
geometrically,
33:58.174 --> 34:02.254
the work done by a force when
it moves a body a distance
34:02.252 --> 34:04.572
dr,
a vector dr,
34:04.572 --> 34:07.882
is the length of the force,
the distance traveled,
34:07.878 --> 34:11.518
times the cosine of the angle
between the force and the
34:11.522 --> 34:13.212
displacement vector.
34:13.210 --> 34:18.480
Okay, I'm almost ready for
business because,
34:18.475 --> 34:22.765
what is my goal?
I find out that in a tiny
34:22.772 --> 34:24.872
displacement,
F.dr,
34:24.865 --> 34:29.125
I start from somewhere,
I go to a neighboring place,
34:29.134 --> 34:31.734
a distance dr away.
34:31.730 --> 34:34.630
The change in kinetic energy is
F.dr.
34:34.630 --> 34:37.650
So, let me make a big trip,
okay, let me make a trip in the
34:37.654 --> 34:40.684
xy plane made up of a
whole bunch of little segments
34:40.679 --> 34:42.869
in each one of which I calculate
this,
34:42.870 --> 34:46.940
and I add them all up.
34:46.940 --> 34:50.110
On the left-hand side,
the ΔKs,
34:50.112 --> 34:55.212
this whole thing was defined so
that it's equal to ΔK,
34:55.205 --> 34:57.405
right?
ΔK was equal to this,
34:57.410 --> 34:58.640
that's equal to all that.
34:58.639 --> 35:00.679
So, if you add all the
ΔKs, it's very clear
35:00.682 --> 35:01.502
what you will get.
35:01.500 --> 35:04.640
You will get the kinetic energy
at the end minus kinetic energy
35:04.638 --> 35:05.598
at the beginning.
35:05.599 --> 35:09.169
On the right-hand side,
you are told to add F.dr
35:09.174 --> 35:10.834
for every tiny segment.
35:10.829 --> 35:16.919
Well, that is written
symbolically as this.
35:16.920 --> 35:19.740
This is the notation we use in
calculus.
35:19.739 --> 35:22.619
That just means,
if you want to go from A
35:22.616 --> 35:26.166
to B along some path,
you chop up the path into tiny
35:26.166 --> 35:28.046
pieces.
Each tiny segment,
35:28.045 --> 35:31.725
if it's small enough to be
approximated by a tiny vector
35:31.729 --> 35:34.479
dr,
then take the dot product of
35:34.477 --> 35:38.027
that little dr,
with the force at that point,
35:38.030 --> 35:40.390
which means length of F,
times the length of the
35:40.385 --> 35:41.995
segment, times cosine of the
angle;
35:42.000 --> 35:45.180
add them all up.
Then somebody will say,
35:45.181 --> 35:49.111
"Well, your segments are not
short enough for me."
35:49.110 --> 35:51.320
We'll chop it up some more,
then chop it up some more,
35:51.324 --> 35:53.584
chop it up till your worst
critic has been silenced.
35:53.579 --> 35:57.029
That's the limit in which you
can write the answer as this
35:57.033 --> 35:59.823
integral.
Just like in calculus when you
35:59.817 --> 36:03.837
integrate a function you take
tiny intervals Δx,
36:03.840 --> 36:06.220
multiply F by
Δx, but then you make
36:06.222 --> 36:09.042
the intervals more and more
numerous but less and less wide
36:09.043 --> 36:11.913
and the limit of that is the
area under the graph and that's
36:11.912 --> 36:14.892
called "integral,"
you do the same thing now in
36:14.887 --> 36:18.657
two dimensions.
So, now maybe it'll be true,
36:18.658 --> 36:24.128
just like in one dimension,
the integral of this function
36:24.134 --> 36:29.224
will be something that depends
on the end points.
36:29.219 --> 36:32.259
I'm just going to call it
U(1) - U(2),
36:32.258 --> 36:35.638
of some function U,
just like it was in one
36:35.643 --> 36:36.613
dimension.
36:36.610 --> 36:39.690
36:39.690 --> 36:42.130
If that is true,
then my job is done because
36:42.133 --> 36:45.603
then I have K_1 +
U_1 = K_2 +
36:45.599 --> 36:46.849
U_2.
36:46.850 --> 36:56.060
36:56.059 --> 36:58.929
So, when I'm looking for the
Law of Conservation of Energy,
36:58.933 --> 37:01.713
I've got to go to some calculus
book and I got to ask the
37:01.708 --> 37:04.348
calculus book,
"Look, in one dimension you
37:04.349 --> 37:08.169
told me integral of F
from start to end is really the
37:08.173 --> 37:12.003
difference of another function
G of the end minus the
37:11.997 --> 37:13.877
start,
with G as that function
37:13.875 --> 37:15.075
whose derivative is F."
37:15.079 --> 37:18.339
Maybe there is going to be some
other magic function you knew in
37:18.343 --> 37:20.263
two dimensions related to
F,
37:20.260 --> 37:22.630
in some way,
so that this integral is again
37:22.629 --> 37:25.729
given by a difference of
something there minus something
37:25.732 --> 37:27.282
here.
If that is the case,
37:27.280 --> 37:29.360
then you can rearrange it and
get this.
37:29.360 --> 37:37.280
But you will find that it's not
meant to be that simple.
37:37.280 --> 37:40.740
So, again, has anybody heard
rumors about why it may not be
37:40.735 --> 37:41.565
that simple?
37:41.570 --> 37:45.160
37:45.160 --> 37:46.310
What could go wrong?
37:46.310 --> 37:51.160
37:51.160 --> 37:53.940
Okay so, yes?
Student: [inaudible]
37:53.936 --> 37:55.856
Professor Ramamurti
Shankar: Okay,
37:55.856 --> 37:58.876
that's probably correct but say
it in terms of what we know.
37:58.880 --> 37:59.730
What can go wrong?
37:59.730 --> 38:01.930
I'm saying in one end,
I did an integral;
38:01.929 --> 38:06.679
the integral was the difference
of two numbers and therefore I
38:06.678 --> 38:08.778
got K + U = K + U.
38:08.780 --> 38:11.900
So, something could go wrong
somewhere here when I say this
38:11.903 --> 38:15.033
integral from start to finish is
the difference only of the
38:15.027 --> 38:17.287
ending point minus the starting
point.
38:17.289 --> 38:20.069
Is that reasonable or could you
imagine it depending on
38:20.073 --> 38:21.783
something else?
Yes?
38:21.780 --> 38:24.470
Student: Well,
say you have one of your forces
38:24.474 --> 38:26.984
is friction.
If you take a certain path from
38:26.984 --> 38:30.044
point one to point two,
you should have the same thing
38:30.035 --> 38:31.355
in potential energy.
38:31.360 --> 38:34.950
But if you take a longer path
and there's friction involved,
38:34.954 --> 38:37.334
your kinetic energy would be
reduced.
38:37.329 --> 38:37.839
Professor Ramamurti
Shankar: Absolutely correct.
38:37.840 --> 38:40.570
It is certainly true that if
you've got friction,
38:40.568 --> 38:42.158
this is not going to work.
38:42.160 --> 38:44.540
It doesn't even work in 1D.
38:44.539 --> 38:47.099
In 1D, if I start here and end
here, and I worry about
38:47.100 --> 38:50.050
friction, if it went straight
from here to here there's amount
38:50.047 --> 38:52.677
of friction.
