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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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were used.

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And I'm going to
explain both of them

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so that when you encounter
them in the videos

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of the recitations of the
videos of the lectures,

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you'll be able to use either
of the notation systems

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interchangeably.

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So the two notation
systems would

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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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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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And that's reference frame AXYZ.

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And the whole system is
translating and rotating

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in an inertial frame O, capital
X, capital Y, capital Z.

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And I need to be
able to describe

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the position and the
velocity of this rigid body,

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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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with respect to the rigid body.

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So the position of
this reference frame

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in System I-- this is Notation
System I-- we designate

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as R of A in reference frame
O. And the O is in superscript

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that precedes the R.

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Point B is R of B in
O. And this vector

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is R of B with respect
to A. And we write it B.

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And with respect to
A is a superscript.

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So this essentially,
the superscript version

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of the notation.

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We do the same thing in
a slightly different way

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in which we say with
respect to is a slash.

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So A, with respect
to frame O, is

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written RA/O. Point B
is RB/O. And the vector

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that goes from A
to B is R of B/A.

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So the two are
exactly equivalent.

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And we'll go one step farther.

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And that is to take the time
derivative of this vector B

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and use it to derive expressions
for velocities in a rotating

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and translating frame.

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The position RB, for
example, using this notation,

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with respect to O,
is RA with respect

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to O plus RB with respect to A.

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So it's just a vector sum,
this vector plus this vector

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equals that vector.

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And we want to take the time
derivative of this expression

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for RB with respect to
O. And it will give us

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an expression for the velocity
of point B with respect

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to the O frame.

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And here, I've written this
out in both notation systems.

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So in the notation system where
we use slash O as the with

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respect to, the velocity
of B, with respect to O,

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would be the velocity
of that frame,

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translational velocity
of A with respect to O,

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plus the time derivative of the
vector RB with respect to A.

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And that time derivative must
be taken in the inertial frame,

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So /O.

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This term expands
into two pieces.

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So this is equal to, again,
V of A with respect to O,

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but now a derivative of RBA,
with respect to the A frame.

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This is as if you were
sitting on that rigid body.

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Is that vector getting
any longer or shorter?

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Plus the rotation, omega of
the body with respect to O

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cross-product with RBA.

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I've left out underscores
here to emphasize

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that these are vectors.

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But these are all vectors.

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And in the alternative
notation system,

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the /Os become superscripts.

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So the velocity of B
and O is the velocity

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of A with respect to
O plus the velocity

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as seen from the
point of view of being

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on the rigid body plus-- this
is the contribution to velocity

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as seen in the
inertial frame caused

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by the rotation of the body.

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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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You'll see them in
either of the two

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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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and in these recitations.