Transforming Velocities Between Rotating Coordinate Systems
Let
and
be coordinate systems with the same origin
with systemrotating with angular velocity
with respect to
Let
be the position vector of a particle in
then
In the coordinate system s,
are changing with time so
Since
so
and
are perpendicular. Similarly for
and
and
and
Hence
(1)
(2)
,
(3)
Differentiating
gives
(4)
Differentiating
gives
(5)
Differentiating
gives
(6)
Taking the dot product of (1) withgives
Taking the dot product of (2) withgives
Hence
Similarly
and
Hence
Let
then we can write
Then
