PC1672 Advanced dynamics

2.2 Rotating frames

If we choose to work in a frame which is rotating at angular velocity $\omega$ (which for simplicity we will chose to be constant), then for any vector ${\bf Q}$ the rate of change which we observe will differ from the rate of change seen by an inertial observer:

$\begin{displaymath}\left({{\rm d}{\bf Q}\over{\rm d}t}\right)_{\! 0}=\left({{\rm... ...{\rm d}t}\right)_{\! r}+\hbox{\boldmath $\omega$}\times {\bf Q}\end{displaymath}$

where the subscript

refers to the inertial frame and

to the rotating one. The term $\hbox{\boldmath$\omega$ }\times {\bf Q}$ is the rate at which the vector is carried round by the rotation.

For an object with position ${\bf x}$ , its velocity in the inertial frame ${\bf v}_0$ is related to its velocity ${\bf v}_r$ in the rotating frame by

$\begin{displaymath}{\bf v}_0={\bf v}_r+\hbox{\boldmath $\omega$}\times {\bf x}\end{displaymath}$

The acceleration in the inertial frame is related to that in our rotating frame by

$\begin{displaymath}{\bf a}_0={\bf a}_r+2\hbox{\boldmath $\omega$}\times {\bf v}_... ...dmath $\omega$}\times (\hbox{\boldmath $\omega$}\times {\bf x})\end{displaymath}$

In the rotating frame Newton's second law becomes

$\begin{displaymath}m{\bf a}_r={\bf F}-2m\hbox{\boldmath $\omega$}\times {\bf v}_... ...dmath $\omega$}\times (\hbox{\boldmath $\omega$}\times {\bf x})\end{displaymath}$

There are two inertial forces in this case. The first, which depends on the object's velocity in the rotating frame, is the Coriolis force. The second, which depends on the object's position, is the centrifugal force.

Some instructive movies of motion from inertial and rotating points of view have been produced by P. Flament and coworkers (University of Hawaii).

Centrifugal force

The centrifugal force

$\begin{displaymath}{\bf F}_{\rm cent}=-m\hbox{\boldmath $\omega$}\times (\hbox{\boldmath $\omega$}\times {\bf x})\end{displaymath}$

is the more familiar of these two inertial forces. When the double vector product is unpacked, it gives of force pointing radially outwards from the axis of rotation, with magnitude $mr\omega^2$ where

is the distance from the axis. This contains $r\omega^2$ which is just the centripetal acceleration of an object moving in a circle. (To keep an object moving in a circle we need to provide a real centripetal force that exactly cancels the centrifugal force which appears to act on it in the rotating frame.)

Coriolis force

The Coriolis force

$\begin{displaymath}{\bf F}_{\rm Cor}=-2m\hbox{\boldmath $\omega$}\times {\bf v}_r\end{displaymath}$

is more subtle in its effects. It is proportional to the object's velocity and it always acts perpendicularly to the velocity (and to the axis of rotation). In a frame rotating right-handedly (for example a turntable spinning anticlockwise when seen from above), the Coriolis force tries to deflect the motion of an object to the right. In a left-handed (clockwise) frame, the Coriolis deflection is to the left.

Textbook references

K&K 8.5, 7.2 example 7.1
F&C 5.2, 5.3
B&O 7.1, 7.2
M 11.1, 11.2
M&T 10.1, 10.2

Mike Birse
23rd March 2001