If I just went back and forth
38:52.681 --> 38:56.661
97 times and then I ended up
there is 97 times more friction.
38:56.659 --> 38:59.149
So, we agreed that if there's
friction, this is not going to
38:59.154 --> 39:00.754
work.
But the trouble with friction
39:00.745 --> 39:03.105
was, the force was not a
function only of x and
39:03.113 --> 39:05.463
y.
It depended on the direction of
39:05.462 --> 39:07.952
motion.
But now, I grant you that the
39:07.946 --> 39:10.516
force is not a function of
velocity.
39:10.519 --> 39:12.219
It's only a function of where
you are.
39:12.220 --> 39:13.680
Can something still be wrong?
39:13.680 --> 39:19.060
39:19.059 --> 39:22.409
Well, let me ask you the
following question.
39:22.410 --> 39:24.240
Another person does this.
39:24.240 --> 39:27.630
39:27.630 --> 39:29.860
[Shows a different path from 1
to 2]
39:29.861 --> 39:33.501
Do you think that person should
do the same amount of work
39:33.495 --> 39:37.125
because the force is now
integrated on a longer path?
39:37.130 --> 39:39.440
So, you see,
in one dimension there's only
39:39.438 --> 39:41.408
one way to go from here to
there.
39:41.410 --> 39:44.410
Just go, right?
That way, when you write an
39:44.408 --> 39:46.618
integral you write the lower
limit and the upper limit and
39:46.622 --> 39:48.912
you don't say any more because
it's only one way in 1D to go
39:48.913 --> 39:51.053
from x_1 to
x_2.
39:51.050 --> 39:53.260
In two dimensions,
there are thousands of ways to
39:53.263 --> 39:54.973
go from one point to another
point.
39:54.969 --> 39:58.599
You can wander all over the
place and you end up here.
39:58.599 --> 40:01.219
Therefore, this integral,
even if I say the starting
40:01.221 --> 40:03.891
point is r_1
and the ending point is
40:03.894 --> 40:05.884
r_2,
that this is
40:05.880 --> 40:08.910
r_1 and that's
r_2,
40:08.914 --> 40:10.144
it's not adequate.
40:10.139 --> 40:12.649
What do you think I should
attach to this integration?
40:12.650 --> 40:14.200
What other information should I
give?
40:14.200 --> 40:17.260
What more should I specify?
40:17.260 --> 40:19.840
Yep?
Student: Along a closed
40:19.843 --> 40:20.543
path.
Professor Ramamurti
40:20.539 --> 40:21.319
Shankar: No,
it's not a closed path.
40:21.320 --> 40:23.020
I'm going from 1 to 2.
40:23.019 --> 40:27.199
What more should I tell you
before you can even find the
40:27.198 --> 40:31.488
work done?
Yes?
40:31.489 --> 40:32.429
Student: What path
you're going on.
40:32.429 --> 40:34.929
Professor Ramamurti
Shankar: You have to say
40:34.925 --> 40:37.615
which path you're going on
because if you only have two
40:37.620 --> 40:40.360
points,
1 and 2, then there is a work
40:40.363 --> 40:42.863
done but it depends on the path.
40:42.860 --> 40:45.970
If the work depends on the
path, then the answer cannot be
40:45.970 --> 40:48.970
a function of U(1) -
U(2), cannot just be U(1)
40:48.971 --> 40:50.931
- U(2).
U(1) - U(2) says,
40:50.931 --> 40:53.261
tell me what you entered,
tell me where you start,
40:53.261 --> 40:56.021
and that difference of some
function U between those
40:56.020 --> 40:57.590
two points is the work done.
40:57.590 --> 40:59.550
In other words,
I'm asking you to think
40:59.547 --> 41:02.737
critically about whether this
equality really could be true.
41:02.739 --> 41:07.169
This is some function U
in the xy plane evaluated
41:07.173 --> 41:09.823
at one point minus the other
point.
41:09.820 --> 41:13.510
This is on a path joining those
two points but I have not told
41:13.511 --> 41:17.141
you which path and I can draw
any path I like with those same
41:17.142 --> 41:19.262
end points.
And you got to realize that
41:19.260 --> 41:21.010
it's very unreasonable to expect
that.
41:21.010 --> 41:23.620
No matter which path you take
you will get the same answer.
41:23.619 --> 41:28.999
Okay, so you might think that
I'm creating a straw man because
41:29.000 --> 41:32.440
it's going to turn out by some
magic,
41:32.440 --> 41:35.210
that no matter what force you
take somehow,
41:35.208 --> 41:37.448
due to the magic of
mathematics,
41:37.449 --> 41:40.179
the integral will depend only
on the end points.
41:40.180 --> 41:42.170
But that's not the case.
41:42.170 --> 41:44.020
In general, that won't happen.
41:44.020 --> 41:46.940
So, I have to show you that.
41:46.940 --> 41:51.810
So, I'm going to start by
asking you, give me a number
41:51.812 --> 41:54.202
from 1 to 3.
2?
41:54.200 --> 41:58.380
Okay, 2.
Then, I want a few more
41:58.381 --> 42:02.261
numbers, another number from 1
to 3.
42:02.260 --> 42:03.750
Professor Ramamurti
Shankar: From 1 to 3,
42:03.747 --> 42:04.377
[audience laughs].
42:04.380 --> 42:06.970
All right then, two more.
42:06.970 --> 42:11.100
Mark, you pick a number.
42:11.099 --> 42:12.009
Student: [inaudible]
Professor Ramamurti
42:12.006 --> 42:12.576
Shankar: Good,
thank you.
42:12.580 --> 42:17.430
Then I need one more, yes?
42:17.430 --> 42:20.520
1 to 3, a number from 1 to 3.
42:20.519 --> 42:21.959
Student: [inaudible]
Professor Ramamurti
42:21.958 --> 42:22.778
Shankar: 2,
very good.
42:22.780 --> 42:28.420
Okay, so you pick these numbers
randomly, and I'm going to take
42:28.415 --> 42:33.315
a force which looks like
I times x^(2)y^(3) +
42:33.323 --> 42:36.633
J times xy^(2),
okay?
42:36.630 --> 42:40.230
I put the powers based on what
you guys gave me.
42:40.230 --> 42:43.490
So, we picked the force in two
dimensions out of the hat.
42:43.489 --> 42:46.589
Now, let's ask,
"Is it true for this force that
42:46.589 --> 42:50.829
the work done in going from one
point to another depends only on
42:50.834 --> 42:52.604
the path,
or does it depend,
42:52.598 --> 42:55.608
I mean, it depends only on the
end points or does it depend in
42:55.609 --> 42:57.829
detail on how you go from the
end points?"
42:57.829 --> 43:01.969
You all have to understand,
before you copy anything down,
43:01.968 --> 43:04.218
where we are going with this.
43:04.220 --> 43:05.810
What's the game plan?
43:05.809 --> 43:09.139
So, let me tell you one more
time because you can copy this
43:09.136 --> 43:11.426
all you want,
it will get you nowhere.
43:11.429 --> 43:14.089
You should feel that you know
where I'm going but the details
43:14.094 --> 43:16.984
remain to be shown or you should
have an idea what's happening.
43:16.980 --> 43:21.150
If you try to generalize the
Work Energy Theorem to two
43:21.149 --> 43:24.469
dimensions, this is what
happened so far.
