Rotating reference frames

(6.1) |

Suppose, now, that our object appears to move in the rotating reference frame with instantaneous velocity . It is fairly obvious that the appropriate generalization of the preceding equation is simply

Let and denote apparent time derivatives in the non-rotating and rotating frames of reference, respectively. Because an object that 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 , Equation (6.2) takes the form

or

because is a general position vector. Equation (6.4) 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 in Equation (6.4), we get

(6.5) |

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

Operating twice on the position vector with the time derivative in Equation (6.4), we obtain

(6.6) |

or

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

(6.8) |

Here, is the mass of our object, and is the (non-fictitious) force acting on it. These quantities are the same in both reference frames. Making use of Equation (6.7), the apparent equation of motion of our object in the rotating reference frame takes the form

The last two terms in this equation are so-called