S.
Widnall,
J.
Peraire
16.07
Dynamics
Fall
2009
Version
2.0
Lecture
L8
 Relative
Motion
using
Rotating
Axes
In
the
previous
lecture,
we
related
the
motion
experienced
by
two
observers
in
relative
translational
motion
with
respect
to
each
other.
In
this
lecture,
we
will
extend
this
relation
to
another
type
of
observer.
That
is,
an
observer
who
rotates
relative
to
a
stationary
observer.
Later
in
the
term,
we
will
consider
the
more
general
case
of
a
translating
(includes
accelerating)
and
rotating
observer
as
well
as
multiple
observers
who
may
be
translating
and
rotating
with
respect
to
one
another.
Although
governed
by
the
same
equations,
it
is
useful
to
distinguish
two
cases:
first
is
the
rotating
observer
that
observes
a
phenomena
that
a
stationary
observer
would
categorize
as
nonaccelerating
motion
such
as
a
fixed
object
or
one
moving
at
constant
velocity
(and
direction).
Such
an
observer
might
be
a
child
riding
on
a
merrygoround
or
playground
turntable.
Such
a
rotating
observer
would
observe
a
fixed
object
or
one
with
constant
velocity
motion
as
tracing
a
curved
trajectory
with
changing
velocity
and
therefore
accelerating
motion.
Our
transformation
from
the
rotating
into
the
fixed
system,
must
show
that
the
observed
motion
was
nonaccelerating
and
thus
forcefree
by
Newton’s
law.
We
should
also
be
able
to
describe
the
motion
seen
by
the
rotating
observer.
The
second
case
is
where
a
physical
phenomena
takes
place
on
a
platform
that
is
rotating,
such
as
a
mass
spring
system
fixed
to
a
turntable.
In
this
case,
the
motion,
the
forces
and
the
frequency
of
oscillation
will
be
a
ff
ected
by
the
rotation
and
will
di
ff
er
from
the
system
behavior
in
an
inertial
coordinate
system.
Our
approach
must
demonstrate
this
di
ff
erence
and
provide
the
tools
for
analyzing
the
system
dynamics
in
the
rotating
frame.
As
a
matter
of
illustration,
let
us
consider
a
very
simple
situation,
in
which
a
point
a
at
rest
with
respect
to
the
fixed
observer
A
located
at
the
origin
of
the
coordinates
x, y, z
,
point
O
,
is
also
observed
by
a
rotating
observer,
B
,
who
is
also
located
at
point
O
.
The
coordinate
system
used
by
B
,
x
, y
, z
is
instantaneously
aligned
with
x, y, z
but
rotating
with
angular
velocity
Ω
.
In
a),
the
coordinate
system
x, y, z
and
x
, y
, z
are
shown
slightly
o
ff
set
for
emphasis.
1
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Consider
first
case
shown
in
a)
,
in
which
the
point
a
located
at
position
r
does
not
move
in
inertial
space.
Observer
A
will
observe
v
a
=
0
for
the
point
r
.
Now
consider
the
same
situation
observed
by
B
.
In
the
rotating
coordinate
system,
B
will
observe
that
the
point
r
,
which
is
fixed
in
inertial
space
appears
to
move
backwards
due
to
the
rotation
of
B
s
coordinate
system.
v
a/B
=
−
Ω
×
r
.
(1)
To
reconcile
these
two
observations,
it
is
clear
that
to
obtain
the
velocity
in
inertial
coordinates,
A
must
correct
for
the
”artificial”
velocity
seen
by
B
due
to
the
rotation
of
B
s
coordinate
system
by
adding
Ω
×
r
to
the
velocity
reported
by
B
.
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 Velocity, observer, Frame of reference, Polar coordinate system

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