43:24.469 --> 43:27.599
You found a definition of work
which has the property that the
43:27.601 --> 43:29.861
work done is the change in
kinetic energy.
43:29.860 --> 43:33.010
Then, you added up all the
changes of kinetic energy and
43:33.013 --> 43:36.283
added up all the work done and
you said K_2 -
43:36.281 --> 43:39.321
K_1 is the
integral of F.dr;
43:39.320 --> 43:42.750
that's guaranteed to be true;
that's just based on Newton's
43:42.751 --> 43:44.841
laws.
What is tricky is the second
43:44.841 --> 43:48.331
equality that that integral is
the difference in a function
43:48.326 --> 43:51.746
calculated at one end point
minus the other end point.
43:51.750 --> 43:55.330
If that was true,
if the integral depended only
43:55.333 --> 43:58.923
on the end points,
then it cannot depend on the
43:58.918 --> 44:00.708
path that you take.
44:00.710 --> 44:03.080
If it depends on the path,
every path you take within the
44:03.076 --> 44:05.186
same two end points will give
different numbers.
44:05.190 --> 44:07.510
So, the answer cannot simply be
U_1 -
44:07.512 --> 44:08.462
U_2.
44:08.460 --> 44:11.540
First, I'm trying to convince
you, through an example that I
44:11.536 --> 44:13.996
selected randomly,
that if you took a random force
44:13.996 --> 44:16.596
and found the work done along
one path, or another path,
44:16.599 --> 44:19.719
you will in fact get two
different answers,
44:19.719 --> 44:21.519
okay?
That's the first thing,
44:21.518 --> 44:23.628
to appreciate that there's a
problem.
44:23.630 --> 44:25.840
So generally,
if we took a random force,
44:25.843 --> 44:28.743
not a frictional force,
a force that depends only on
44:28.737 --> 44:32.017
location and not velocity,
it will not be possible to
44:32.016 --> 44:35.646
define a potential energy nor
will it be possible to define a
44:35.653 --> 44:38.203
K + U so that it doesn't
change.
44:38.199 --> 44:41.729
So, it's going to take a very
special force for which the
44:41.730 --> 44:45.200
answer depends only on the
starting and ending point and
44:45.198 --> 44:48.108
not on the path.
To show you that that's a
44:48.108 --> 44:51.468
special situation,
I'm taking a generic situation,
44:51.469 --> 44:55.399
namely, a force manufactured by
this class, without any prior
44:55.402 --> 44:58.042
consultation with me,
and I will show you that for
44:58.036 --> 45:00.286
that force the answer is going
to depend on how you go.
45:00.289 --> 45:04.509
So, let's take that force and
let's find the work done in
45:04.510 --> 45:07.600
going from the origin to the
point 1,1.
45:07.600 --> 45:11.690
So, I'm going to take two paths.
45:11.690 --> 45:14.660
One path I'm going to go
horizontally till I'm below the
45:14.658 --> 45:20.058
point.
Then I'm going straight up,
45:20.064 --> 45:23.704
okay.
So, let's find the work done
45:23.701 --> 45:25.511
when I go this way.
45:25.510 --> 45:29.780
So again, you should be
thinking all the time.
45:29.780 --> 45:32.990
You should say if this guy got
struck by lightning can I do
45:32.986 --> 45:36.406
anything, or am I just going to
say "Well, I don't know what he
45:36.413 --> 45:37.743
was planning to do."
45:37.739 --> 45:39.579
You've got to have some idea
what I'm going to do.
45:39.579 --> 45:42.449
I'm going to integrate
F.dr, first on the
45:42.452 --> 45:45.572
horizontal segment,
then on the vertical segment.
45:45.570 --> 45:47.780
F is some vector you
give me at the point x
45:47.778 --> 45:49.528
and y;
I'll plug in some numbers,
45:49.528 --> 45:51.848
I'll get something times
I and something times
45:51.845 --> 45:54.025
J.
It may be pointing like that at
45:54.025 --> 45:56.515
this point, and that's F,
that's given.
45:56.519 --> 46:00.099
Now, when I'm moving
horizontally my displacement
46:00.101 --> 46:03.311
dr, has only got a
dx part.
46:03.310 --> 46:04.690
I hope you see that.
46:04.690 --> 46:07.730
Every step I move,
there's no dy in it,
46:07.732 --> 46:09.222
it's all horizontal.
46:09.219 --> 46:15.749
So, F.dr just becomes
F_xdx because
46:15.748 --> 46:21.488
there is no dy when you
move horizontally.
46:21.489 --> 46:23.199
So, when you do the integral,
you have
46:23.202 --> 46:24.362
F_xdx.
46:24.360 --> 46:29.240
F_x happens to
be x^(2)y^(3)dx,
46:29.239 --> 46:31.859
x going from 0 to 1.
46:31.860 --> 46:36.140
Now, what do I do with
y^(3)?
46:36.139 --> 46:38.339
You all know how to integrate a
function of x times
46:38.337 --> 46:41.067
dx.
What do I do with y^(3)?
46:41.070 --> 46:43.010
What do you think it means?
46:43.010 --> 46:44.350
Student: [inaudible]
Professor Ramamurti
46:44.347 --> 46:45.027
Shankar: Pardon me?
46:45.030 --> 46:47.970
What should I do with y?
46:47.969 --> 46:52.749
Evaluate y on that path
at that point.
46:52.750 --> 46:56.560
Well, it turns out,
throughout this horizontal
46:56.561 --> 46:59.951
segment y = 0,
so this is gone.
46:59.949 --> 47:02.109
Basically, the point is very
simple.
47:02.110 --> 47:05.090
When you move horizontally,
you're working against
47:05.086 --> 47:08.416
horizontal forces doing work,
but on the x axis when
47:08.423 --> 47:10.633
y is 0 there is no
horizontal force;
47:10.630 --> 47:12.510
that's why there's nothing to
do.
47:12.510 --> 47:14.160
Now, you come to this segment.
47:14.159 --> 47:17.069
I think you all agree,
the distance traveled is
47:17.071 --> 47:18.781
J times dy.
47:18.780 --> 47:22.070
So, I have to have another
segment, the second part of my
47:22.068 --> 47:24.418
trip, which is
F_y times
47:24.417 --> 47:27.747
dy,
and y goes from 0 to 1,
47:27.745 --> 47:31.665
and the y component is
xy^(2)dy,
47:31.670 --> 47:33.190
where y goes from 0 to 1.
47:33.190 --> 47:39.010
But now, on the entire line
that I'm moving up and down,
47:39.012 --> 47:41.722
x = 1.
Do you see that?
47:41.719 --> 47:45.799
x is a constant on the
line so you can replace x
47:45.796 --> 47:49.136
by 1 and integral of
y^(2)dy is y^(3)
47:49.138 --> 47:52.508
over 3.The work done is,
therefore, 1/3 joules.
47:52.510 --> 47:56.550
So, the work done in going
first to the right and then to
47:56.552 --> 47:57.782
the top is 1/3.
47:57.780 --> 48:04.460
48:04.460 --> 48:09.540
You can also go straight up and
then horizontally.
48:09.539 --> 48:11.309
By a similar trick,
I won't do that now,
48:11.307 --> 48:13.937
because I want to show you
something else that's useful for
48:13.937 --> 48:15.797
you,
I'm going to pick another path
48:15.803 --> 48:18.863
which is not just made up of
x and y segments.
