5Rotating frames
IA Dynamics and Relativity
5.1 Motion in rotating frames
Now suppose that
S
is an inertial frame, and
S
0
is rotating about the
z
axis
with angular velocity ω =
˙
θ with respect to S.
Definition
(Angular velocity vector)
.
The angular velocity vector of a rotating
frame is ω = ω
ˆ
z, where
ˆ
z is the axis of rotation and ω is the angular speed.
First we wish to relate the basis vectors
{e
i
}
and
{e
0
i
}
of
S
and
S
0
respectively.
Consider a particle at rest in S
0
. From the perspective of S, its velocity is
dr
dt
S
= ω × r,
where
ω
=
ω
ˆ
z
is the angular velocity vector (aligned with the rotation axis).
This formula also applies to the basis vectors of S
0
.
de
0
i
dt
S
= ω × e
0
i
.
Now given a general time-dependent vector
a
, we can express it in the
{e
0
i
}
basis
as follows:
a =
X
a
0
i
(t)e
0
i
.
From the perspective of
S
0
,
e
0
i
is constant and the time derivative of
a
is given
by
da
dt
S
0
=
X
da
0
i
dt
e
0
i
.
In
S
, however,
e
0
i
is not constant. So we apply the product rule to obtain the
time derivative of a:
da
dt
S
=
X
da
i
dt
e
0
i
+
X
a
0
i
ω × e
0
i
=
da
dt
S
0
+ ω × a.
This key identity applies to all vectors and can be written as an operator equation:
Proposition.
If
S
is an inertial frame, and
S
0
is rotating relative to
S
with
angular velocity ω, then
d
dt
S
=
d
dt
S
0
+ ω × .
Applied to the position vector r(t) of a particle, it gives
dr
dt
S
=
dr
dt
S
0
+ ω × r.
We can interpret this as saying that the difference in velocity measured in
the two frames is the relative velocity of the frames.
We apply this formula a second time, and allow
ω
to depend on time. Then
we have
d
2
r
dt
2
S
=
d
dt
S
0
+ ω×
dr
dt
S
0
+ ω × r
.
=
d
2
r
dt
2
S
0
+ 2ω ×
dr
dt
S
0
+
˙
ω × r + ω × (ω × r)
Since S is inertial, Newton’s Second Law is
m
d
2
r
dt
2
S
= F.
So
Proposition.
m
d
2
r
dt
2
S
0
= F − 2mω ×
dr
dt
S
0
− m
˙
ω × r − mω × (ω × r).
Definition
(Fictious forces)
.
The additional terms on the RHS of the equation
of motion in rotating frames are fictitious forces, and are needed to explain the
motion observed in S
0
. They are
– Coriolis force: −2mω ×
dr
dt
S
0
.
– Euler force: −m
˙
ω × r
– Centrifugal force: −mω × (ω × r).
In most cases, ω is constant and can neglect the Euler force.