In
order
to
determine
the
motion
of
a
point
P
in
the
body,
we
consider
a
second
set
of
axes
x
y
,
always
parallel
to
xy
,
with
origin
at
O
,
and
write,
r
P
=
r
O
+
r
P
(1)
v
P
=
v
O
+ (
v
P
)
O
(2)
a
P
=
a
O
+ (
a
P
)
O
.
(3)
Here,
r
P
,
v
P
and
a
P
are
the
position,
velocity
and
acceleration
vectors
of
point
P
,
as
observed
by
O
;
r
O
is
the
position
vector
of
point
O
;
and
r
, (
v
P
)
O
and
(
a
P
)
O
are
the
position,
velocity
and
acceleration
P
vectors
of
point
P
,
as
observed
by
O
.
Relative
to
point
O
,
all
the
points
in
the
body
describe
a
circular
orbit
(
r
P
=
constant),
and
hence
we
can
easily
calculate
the
velocity,
(
v
P
)
O
=
r
P
θ
˙
=
r
ω
,
or,
in
vector
form,
(
v
P
)
O
=
ω
×
r
,
P
where
ω
is
the
angular
velocity
vector.
The
acceleration
has
a
circumferential
and
a
radial
component,
((
a
P
)
O
)
θ
=
r
θ
¨
=
r
˙
((
a
P
)
O
)
r
=
−
r
P
θ
˙
2
P
ω
2
.
P
P
ω
,
=
−
r
Noting
that
ω
and
ω
˙
are
perpendicular
to
the
plane
of
motion
(i.e.
ω
can
change
magnitude
but
not
direction),
we
can
write
an
expression
for
the
acceleration
vector
as,
(
a
P
)
O
=
ω
˙
×
r
P
)
.
P
+
ω
×
(
ω
×
r
Recall
here
that
for
any
three
vectors
A
,
B
and
C
,
we
have
A
×
(
B
×
C
) = (
A C
)
B
−
(
A B
)
C
.
Therefore
·
·
ω
×
(
ω
×
r
) = (
ω
r
)
ω
−
ω
2
r
=
−
ω
2
r
.
Finally,
equations
2
and
3
become,
P
P
P
P
·
v
P
=
v
O
+
ω
×
r
(4)
P
a
P
=
a
O
+
ω
˙
×
r
P
)
.
(5)
P
+
ω
×
(
ω
×
r
Body
Axes
An
alternative
description
can
be
obtained
using
body
axes.
Now,
let
x
y
be
a
set
of
axes
which
are
rigidly
attached
to
the
body
and
have
the
origin
at
point
O
.