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.