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hello
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let's recap week number nine of dynamics
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progressing fast indeed christmas is
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almost on the radar
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what we discussed last time was inertial
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forces we talked about moving and
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rotating frames and how in those frames
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inertial forces appear these corollas
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euler centrifugal forces and they give
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rise to gave rise to interesting
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dynamics
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what we discussed this week is that we
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also see torques being produced by these
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forces and so because of that we also
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need to talk about angular momentum
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balance in a rotating or moving frame
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that's exactly what we discussed here so
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this week's topic is amv
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in the moving and for us that usually
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means a rotating frame
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and
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the main equation is pretty much the
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same as before namely the net torque
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with respect to a certain point equals
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the rate of change of angular momentum
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plus potentially extra terms we made one
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simplification here that we want to
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stick to which is we always choose as
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the reference point b the center of mass
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or a fixed point
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that's what we would like to choose
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because this simplifies things
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so if this is the case then angular
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momentum balance reads how
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the net torque m with respect to point b
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equals what we've seen before is
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ib
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times omega these extra terms disappear
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because of that assumption now the catch
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is because we're in a moving frame i
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need to differentiate it with respect to
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time so d by dt but in a rotating frame
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this becomes
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the derivative
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seen by the rotating frame plus
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omega m the angular velocity of the
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rotating frame cross
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in this case this would again be ib
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times omega and this is angular momentum
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balance and this over here is nothing
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else but the angular velocity of the
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rotating frame so
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if you pick a reference frame area
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coordinates e1m
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e2 m and 3m
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e3m
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are moving this is the angular velocity
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of that frame
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now
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we discussed one
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important case namely we have one
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special frame the m frame is a moving
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one we have one special one that we
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highlighted this is what we call the
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body frame
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and we called it m hat because it's a
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special
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and this is the frame which satisfies
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two conditions first of all it rotates
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with the body of interest meaning that
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the angular velocity of the frame is the
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same as the angular velocity of the body
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that we're considering in this case
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these two guys will be the same just
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simplifying things
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and the second condition
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and this is important is that and
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this m hat is a principal frame
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remember
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every frame had a particular moment of
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inertia tensor it looked different in
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all kinds of frames but if we were in a
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principle frame meaning our axes align
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with the principal symmetry axis of the
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body then our ib moment of inertia
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tensor became diagonal
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and that's what we want to explore here
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so we want to seek a principle frame in
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which ib is diagonal and we make sure
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that we rotate with the body and if
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these two conditions are satisfied then
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in the body frame we derive what we know
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is the euler equations
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and what are these euler equations well
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i'm just going to write them down they
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look like this
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i 1 hat times omega 1 dot plus and here
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i have to look at the cheat sheet
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because i keep forgetting this myself i3
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minus i2 and then comes the two omegas
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with these two indices two and three and
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this is nothing else but the moment
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respect to
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point b
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around the one axis and the second one
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is i two hat omega two dot
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plus
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here i think they flip yes
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i one hat
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minus i three hat
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times omega one omega three and this is
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the net torque
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about the two axis and last not least we
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have i three hat
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times omega three dot plus
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and here we have i2 hat
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minus i1 hat times omega 1
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omega 2 and this is nothing else but the
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net torque
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about the three axis and these over here
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are the famous euler equations
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that's the main achievement
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of hours of this week
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now a few comments that we have to be
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aware of here these guys are twerks with
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respect to point b about the three axes
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but what's essential is that these guys
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are evaluated in the m frame so these
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over here
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are
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the components of this net torque
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evaluated in
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this m hat frame so we need to make sure
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that we get the components in the right
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frame
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next
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these
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eyes that we see over here these are
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nothing else but the diagonal values of
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my eye tensor we mentioned before that
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we're in the principle frame which means
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our moment of inertia tensor now looks
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like that i 1 hat
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i 2 hat my 3 hat
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and this is the moment of inertia tensor
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in the principle frame right
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so this is in the
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hat
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frame
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and then we have the omegas in here one
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comment about those these are nothing
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else but the angular velocities off the
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body and of the frame because they're