48:18.860 --> 48:20.600
Then it's very easy to do this.
48:20.599 --> 48:24.139
So, I'm going to pick another
way to go from 0,0 to 1,1,
48:24.140 --> 48:27.700
which is on this curve;
this is the curve y =
48:27.697 --> 48:28.587
x^(2).
48:28.590 --> 48:33.230
48:33.230 --> 48:35.040
First of all,
you've got to understand the
48:35.038 --> 48:37.378
curve y = x^(2) goes
through the two points I'm
48:37.376 --> 48:39.506
interested in.
If you took y =
48:39.506 --> 48:41.446
5x^(2),
it doesn't work.
48:41.449 --> 48:44.759
But this guy goes through 0,0
and goes through 1,1.
48:44.760 --> 48:48.060
And I'm asking you,
if I did the work done by the
48:48.059 --> 48:51.359
force along that segment,
what is the integral of
48:51.360 --> 48:55.160
F.dr?
So, let's take a tiny portion
48:55.162 --> 48:59.732
of that, looks like this,
right, that's dr,
48:59.729 --> 49:04.389
it's got a dx,
and it's got a dy.
49:04.389 --> 49:11.519
Now, you notice that as this
segment becomes very small
49:11.518 --> 49:19.918
dy/dx is a slope;
therefore, dy will be
49:19.917 --> 49:25.237
dy/dx times dx.
49:25.239 --> 49:27.039
In other words,
dx and dy are not
49:27.035 --> 49:29.295
independent if they're moving in
a particular direction.
49:29.300 --> 49:30.620
I hope you understand that.
49:30.619 --> 49:32.139
You want to follow a certain
curve.
49:32.139 --> 49:34.959
If you step to the right by
some amount, you've got to step
49:34.959 --> 49:38.069
vertically by a certain amount
so you're moving on that curve.
49:38.070 --> 49:40.920
That's why, when you calculate
the work done,
49:40.920 --> 49:43.900
dx and dy are not
independent.
49:43.900 --> 49:48.170
So, what you really want is
F_xdx +
49:48.165 --> 49:52.695
F_ydy,
but for dy I'm going to
49:52.702 --> 49:56.062
use dy/dx times
dx.
49:56.059 --> 49:59.049
In other words,
every segment Δy that
49:59.053 --> 50:03.003
you have is related to the
Δx you took horizontally
50:02.998 --> 50:05.378
so that you stay on that curve.
50:05.380 --> 50:07.890
So, everything depends only on
dx.
50:07.889 --> 50:11.229
But what am I putting inside
the integral?
50:11.230 --> 50:13.350
Let's take
F_x,
50:13.351 --> 50:16.501
the x component is
x^(2)y^(3).
50:16.500 --> 50:21.030
On this curve y = x^(2),
so you really write
50:21.027 --> 50:24.107
x^(2),
and y = x^(2),
50:24.106 --> 50:26.366
that is x^(6).
50:26.369 --> 50:28.739
This x^(3) [should have
said x^(6)]
50:28.739 --> 50:31.859
is really y^(3) written
in terms of x.
50:31.860 --> 50:34.960
50:34.960 --> 50:38.420
Then, I have to write
F_y,
50:38.418 --> 50:41.628
which is x times
y^(2),
50:41.630 --> 50:45.080
which is x^(4) times
dy/dx which is 2x.
50:45.080 --> 50:49.460
50:49.460 --> 50:55.750
So, I get here,
from all of this,
50:55.748 --> 51:01.248
(x^(8) + 2x^(5))dx.
51:01.250 --> 51:02.750
Did I make a mistake somewhere?
51:02.750 --> 51:06.260
Pardon me?
Student: The second part
51:06.262 --> 51:07.962
of this.
Professor Ramamurti
51:07.957 --> 51:09.347
Shankar: Did I,
here?
51:09.350 --> 51:12.950
Here?
Student: No.
51:12.949 --> 51:15.979
Professor Ramamurti
Shankar: Oh here?
51:15.980 --> 51:17.430
Student: [inaudible]
Professor Ramamurti
51:17.425 --> 51:18.275
Shankar: Oh,
x^(6),
51:18.280 --> 51:18.900
right, thank you.
51:18.900 --> 51:20.850
How is that?
Thanks for watching that;
51:20.850 --> 51:22.530
you have to watch it.
51:22.530 --> 51:24.570
So, now what do I get?
51:24.570 --> 51:28.180
x^(8) integral is
x^(9)/9.
51:28.179 --> 51:31.539
That gives me 1,9 because x is
going from 0 to 1.
51:31.539 --> 51:35.799
The next thing is
x^(7)/7,
51:35.797 --> 51:38.987
which is 2 times that.
51:38.989 --> 51:42.079
Well, I'm not paying too much
attention to this because I know
51:42.077 --> 51:43.137
it's not 1/3,
okay?
51:43.139 --> 51:47.069
There's no way this guy's going
to be 1/3, that's all I care
51:47.067 --> 51:49.887
about.
So, I've shown you that if we
51:49.887 --> 51:53.747
took a random force,
the work done is dependent on
51:53.753 --> 51:55.593
the path.
For this force,
51:55.590 --> 51:59.150
you cannot define a potential
energy whereas in one dimension
51:59.150 --> 52:02.350
any force that was not friction
allowed you to define a
52:02.354 --> 52:03.664
potential energy.
52:03.659 --> 52:05.399
In higher dimensions,
you just cannot do that;
52:05.400 --> 52:06.810
that's the main point.
52:06.809 --> 52:09.839
So, if you're looking for a
conservative,
52:09.844 --> 52:12.884
this is called a "conservative
force."
52:12.880 --> 52:15.550
It's a force for which you can
define a potential energy.
52:15.550 --> 52:18.880
It has the property of the work
done in going from A to
52:18.880 --> 52:21.560
B, or 1 to 2,
is independent of how you got
52:21.556 --> 52:24.106
from 1 to 2.
And the one force that the
52:24.109 --> 52:27.789
class generated pretty much
randomly is not a conservative
52:27.790 --> 52:31.020
force because the work done was
path dependent.
52:31.019 --> 52:35.909
So, what we have learned,
in fact I'll keep this portion
52:35.912 --> 52:40.452
here, if you're looking for a
conservative force,
52:40.449 --> 52:44.749
a force whose answer does not
depend on how you went from
52:44.752 --> 52:48.292
start to finish,
then you have to somehow dream
52:48.285 --> 52:52.965
up some force so that if you did
this integral the answer does
52:52.972 --> 52:55.202
not depend on the path.
52:55.199 --> 52:57.469
You realize,
that looks really miraculous
52:57.471 --> 53:00.651
because we just wrote down an
arbitrary force that we all
53:00.651 --> 53:03.151
cooked up together with these
exponents,
53:03.150 --> 53:05.060
and the answer depends on the
path.
53:05.059 --> 53:07.789
And I guarantee you,
if you just arbitrarily write
53:07.790 --> 53:09.630
down some force,
it won't work.
53:09.630 --> 53:12.880
So, maybe there is no Law of
Conservation of Energy in more
53:12.882 --> 53:14.062
than one dimension.
53:14.060 --> 53:18.030
53:18.030 --> 53:21.990
So, how am I going to search
for a force that will do the
53:21.986 --> 53:23.966
job?
