Rotating reference frames
Suppose that a given object has position vector in some inertial (i.e., nonrotating) reference frame. Let us observe the motion of
this object in a noninertial reference frame that rotates with constant angular
velocity
about
an axis passing through the origin of the inertial frame. Suppose, first of all, that our object appears stationary in the rotating reference frame. Hence, in the nonrotating frame,
the object's position vector will appear to precess about the origin with
angular velocity
. It follows from Section A.7
that, in the nonrotating reference frame,
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 nonrotating and rotating frames of reference, respectively. Because an object that is
stationary in the rotating reference frame appears to move in the nonrotating
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 nonrotating and
rotating reference frames.
Operating on the general position vector with the time derivative in Equation (6.4), we get
This equation relates the apparent velocity,
, of an object with
position vector in the nonrotating 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 nonrotating reference frame to its
apparent acceleration,
, in the rotating reference frame.
Applying Newton's second law of motion in the inertial (i.e., nonrotating) reference frame, we obtain

(6.8) 
Here, is the mass of our object, and is the (nonfictitious) 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 socalled fictitious forces; such forces
are always needed to account for motion observed in noninertial reference
frames. Fictitious forces can always be distinguished from
nonfictitious forces in Newtonian mechanics because the former
have no associated reactions.
Let us now investigate the two fictitious forces appearing in Equation (6.9).