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PROFESSOR: Because some of this course
was shot in two different years, two different notations systems
were used.
And I'm going to explain both of them
so that when you encounter them in the videos
of the recitations of the videos of the lectures,
you'll be able to use either of the notation systems
interchangeably.
So the two notation systems would
refer to how we explain position vectors, velocity vectors,
and vectors of any kind that might
be associated with translating and rotating reference frames.
So in this diagram, I've got a rigid body.
And attached to that rigid body is a reference frame.
I'll call it XYZ.
Attached and moves with the body.
And that's reference frame AXYZ.
And the whole system is translating and rotating
in an inertial frame O, capital X, capital Y, capital Z.
And I need to be able to describe
the position and the velocity of this rigid body,
and a point on this rigid body, which I'll
call B, which might actually even be moving
with respect to the rigid body.
So the position of this reference frame
in System I-- this is Notation System I-- we designate
as R of A in reference frame O. And the O is in superscript
that precedes the R.
Point B is R of B in O. And this vector
is R of B with respect to A. And we write it B.
And with respect to A is a superscript.
So this essentially, the superscript version
of the notation.
We do the same thing in a slightly different way
in which we say with respect to is a slash.
So A, with respect to frame O, is
written RA/O. Point B is RB/O. And the vector
that goes from A to B is R of B/A.
So the two are exactly equivalent.
And we'll go one step farther.
And that is to take the time derivative of this vector B
and use it to derive expressions for velocities in a rotating
and translating frame.
The position RB, for example, using this notation,
with respect to O, is RA with respect
to O plus RB with respect to A.
So it's just a vector sum, this vector plus this vector
equals that vector.
And we want to take the time derivative of this expression
for RB with respect to O. And it will give us
an expression for the velocity of point B with respect
to the O frame.
And here, I've written this out in both notation systems.
So in the notation system where we use slash O as the with
respect to, the velocity of B, with respect to O,
would be the velocity of that frame,
translational velocity of A with respect to O,
plus the time derivative of the vector RB with respect to A.
And that time derivative must be taken in the inertial frame,
So /O.
This term expands into two pieces.
So this is equal to, again, V of A with respect to O,
but now a derivative of RBA, with respect to the A frame.
This is as if you were sitting on that rigid body.
Is that vector getting any longer or shorter?
Plus the rotation, omega of the body with respect to O
cross-product with RBA.
I've left out underscores here to emphasize
that these are vectors.
But these are all vectors.
And in the alternative notation system,
the /Os become superscripts.
So the velocity of B and O is the velocity
of A with respect to O plus the velocity
as seen from the point of view of being
on the rigid body plus-- this is the contribution to velocity
as seen in the inertial frame caused
by the rotation of the body.
And that's the two different notation systems.
And you'll see these used on solutions to problems.
You'll see them in either of the two
frames are either of the two notation systems.
And you will see these notation systems used in lecture
and in these recitations.