Are there at least some forces
53:23.971 --> 53:25.481
for which this will be true?
53:25.480 --> 53:27.190
Yes?
Student: The force is
53:27.187 --> 53:29.097
always parallel to the path
along which you have [inaudible]
53:29.099 --> 53:30.459
Professor Ramamurti
Shankar: No,
53:30.460 --> 53:34.980
but see the point is you've got
to first write down a force in
53:34.979 --> 53:36.609
the xy plane.
53:36.610 --> 53:39.590
Then, I should be free to pick
any two points and connect them
53:39.587 --> 53:42.267
any way I like and the answer
shouldn't depend on how we
53:42.272 --> 53:44.432
connected them.
Only then, you can have
53:44.433 --> 53:45.723
conservation of energy.
53:45.719 --> 53:48.389
I hope you all understand that
fact.
53:48.389 --> 53:50.619
So, I'm saying,
a generic force on the
53:50.615 --> 53:52.655
xy plane doesn't do that.
53:52.659 --> 53:54.929
Then we ask,
"Can there ever be an answer?"
53:54.930 --> 53:56.060
It is so demanding.
53:56.060 --> 53:57.870
Yes?
Student: Well,
53:57.867 --> 54:00.227
in the inverse square force
they've got it.
54:00.230 --> 54:01.230
Professor Ramamurti
Shankar: Right,
54:01.233 --> 54:02.093
so he's saying,
"I know a force."
54:02.090 --> 54:04.320
The force of gravity,
in fact, happens to have the
54:04.323 --> 54:07.013
property that the work done by
the force of gravity does not
54:07.011 --> 54:08.061
depend on the path.
54:08.060 --> 54:10.650
We will see why that is true.
54:10.650 --> 54:13.150
But you see,
what I want is to ask,
54:13.145 --> 54:16.885
"Is there a machine that'll
manufacture conservative
54:16.889 --> 54:19.479
forces?"
and I'm going to tell you there
54:19.479 --> 54:21.329
is.
I will show you a machine
54:21.327 --> 54:24.707
that'll produce a large number,
an infinite number of
54:24.713 --> 54:27.583
conservative forces,
and I'll show you how to
54:27.578 --> 54:30.258
produce that.
Here is the trick.
54:30.260 --> 54:32.930
The trick is,
instead of taking a force and
54:32.934 --> 54:36.824
finding if there's a potential
that will come from it by doing
54:36.818 --> 54:38.978
integrals,
let me assume there is a
54:38.978 --> 54:42.088
potential.
Then, I will ask what force I
54:42.086 --> 54:46.266
can associate with the
potential, and here is the
54:46.268 --> 54:48.628
answer.
The answer is,
54:48.626 --> 54:54.356
step one, pick any U of
x and y.
54:54.360 --> 54:58.460
You pick the function first.
54:58.460 --> 55:01.660
That function is going to be
your potential energy;
55:01.659 --> 55:04.889
it's been anointed even before
we do anything.
55:04.889 --> 55:08.969
But then, from the potential,
I want you to manufacture the
55:08.972 --> 55:12.362
following force.
I want the force that I'm going
55:12.356 --> 55:16.036
to find to have an x
component, which is [minus]
55:16.044 --> 55:19.804
the derivative of U with
respect to x,
55:19.800 --> 55:21.810
and is going to have a y
component which is [minus]
55:21.807 --> 55:23.707
the derivative of U with
respect to y.
55:23.710 --> 55:27.680
55:27.680 --> 55:29.870
That is step two.
55:29.869 --> 55:33.289
In fact, the claim now is this
force is going to be a
55:33.289 --> 55:34.669
conservative force.
55:34.670 --> 55:37.150
And in fact,
this potential energy
55:37.148 --> 55:41.578
associated with it will be the
function you started with.
55:41.580 --> 55:44.050
So, how do I know that?
55:44.050 --> 55:48.220
So, for example,
I'm saying suppose U =
55:48.218 --> 55:53.198
xy^(3), then
F_x = -dU/dx,
55:53.199 --> 55:55.809
which is equal to
-y^(3),
55:55.807 --> 56:00.517
and F_y,
which is -dU/dy would be
56:00.518 --> 56:02.788
-3xy^(2).
56:02.789 --> 56:04.049
The claim is,
if you put I times
56:04.046 --> 56:05.296
F_x and
J times
56:05.303 --> 56:08.083
F_y,
the answer will not depend on
56:08.080 --> 56:10.300
how you go from start to finish.
56:10.300 --> 56:13.650
So, let me prove that to you.
56:13.650 --> 56:15.220
Here is the proof.
56:15.219 --> 56:19.549
The change in the function
U, in the xy
56:19.551 --> 56:23.051
plane, you guys remember me
telling you,
56:23.050 --> 56:29.760
is dU/dx times dx +
dU/dy times dy.
56:29.760 --> 56:33.390
That's the whole thing of
mathematical preliminaries which
56:33.386 --> 56:35.736
was done somewhere,
I forgot, here.
56:35.739 --> 56:38.869
It says, here,
ΔF is dF/dx
56:38.873 --> 56:42.323
times Δx + dF/dy times
Δy.
56:42.320 --> 56:46.450
I'll apply that to the function
U, but who is this?
56:46.450 --> 56:48.500
You notice what's going on here?
56:48.500 --> 56:52.360
dU/dx times dx,
this is F_x,
56:52.359 --> 56:55.349
with a minus sign,
that's F_y
56:55.353 --> 56:56.953
with the minus sign.
56:56.949 --> 57:00.739
Therefore this is equal to
-F.dr.
57:00.740 --> 57:06.220
57:06.220 --> 57:08.500
Right?
Because we agree,
57:08.502 --> 57:10.502
the force that I want to
manufacture is related to
57:10.499 --> 57:11.639
U in this fashion.
57:11.639 --> 57:17.099
Now, if you add all the
changes, the right-hand side
57:17.095 --> 57:21.155
becomes the integral of
F.dr,
57:21.159 --> 57:25.569
and the left-hand side becomes
the sum of all the ΔUs
57:25.570 --> 57:27.040
with a minus sign.
57:27.039 --> 57:31.339
Add all the ΔUs with a
minus sign, that'll just become
57:31.343 --> 57:33.533
U at 1 -U at 2.
57:33.530 --> 57:40.720
57:40.719 --> 57:44.069
It is cooked up so that
F.dr is actually at the
57:44.070 --> 57:46.220
change in the function U.
57:46.219 --> 57:48.729
That's if you say,
what was the trick that you
57:48.725 --> 57:51.115
did?
I cooked up a force by design
57:51.123 --> 57:55.753
so that F.dr was a change
in a certain function U.
57:55.750 --> 57:57.910
If I add all the F.drs,
I'm going to get a change in
57:57.909 --> 58:00.139
the function U from start
to finish and it's got to be
58:00.143 --> 58:01.673
U_1 -
U_2.
58:01.670 --> 58:07.460
58:07.460 --> 58:10.500
So, I don't know how else I can
say this.
58:10.500 --> 58:16.810
Maybe the way to think about it
is, why do certain integrals not
58:16.806 --> 58:20.906
depend on how you took the path,
right?
58:20.909 --> 58:22.139
Let me ask you a different
question;
58:22.140 --> 58:23.580
forget about integrals.