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the same but the important point is
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again these guys are the components of
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my omega evaluated in the m frame so if
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you have some general moment of angular
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velocity vector always make sure to
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compose it in the components in this m
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frame and here of course we're in the m
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hat frame and then last not least when
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you have these components we can also
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compute those guys here and these there
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are many different notations you can
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find in the notes on how to derive this
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please remember just the one thing if
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you know these omegas they are nothing
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else but their derivative so
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omega i
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dot for each of these three is nothing
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else but the time derivative
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of the omega i from over here so if you
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know the angular velocities all we need
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to do is take one time derivative and
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this gives you the omega dots over here
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okay
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now these are the euler equations which
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we can use and what they give us is
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three equations for you know rotations
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about the three axis what's special as
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compared to previous cases are these
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terms here in the middle because usually
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we have i times v double dot equals some
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torque and we principle half that if we
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consider this here as the angular
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acceleration we have i times angular
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acceleration equals torque and that
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looks as always but we have these extra
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terms that come in and that's because
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we're in a rotating frame
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what are those well think about it this
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way
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we will look at lmv and during a
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rotating frame we see not only the real
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forces but we also see inertial forces
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corollas euler centrifugal these are
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extra terms that come in as inertial or
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fictitious apparent forces because we're
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on the rotating frame and here it's
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exactly the same business we are in a
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rotating reference frame where these
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forces exist
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but whenever there are forces these
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forces can produce torques
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and so the euler centrifugal and
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coriolis forces can all produce torques
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on the system and that's exactly what
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these two extra terms are in a nutshell
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we've seen examples in class
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where these come in and we can actually
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interpret the extra terms as coming from
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for example coriolis or centrifugal
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forces if you use these equations we
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don't have to worry about that all we
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need are the three torque components in
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the moving frame
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the angular velocity and the moving
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frame the moment of inertia tensor
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components in that moving frame and the
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omega dots and if you have those we plug
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it in and there's no need to worry about
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real or inertial forces or whatnot we
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just use these equations
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now note one thing sometimes it is
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convenient to not use the body frame
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especially if we're an axis symmetric
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body for example one that is rotational
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symmetric right if you have a circle
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circular cross section
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and you're rotating about this axis you
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could pick
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an axis that looks like this or you
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could pick a coordinate system or you
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could pick one that rotates with the
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body
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what's special is because this is
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axosymmetric
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which means
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no matter which of the two ones you use
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the cross section always looks the same
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to you it doesn't change as a function
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of angle which means that we're still in
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the principle frame
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no matter how far and how we rotate
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we're always in the principle frame the
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only difference if we choose the green
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one here is that we're still in the
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principle frame but
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if we choose this green thing over here
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then our omega m
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is not necessarily the same as the omega
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of the body so if the omega the body is
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spinning
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the body is spinning with omega
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and your frame is spinning with omega m
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these two may not necessarily be the
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same
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and in this particular case we cannot
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use the euler equations but we can use
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the equation that led to the order
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equations and the only difference
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between this guy up here and these
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equations down here is that here we had
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to sneak in that capital omega as little
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omega
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this
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for the axosymmetric system i'm talking
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about simplified in the first term so
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this over here becomes
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this column
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and this over here
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is what becomes
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that column over here and so the only
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difference if you have to do it with a
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frame where capital omega m the frame of
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the rotating coordinate system is not
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the same as the body we need to replace
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this one which means here we're not
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going to have two little omegas we're
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going to have one little omega and one
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capital omega if you need that just take
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a look at the formula collection or the
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lecture notes where it's explained quite
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neatly what the equations look like in
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both cases i don't want to write them
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down here in a full glory okay
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so as a last comment where do we need
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this and why do we need it well
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we need it typically when a body is
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rotating but not about a principal axis
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so
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for example when we are rotating
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but not
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about a principal axis
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and the one example we showed in class
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was the cylinder
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that
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looks something like this imagine this
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is a cylinder with a circular cross
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section right it has an axis of symmetry
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here
