Vector Transformation in Rotating Coordinate Systems

Consider a vector rotating about an axis with an angular velocity such that its tip sweeps a circular path.

The time derivative of this vector equals its position change during an infinitesimal time interval. This expression is rewritten in terms of the cross product of the angular velocity and the vector.

Consider an arbitrary vector and its time derivative in an inertial frame. When observed from a rotating frame, the scalar components and the unit vectors change.

The vector's time derivative in the rotational frame yields two terms. The first term is the time derivative of the vector measured in the rotating frame.

As the unit vectors rotate, their rate of change expression is used to simplify the second term.

So, the time derivative of a vector in the inertial frame equals its time derivative in the rotating frame plus the cross product of the angular velocity and the vector.

The transformation equation for a position vector gives the velocity, while that for a velocity vector yields the acceleration in a rotating system.