58:23.579 --> 58:26.709
You are on top of some hilly
mountain.
58:26.710 --> 58:29.950
We have a starting point,
you have an ending point,
58:29.946 --> 58:32.126
okay.
I started the starting point,
58:32.126 --> 58:34.146
and I walked to the ending
point.
58:34.150 --> 58:38.270
At every portion of my walk,
I keep track of how many feet
58:38.274 --> 58:41.664
I'm climbing;
that's like my ΔU.
58:41.660 --> 58:43.060
I add them all up.
58:43.059 --> 58:45.669
At the end of the day,
the height change will be the
58:45.672 --> 58:48.132
top of the mountain minus bottom
of the mountain,
58:48.131 --> 58:49.721
the height of the mountain.
58:49.719 --> 58:52.119
You go on a different path,
you don't go straight for the
58:52.123 --> 58:54.573
summit, you loop around and you
coil and you wind down,
58:54.570 --> 58:57.510
you go up, you do this,
and you also end up at the
58:57.510 --> 58:59.780
summit.
If you kept track of how long
58:59.777 --> 59:02.157
you walked, it won't be the same
as me.
59:02.159 --> 59:04.859
But if you also kept track of
how many feet you climbed and
59:04.861 --> 59:07.191
you added them all up,
what answer will you get?
59:07.190 --> 59:09.210
You'll get the same answer I
got.
59:09.210 --> 59:13.590
Therefore, if what you were
keeping track of was the height
59:13.592 --> 59:16.682
change in a function,
then the sum of all the height
59:16.676 --> 59:18.826
changes will be simply the total
height change,
59:18.829 --> 59:21.389
which is the height at the end
minus height at the beginning.
59:21.389 --> 59:24.109
Therefore, starting with the
height function,
59:24.113 --> 59:27.583
by taking its derivatives,
if you manufacture a force,
59:27.579 --> 59:31.669
this will be none other than
the fact of adding up the
59:31.668 --> 59:34.278
changes.
And that's why this will be
59:34.283 --> 59:36.743
K_2 -
K_1,
59:36.741 --> 59:40.361
and then you will get
K_1 + U_1
59:40.360 --> 59:43.160
= K_2 +
U_2.
59:43.159 --> 59:46.589
I hope you understand how
conservative forces are not
59:46.590 --> 59:47.910
impossible to get.
59:47.909 --> 59:50.219
In fact, for every function
U of x and
59:50.219 --> 59:52.349
y you can think of,
you can manufacture a
59:52.348 --> 59:53.388
conservative force.
59:53.389 --> 59:57.009
So, you may ask the following
question.
59:57.010 --> 1:00:01.010
Maybe there are other ways to
manufacture the conservative
1:00:01.014 --> 1:00:05.234
force and you just thought of
one, and the answer is "no."
1:00:05.230 --> 1:00:08.600
Not only is this a machine that
generates conservative forces,
1:00:08.602 --> 1:00:11.562
my two step algorithm,
pick a U and take its
1:00:11.560 --> 1:00:15.000
derivatives, every conservative
force you get is necessarily
1:00:14.995 --> 1:00:18.365
obtained by taking derivatives
with respect to x and
1:00:18.372 --> 1:00:21.632
y of some function
U and that U will
1:00:21.633 --> 1:00:25.013
be the potential energy
associated with that force.
1:00:25.010 --> 1:00:28.660
So, when that force alone acts
on a body the kinetic plus that
1:00:28.656 --> 1:00:30.326
potential will not change.
1:00:30.329 --> 1:00:37.019
Finally, you remember that when
the class picked a certain
1:00:37.015 --> 1:00:42.385
force, which I wrote here,
I went on a limb and I said I'm
1:00:42.386 --> 1:00:46.556
going to do the integral of this
force along this path and that
1:00:46.559 --> 1:00:50.259
path and I'm going to get
different answers and show you
1:00:50.261 --> 1:00:51.811
we have a problem.
1:00:51.809 --> 1:00:55.639
What if the force you had given
me was actually a conservative
1:00:55.637 --> 1:00:58.337
force?
Then, I would be embarrassed
1:00:58.335 --> 1:01:01.625
because then I'll find,
after all the work,
1:01:01.627 --> 1:01:04.447
this'll turn out to be again
1/3.
1:01:04.449 --> 1:01:07.029
So, I have to make sure right
away that the force is not
1:01:07.027 --> 1:01:11.767
conservative.
How can you tell?
1:01:11.769 --> 1:01:14.489
One way to say it is,
ask yourself,
1:01:14.489 --> 1:01:18.569
"Could that be some function
U whose x
1:01:18.570 --> 1:01:22.740
derivative of this was this,
and whose y derivative
1:01:22.739 --> 1:01:24.519
was that?"
You can probably convince
1:01:24.523 --> 1:01:27.343
yourself no function is going to
do it for you because if you
1:01:27.339 --> 1:01:29.969
took an x derivative you
should've lost a power of
1:01:29.968 --> 1:01:32.108
x here.
That means, if you just took
1:01:32.107 --> 1:01:34.557
the y derivative to go
here, you should have more
1:01:34.557 --> 1:01:37.097
powers of x and less
powers of y but this is
1:01:37.097 --> 1:01:38.297
just the opposite way.
1:01:38.300 --> 1:01:41.480
So, we know this couldn't have
come from a U.
1:01:41.480 --> 1:01:43.680
But there's a better test.
1:01:43.679 --> 1:01:46.679
Instead of doing all that,
instead of saying I'm satisfied
1:01:46.675 --> 1:01:49.405
that this doesn't come from
taking derivatives of the
1:01:49.408 --> 1:01:52.248
U by looking at possible
Us are not being
1:01:52.246 --> 1:01:54.976
satisfied because maybe I'm not
clever enough.
1:01:54.980 --> 1:01:57.010
There is a mechanical way to
tell.
1:01:57.010 --> 1:02:00.640
The mechanical way to tell is
the following.
1:02:00.639 --> 1:02:04.079
Maybe I want you guys to think
for a second about what the
1:02:04.083 --> 1:02:07.313
recipe may be.
If a force came from a function
1:02:07.312 --> 1:02:11.182
U by taking derivatives,
as written up there,
1:02:11.179 --> 1:02:15.039
what can you say about the
components of that force,
1:02:15.040 --> 1:02:16.630
from the fact--yes?
1:02:16.630 --> 1:02:18.970
Student: What is the
cross derivative [inaudible]
1:02:18.968 --> 1:02:20.838
Professor Ramamurti
Shankar: Right,
1:02:20.840 --> 1:02:23.950
we know that for every function
U the cross derivatives
1:02:23.950 --> 1:02:26.500
are equal but the ordinary
derivatives are just the
1:02:26.500 --> 1:02:29.050
F_x and
F_y;
1:02:29.050 --> 1:02:34.100
therefore, d^(2)U over
dydx is really d
1:02:34.101 --> 1:02:37.441
by dy of
F_x.
1:02:37.440 --> 1:02:41.300
And I want that to be equal to
d by dx of
1:02:41.304 --> 1:02:45.604
F_y because
that would then be d^(2)U
1:02:45.597 --> 1:02:47.097
over dxdy.
1:02:47.099 --> 1:02:49.429
In other words,
if the force satisfies this
1:02:49.433 --> 1:02:51.433
condition, the y
derivative of
1:02:51.433 --> 1:02:53.993
F_x is the
x derivative
1:02:53.989 --> 1:02:55.489
F_y.