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so we could easily draw
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for example a principle frame e1m
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e2m e3m out of the board now this is a
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principal frame for a cylinder
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now if you rotate about this axis
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fantastic in a principal frame you're
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done but what if the thing you know is
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supported like this
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and now you're rotating about this axis
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let's imagine this is how you're
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rotating
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that's exactly one of the scenarios we
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discussed in class
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or the other case we can think about the
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spinning tops which we also talked about
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so i'm bad at drawing but imagine that
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this
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was a spinning top kegel
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and what happens here is it stands on
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the ground but you know this point
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undergoes a circular motion while also
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spinning about its own axis
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this is actually symmetric so in this
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case of course you could easily say it's
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axis symmetric it's spinning about its
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axis so why not introduce a coordinate
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system here
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e1 m
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e2 m
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e3 m you know perpendicular to that
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and that makes things easy but again
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this thing is not only spinning about
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its own axis but it's also rotating
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about the central axis it's undergoing
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complicated motion going on a circular
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motion plus spinning plus possibly
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wobbling up and down that's what we call
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notation and so in all of these cases
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it's easy to find the principal axis but
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we're just not rotating about these
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principal axes right in this case it's
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even more complicated than there because
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there's some rotation component about
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that axis there's some rotation
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component about that axis
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we choose this as a reference frame we
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can easily do that because we align with
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the principal axis and if we're rotating
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with the body this could be a body frame
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we could do the same thing here if we're
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actually spinning and rotating with the
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body
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but
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these are rotating reference frames
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they're moving this is not standing
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still and that's why we need the euler
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equations
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let me just close with one quick remark
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because we did discuss one special case
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and this special case
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is what happens if you have something
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like this that spins very fast so
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something is fast spinning
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and accessometric
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wheels spinning tops all these kind of
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things
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then what we concluded for those was we
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can use what we call the tsp rule and
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this means that
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we call the precession rate across
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the spin rate both being vectors of
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angular velocity
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equals approximately the applied torque
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divided by
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the
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moment of inertia i3 about the spin axis
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this is the key equation i think my pen
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is giving up soon
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and this is what we know is the tsp rule
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what we need for this is okay this is
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the torque that produces
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the motion
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the precession of the object this over
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here is nothing else but the spin
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so in this particular example it's
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nothing else but the spinning off the
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object about its own axis so the spin
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around this axis is what we call c dot
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and these feed out is what we call the
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precession for a steady precession what
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this means is that we're also undergoing
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a rotation
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with the whole object about this axis
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over here
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and this denotes my
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feed dot so the rotation
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about this guy
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and what this equation tells me is that
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if i take this precession v dot
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i cross it into the spin rate c dot
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then the cross part of those which in
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this case comes out of the plane goes
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towards you this must be indicating the
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net torque
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with some normalization constant that
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comes from the moment of inertia and
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this we can interpret physically because
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what actually causes the procession here
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precession
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cross spin equals torque the torque
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comes out of the plane which means it
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must be a torque that somehow tries to
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drag this thing down
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and this is exactly what gravity is
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doing here so the gravitational force is
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dragging us down so for example with
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respect to point o down here which is
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fixed on the ground
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this thing produces a torque and this is
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why if you try to spin it it's not going
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to stand still but it's going to proceed
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it's going to do the circular dance
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if you want to spin the other way around
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you must still satisfy that equation the
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only way for this to work with the same
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torque is you must also change the
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orientation of your feed dot and then
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minus and minus gives the same again
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which means
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if you spin it the other way around it's
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not going to do the same dance but it's
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going to dance in the opposite direction
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as a consequence of the tsp
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so this is the spin that's what we
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called the
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precession and so whenever from now you
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see something spinning fast we don't
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necessarily have to use the full-blown
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glorious boiler equations but we can use
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the tsp rule and that's very often quite
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handy if you just want to figure out you
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know what direction is the torque
00:16:08
is is the torque acting onto a rotating
00:16:11
system
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and that's pretty much it for week nine
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we've arrived more or less at the end of
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our rotating non-inertial frame
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discussions
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and we have plenty of nice examples in
00:16:22
the exercises on rotating frames and a
00:16:25
and b in those
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thanks and ciao
