Frame of reference rotating with a constant angular velocity two-dimensional case

Frame of reference rotating with a constant angular velocity (two-dimensional case) 

Assume that origins of two Cartesian systems of coordinates are located at the same point and the frame of reference P rotates about a point 0 of the frame P with constant angular velocity co. Let us imagine two planes, one above another, so that the upper plane P rotates and, correspondingly, unit vectors iiand ji change their direction, Fig. 2.2b. Consider an arbitrary point p, which has coordinates x, y on the plane P and xi, yi on P, and establish relationships between these pairs of coordinates. For the radius vector of the point p in both frames we have 

If a change of the angle 6 is very small, then the unit vector and its change are perpendicular to each other. For instance, for the unit vector ii we have 

This equation establishes the relationship between the velocities in both frames of reference. Performing one more differentiation we obtain for the acceleration in the inertial system P  

We see that the acceleration in the inertial frame P can be represented in terms of the acceleration, components of the velocity and coordinates of the point p in the rotating frame, as well as the angular velocity. This equation is one more example of transformation of the kinematical parameters of a motion, and this procedure does not have any relationship to Newton s laws. Let us rewrite Equation (2.37) in the form 

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