Rotating Reference Frames

(406) |

(407) |

Let and and denote apparent time derivatives in the non-rotating and rotating frames of reference, respectively. Since an object which is
stationary in the rotating reference frame appears to move in the non-rotating
frame, it is clear that
. Writing the apparent velocity, ,
of our object in the rotating reference frame as , the above
equation takes the form

since is a general position vector. Equation (409) expresses the relationship between apparent time derivatives in the non-rotating and rotating reference frames.

Operating on the general position vector with the time derivative (409), we get

(410) |

Operating twice on the position vector with the time
derivative (409), we obtain

(411) |

This equation relates the apparent acceleration, , of an object with position vector in the non-rotating reference frame to its apparent acceleration, , in the rotating reference frame.

Applying Newton's second law of motion in the inertial (*i.e.*, non-rotating) reference frame, we obtain

(413) |

The last two terms in the above equation are so-called ``fictitious forces''. Such forces are always needed to account for motion observed in non-inertial reference frames. Note that fictitious forces can always be distinguished from non-fictitious forces in Newtonian dynamics because the former have no associated reactions. Let us now investigate the two fictitious forces appearing in Equation (414).