1:02:55.489 --> 1:02:58.809
Then, it has the right pedigree
to be a conservative force
1:02:58.809 --> 1:03:02.129
because if a force came from a
function U by taking
1:03:02.128 --> 1:03:04.928
derivatives,
the simple requirement of the
1:03:04.927 --> 1:03:09.047
cross derivatives are equal for
any function U tells me.
1:03:09.050 --> 1:03:11.090
See, if I take the y
derivative of
1:03:11.087 --> 1:03:13.627
F_x,
I'm taking d^(2)U over
1:03:13.634 --> 1:03:15.554
dxdy.
And here, I'm taking
1:03:15.554 --> 1:03:18.594
d^(2)U over dydx
and they must be equal.
1:03:18.590 --> 1:03:21.010
So, that's the diagnostic.
1:03:21.010 --> 1:03:24.150
If I give you a force and I ask
you, "Is it conservative?"
1:03:24.150 --> 1:03:25.910
you simply take the y
derivative of
1:03:25.912 --> 1:03:28.062
F_x and the
x derivative of
1:03:28.061 --> 1:03:30.251
F_y and if they
match you know it's
1:03:30.253 --> 1:03:32.763
conservative.
So, I can summarize by saying
1:03:32.757 --> 1:03:33.977
the following thing.
1:03:33.980 --> 1:03:36.500
In two dimensions,
there are indeed many,
1:03:36.500 --> 1:03:40.280
many forces for which the
potential energy can be defined.
1:03:40.280 --> 1:03:43.510
But every one of them has an
ancestor which is simply a
1:03:43.505 --> 1:03:46.185
function, not a vector,
but a scalar function,
1:03:46.193 --> 1:03:49.183
an ordinary function of
x and y.
1:03:49.179 --> 1:03:51.609
Then, the force is obtained by
taking x and y
1:03:51.610 --> 1:03:54.050
derivatives of that function;
the x derivative with a
1:03:54.051 --> 1:03:55.611
minus sign is called
F_x,
1:03:55.611 --> 1:03:57.971
and the y derivative is
called F_y.
1:03:57.969 --> 1:04:01.979
Okay, so let's take the most
popular example is the force of
1:04:01.980 --> 1:04:04.700
gravity near the surface of the
Earth.
1:04:04.699 --> 1:04:12.629
The force of gravity we know is
-mg times J.
1:04:12.630 --> 1:04:14.740
Is this guy conservative?
1:04:14.739 --> 1:04:17.169
Yes, because the x
derivative of this vanishes and
1:04:17.174 --> 1:04:19.614
the y derivative of
F_x you don't
1:04:19.608 --> 1:04:22.388
even have to worry because there
is no F_x;
1:04:22.390 --> 1:04:23.990
it's clearly conservative.
1:04:23.989 --> 1:04:26.729
Then you can ask,
"What is the potential U
1:04:26.728 --> 1:04:27.868
that led to this?"
1:04:27.869 --> 1:04:34.879
Well, dU/dy with a minus
sign had to be -mg and
1:04:34.883 --> 1:04:38.043
dU/dx had to be 0.
1:04:38.039 --> 1:04:42.239
So, the function that will do
the job is mgy.
1:04:42.239 --> 1:04:46.229
You can also have mgy +
96 but we will not add those
1:04:46.228 --> 1:04:49.488
constants because in the end,
in the Law of Conservation of
1:04:49.487 --> 1:04:51.657
Energy, K_1
+U_1 = K_2 +
1:04:51.663 --> 1:04:54.203
U_2,
adding a 96 to both sides
1:04:54.202 --> 1:04:55.652
doesn't do anything.
1:04:55.650 --> 1:04:59.420
So this means,
when a body's moving in the
1:04:59.419 --> 1:05:04.199
gravitational field ½
mv^(2) + mgy,
1:05:04.199 --> 1:05:10.409
before, is the same as ½
mv^(2) + mgy,
1:05:10.406 --> 1:05:11.506
after.
1:05:11.510 --> 1:05:16.970
1:05:16.969 --> 1:05:19.849
Now, you knew this already when
you're moving up and down the
1:05:19.845 --> 1:05:22.765
y direction but what I'm
telling you is this is true in
1:05:22.769 --> 1:05:24.989
two dimensions.
So, I'll give you a final
1:05:24.988 --> 1:05:27.708
example so you guys can go home
and think about it.
1:05:27.710 --> 1:05:29.550
That is a roller coaster.
1:05:29.550 --> 1:05:32.680
So, here's the roller coaster,
it has a track that looks like
1:05:32.684 --> 1:05:34.214
this.
This is x,
1:05:34.206 --> 1:05:36.926
this is the height of the
roller coaster,
1:05:36.925 --> 1:05:41.005
but at every x there's a
certain height so y is
1:05:41.005 --> 1:05:44.605
the function of x and it
looks like this.
1:05:44.610 --> 1:05:46.710
It's just the profile of the
roller coaster.
1:05:46.710 --> 1:05:49.870
But that is also the potential
energy U,
1:05:49.868 --> 1:05:52.888
because if you multiply this by
mg,
1:05:52.889 --> 1:05:56.529
well, you just scale the graph
by mg it looks the same.
1:05:56.530 --> 1:05:58.860
So, take a photograph of a
roller coaster,
1:05:58.857 --> 1:06:01.237
multiply the height in meters
by mg,
1:06:01.241 --> 1:06:04.251
it's going to still look like
the roller coaster.
1:06:04.250 --> 1:06:08.910
That is my potential energy for
this problem and the claim is
1:06:08.905 --> 1:06:13.865
that kinetic plus potential will
not change, so K + U is a
1:06:13.871 --> 1:06:17.151
constant;
let's call the constant
1:06:17.152 --> 1:06:22.462
E.
So, if a trolley begins here at
1:06:22.456 --> 1:06:27.326
the top, what is its total
energy?
1:06:27.329 --> 1:06:29.929
It's got potential energy equal
to the height,
1:06:29.927 --> 1:06:33.267
it's got no kinetic energy,
so total energy in fact is just
1:06:33.274 --> 1:06:36.174
its height.
And total energy cannot change
1:06:36.173 --> 1:06:38.413
as the trolley goes up and down.
1:06:38.409 --> 1:06:41.479
So, you draw a line at that
height and call it the total
1:06:41.477 --> 1:06:43.607
energy.
You started this guy off at
1:06:43.610 --> 1:06:46.630
that energy, that energy must
always be the same.
1:06:46.630 --> 1:06:50.000
What that means is,
if you are somewhere here,
1:06:49.999 --> 1:06:52.619
so that is your potential
energy,
1:06:52.619 --> 1:06:56.039
that is your kinetic energy,
it's very nicely read off from
1:06:56.039 --> 1:06:58.279
this graph, to reach the same
total.
1:06:58.280 --> 1:07:02.580
So, as you oscillate up and
down, you gain and lose kinetic
1:07:02.582 --> 1:07:04.872
and potential.
When you come here your
1:07:04.871 --> 1:07:07.751
potential energy is almost your
whole energy but you've got a
1:07:07.746 --> 1:07:09.276
little bit of kinetic energy.
1:07:09.280 --> 1:07:12.270
That means, your roller coaster
is still moving when it comes
1:07:12.274 --> 1:07:14.324
here with the remaining kinetic
energy.
