We shall now present a rigorous derivation of centrifugal force, starting with Newton's second law. Our tool will be vector calculus.

A bonus of this derivation will be an expression for the Coriolis force, which is an inertial force that arises for rotational motion, as viewed in suitably chosen rotating reference frames when the linear distance from the rotation axis changes.

Newton's second law in vector notation with respect to an inertial reference frame is

F is the applied force acting on a particle of mass m; a is the particle's acceleration; and subscript i refers to the inertial reference frame.

We wish to express Newton's second law in a reference frame that rotates uniformly and with angular velocity w (units of radians per second) relative to our inertial reference frame. This is accomplished by applying the coordinate transformation

first to the radius vector r from the origin of the rotating reference frame to the position of the particle under consideration:

The symbol x stands for the cross product. v_i is the particle's velocity in the inertial reference frame and v_r is its velocity in the rotating reference frame (subscript r refers to the rotating reference frame).

Upon applying the coordinate transformation a second time, to v_i, we obtain an expression for acceleration a_i in the rotating reference frame:

Substituting this expression into Newton's second law with respect to an inertial reference frame and rearranging terms yields:

This equation is Newton's second law in the rotating reference frame. The expression a_r is the acceleration the particle experiences in that frame, which arises from the effective force

The first term on the right, F_i, is the applied force in the inertial reference frame, which we defined above. The second and third terms are forces that arise in the rotating reference frame: