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PROFESSOR: Because
some of this course
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was shot in two different years,
two different notations systems
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00:00:28,370 --> 00:00:29,370
were used.
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00:00:29,370 --> 00:00:31,300
And I'm going to
explain both of them
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00:00:31,300 --> 00:00:33,750
so that when you encounter
them in the videos
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00:00:33,750 --> 00:00:37,000
of the recitations of the
videos of the lectures,
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00:00:37,000 --> 00:00:41,480
you'll be able to use either
of the notation systems
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00:00:41,480 --> 00:00:43,000
interchangeably.
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00:00:43,000 --> 00:00:44,630
So the two notation
systems would
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00:00:44,630 --> 00:00:52,460
refer to how we explain position
vectors, velocity vectors,
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and vectors of any
kind that might
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be associated with translating
and rotating reference frames.
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00:01:00,070 --> 00:01:03,010
So in this diagram,
I've got a rigid body.
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And attached to that rigid
body is a reference frame.
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I'll call it XYZ.
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Attached and moves
with the body.
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00:01:15,530 --> 00:01:19,360
And that's reference frame AXYZ.
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00:01:19,360 --> 00:01:24,170
And the whole system is
translating and rotating
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00:01:24,170 --> 00:01:32,780
in an inertial frame O, capital
X, capital Y, capital Z.
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00:01:32,780 --> 00:01:34,790
And I need to be
able to describe
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00:01:34,790 --> 00:01:39,180
the position and the
velocity of this rigid body,
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00:01:39,180 --> 00:01:41,540
and a point on this
rigid body, which I'll
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call B, which might
actually even be moving
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00:01:44,540 --> 00:01:46,610
with respect to the rigid body.
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00:01:46,610 --> 00:01:51,510
So the position of
this reference frame
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00:01:51,510 --> 00:01:55,870
in System I-- this is Notation
System I-- we designate
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00:01:55,870 --> 00:02:03,170
as R of A in reference frame
O. And the O is in superscript
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00:02:03,170 --> 00:02:06,750
that precedes the R.
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00:02:06,750 --> 00:02:11,920
Point B is R of B in
O. And this vector
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00:02:11,920 --> 00:02:16,500
is R of B with respect
to A. And we write it B.
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00:02:16,500 --> 00:02:20,060
And with respect to
A is a superscript.
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00:02:20,060 --> 00:02:22,780
So this essentially,
the superscript version
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00:02:22,780 --> 00:02:24,390
of the notation.
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00:02:24,390 --> 00:02:27,790
We do the same thing in
a slightly different way
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00:02:27,790 --> 00:02:32,154
in which we say with
respect to is a slash.
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00:02:32,154 --> 00:02:34,550
So A, with respect
to frame O, is
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00:02:34,550 --> 00:02:42,700
written RA/O. Point B
is RB/O. And the vector
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00:02:42,700 --> 00:02:46,050
that goes from A
to B is R of B/A.
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00:02:46,050 --> 00:02:50,450
So the two are
exactly equivalent.
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00:02:50,450 --> 00:02:53,050
And we'll go one step farther.
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00:02:53,050 --> 00:02:58,920
And that is to take the time
derivative of this vector B
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00:02:58,920 --> 00:03:06,220
and use it to derive expressions
for velocities in a rotating
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00:03:06,220 --> 00:03:08,340
and translating frame.
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00:03:08,340 --> 00:03:21,100
The position RB, for
example, using this notation,
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00:03:21,100 --> 00:03:25,880
with respect to O,
is RA with respect
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00:03:25,880 --> 00:03:30,320
to O plus RB with respect to A.
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00:03:30,320 --> 00:03:33,800
So it's just a vector sum,
this vector plus this vector
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00:03:33,800 --> 00:03:35,110
equals that vector.
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00:03:35,110 --> 00:03:41,460
And we want to take the time
derivative of this expression
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00:03:41,460 --> 00:03:45,890
for RB with respect to
O. And it will give us
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00:03:45,890 --> 00:03:50,460
an expression for the velocity
of point B with respect
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00:03:50,460 --> 00:03:52,960
to the O frame.
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00:03:52,960 --> 00:03:57,930
And here, I've written this
out in both notation systems.
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00:03:57,930 --> 00:04:03,490
So in the notation system where
we use slash O as the with
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00:04:03,490 --> 00:04:08,440
respect to, the velocity
of B, with respect to O,
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00:04:08,440 --> 00:04:11,590
would be the velocity
of that frame,
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00:04:11,590 --> 00:04:15,110
translational velocity
of A with respect to O,
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00:04:15,110 --> 00:04:21,600
plus the time derivative of the
vector RB with respect to A.
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00:04:21,600 --> 00:04:25,980
And that time derivative must
be taken in the inertial frame,
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00:04:25,980 --> 00:04:28,670
So /O.
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00:04:28,670 --> 00:04:32,400
This term expands
into two pieces.
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00:04:32,400 --> 00:04:38,300
So this is equal to, again,
V of A with respect to O,
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00:04:38,300 --> 00:04:45,030
but now a derivative of RBA,
with respect to the A frame.
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00:04:45,030 --> 00:04:48,220
This is as if you were
sitting on that rigid body.
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00:04:48,220 --> 00:04:51,600
Is that vector getting
any longer or shorter?
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00:04:51,600 --> 00:04:58,180
Plus the rotation, omega of
the body with respect to O
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00:04:58,180 --> 00:05:01,140
cross-product with RBA.
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00:05:01,140 --> 00:05:03,850
I've left out underscores
here to emphasize
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00:05:03,850 --> 00:05:05,440
that these are vectors.
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00:05:05,440 --> 00:05:07,650
But these are all vectors.
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00:05:10,560 --> 00:05:14,050
And in the alternative
notation system,
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00:05:14,050 --> 00:05:17,080
the /Os become superscripts.
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00:05:17,080 --> 00:05:19,960
So the velocity of B
and O is the velocity
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00:05:19,960 --> 00:05:25,260
of A with respect to
O plus the velocity
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00:05:25,260 --> 00:05:27,750
as seen from the
point of view of being
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00:05:27,750 --> 00:05:33,180
on the rigid body plus-- this
is the contribution to velocity
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00:05:33,180 --> 00:05:36,490
as seen in the
inertial frame caused
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00:05:36,490 --> 00:05:40,200
by the rotation of the body.
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00:05:40,200 --> 00:05:42,840
And that's the two
different notation systems.
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00:05:42,840 --> 00:05:47,530
And you'll see these used
on solutions to problems.
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00:05:47,530 --> 00:05:50,430
You'll see them in
either of the two
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00:05:50,430 --> 00:05:53,710
frames are either of the
two notation systems.
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00:05:53,710 --> 00:05:59,380
And you will see these notation
systems used in lecture
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00:05:59,380 --> 00:06:01,840
and in these recitations.