1:07:14.320 --> 1:07:19.350
You can have a roller coaster
whose energy is like this.
1:07:19.350 --> 1:07:21.890
This is released from rest here.
1:07:21.890 --> 1:07:23.160
It is released from rest here.
1:07:23.159 --> 1:07:26.869
That's the total energy and
this total energy line looks
1:07:26.870 --> 1:07:29.950
like this.
That means, if you released it
1:07:29.950 --> 1:07:32.680
here, it'll come down,
pick up speed,
1:07:32.675 --> 1:07:34.805
slow down,
pick up speed again,
1:07:34.814 --> 1:07:37.764
and come to this point,
it must stop and turn around
1:07:37.759 --> 1:07:40.819
because at that point the
potential energy is equal to
1:07:40.820 --> 1:07:43.650
total energy and there is no
kinetic energy.
1:07:43.650 --> 1:07:45.630
That means you've stopped,
that means you're turning
1:07:45.632 --> 1:07:47.112
around, it'll rattle back and
forth.
1:07:47.110 --> 1:07:50.330
1:07:50.329 --> 1:07:52.639
By the way, according to laws
of quantum mechanics,
1:07:52.641 --> 1:07:53.891
it can do something else.
1:07:53.890 --> 1:07:54.770
Maybe you guys know.
1:07:54.769 --> 1:07:56.699
You know what else it can do if
it starts here?
1:07:56.700 --> 1:08:00.450
Yes?
Student: It can tunnel.
1:08:00.449 --> 1:08:02.719
Professor Ramamurti
Shankar: It can suddenly
1:08:02.723 --> 1:08:05.503
find itself here and that's not
allowed by classical mechanics
1:08:05.496 --> 1:08:07.766
because to do that it has to go
over the hump.
1:08:07.769 --> 1:08:09.589
Look what's happening at the
hump.
1:08:09.590 --> 1:08:13.020
I've got more kinetic energy
than I have total energy.
1:08:13.019 --> 1:08:15.989
I'm sorry, I've got more
potential energy than I have
1:08:15.992 --> 1:08:18.162
total energy.
That means kinetic energy is
1:08:18.161 --> 1:08:20.881
negative, that's not possible,
because ½ mv^(2) can
1:08:20.876 --> 1:08:21.856
never be negative.
1:08:21.859 --> 1:08:24.829
So, quantum theory allows these
forbidden processes and it's
1:08:24.832 --> 1:08:25.792
called tunneling.
1:08:25.789 --> 1:08:28.799
But for us, the roller coaster
problem, you will just turn
1:08:28.804 --> 1:08:31.184
around.
There's one thing I want you to
1:08:31.177 --> 1:08:32.827
think about before you go.
1:08:32.829 --> 1:08:36.119
You should not have simply
accepted the Law of Conservation
1:08:36.122 --> 1:08:39.302
of Energy in this problem
because gravity is not the only
1:08:39.300 --> 1:08:40.720
force acting on this.
1:08:40.720 --> 1:08:43.250
What else is acting on this
roller coaster?
1:08:43.250 --> 1:08:44.910
Student: No friction.
1:08:44.909 --> 1:08:45.479
Professor Ramamurti
Shankar: Pardon me?
1:08:45.480 --> 1:08:46.310
No friction, then what?
1:08:46.310 --> 1:08:47.970
Student: The normal
force.
1:08:47.970 --> 1:08:49.220
Professor Ramamurti
Shankar: The force of that
1:08:49.217 --> 1:08:50.817
track.
Look, if I didn't want to have
1:08:50.819 --> 1:08:53.119
anything but gravity,
here's a roller coaster ride
1:08:53.122 --> 1:08:55.522
you guys will love,
just push you over the edge.
1:08:55.520 --> 1:08:59.410
That conserves energy and it's
got no track and it's got the
1:08:59.410 --> 1:09:03.630
only gravitational force and you
can happily use this formula.
1:09:03.630 --> 1:09:05.340
Why am I paying all this money?
1:09:05.340 --> 1:09:07.370
Because there's another force
acting on it,
1:09:07.374 --> 1:09:09.364
but that force,
if it's not frictional,
1:09:09.359 --> 1:09:12.559
is necessarily perpendicular to
the motion of the trolley,
1:09:12.560 --> 1:09:15.760
and the displacement of the
trolley is along the track,
1:09:15.760 --> 1:09:17.590
so F.dr vanishes.
1:09:17.590 --> 1:09:20.260
So, I will conclude by telling
you the correct thing to do
1:09:20.264 --> 1:09:22.994
would be to say K_2
- K_1 is the
1:09:22.985 --> 1:09:26.435
integral of all the forces,
divided into force due to the
1:09:26.443 --> 1:09:28.933
track and the force due to
gravity.
1:09:28.930 --> 1:09:32.890
The force due to the track is 0.
1:09:32.890 --> 1:09:34.880
I mean, it's not 0,
but that dot product with
1:09:34.879 --> 1:09:37.189
dr would be 0,
because F and dr
1:09:37.185 --> 1:09:38.175
are perpendicular.
1:09:38.180 --> 1:09:40.650
For that reason,
you'd drop that and the force
1:09:40.653 --> 1:09:43.953
of gravity is cooked up so then
it becomes U_1 -
1:09:43.951 --> 1:09:46.481
U_2 and that's
what we used.
1:09:46.480 --> 1:09:52.220
1:09:52.220 --> 1:09:56.050
Okay, one challenge you guys
can go home and do this,
1:09:56.047 --> 1:09:59.357
take any force in two
dimensions, F;
1:09:59.360 --> 1:10:04.020
1:10:04.020 --> 1:10:06.980
it is parallel to the direction
where you are measured from the
1:10:06.982 --> 1:10:09.852
origin times any function of the
distance from the origin.
1:10:09.850 --> 1:10:13.320
1:10:13.319 --> 1:10:16.409
Show, convince yourself that
this force is a conservative
1:10:16.406 --> 1:10:19.706
force by applying the test I
gave you and the trick is to use
1:10:19.714 --> 1:10:22.254
x and y instead of
r.
1:10:22.250 --> 1:10:23.990
Take this force,
write it now in terms of
1:10:23.986 --> 1:10:26.236
x and y,
take the cross derivatives and
1:10:26.243 --> 1:10:27.723
you can see it's conservative.
1:10:27.720 --> 1:10:30.120
So, any radial force, yep?
1:10:30.119 --> 1:10:33.339
Student: [inaudible]
Professor Ramamurti
1:10:33.341 --> 1:10:34.921
Shankar: Oh,
here.
1:10:34.920 --> 1:10:37.330
This is the force to which you
should try your thing and
1:10:37.330 --> 1:10:39.040
gravity is a special example of
this.
1:10:39.040 --> 1:10:46.460
1:10:46.460 --> 1:10:50.500
Okay, judging from the class
reaction, and the stunned and
1:10:50.499 --> 1:10:53.049
shocked look,
a lot of you people,
1:10:53.050 --> 1:10:56.240
maybe this material is new,
so you should think about it,
1:10:56.244 --> 1:10:58.644
talk about it,
go to discussion section,
1:10:58.640 --> 1:11:01.910
but this is the level at which
you should understand energy
1:11:01.912 --> 1:11:04.002
conservation in a course like
this.