Production of Centrifugal Force

I have a little more to say about theoretical aspects of centrifugal force, before looking at some practical cases next time.

A century-old textbook

A very good textbook, written during the First World War

I sometimes find it useful to check present-day concepts in science and technology against earlier ideas, to see what different insights those earlier writers had, even on apparently straightforward concepts like centrifugal force.

Nearly a century ago, Prof. Andrew Gray, F.R.S. of the University of Glasgow wrote what many think is still the best English-language textbook on gyroscopic theory:  A Treatise on Gyrostatics and Rotational Motion. It was republished by Dover Publications, Inc. in 1959.

A turning vector causes a time-rate production of itself

In discussing how turning the axis of a spinning gyroscope can cause a time-rate of production of angular momentum, i.e. a torque, Prof. Gray writes the following footnote (page 8):—
 

My interpretation of Prof. Gray's footnote on p8 of his book.
A turning vector (momentum) causes a time-rate of production of itself ( = force)

"For example, a particle of mass m is moving at any point P along a curve, that is along the tangent to the curve at P, and therefore the direction of motion is changing at P with angular speed v/r, where r is the radius of curvature at P. We may regard the tangent as an axis with which is associated the momentum mv, and which turns with angular speed v/r, as the point of contact advances along the curve. Thus along the direction towards the center of curvature at P, which direction is fixed for P, and towards which the tangent at P is turning, there is a rate of growth of momentum measured by mvω = mv.v/r = mv²/r, a very well-known result. The same process holds for any directed [i.e. vector] quantity (momentum, angular velocity, angular momentum, etc., associated with an axis)."

Reading this, I understood for the first time that the production of centrifugal force, considered as a time-rate of growth of momentum (d(mv)/dt = force) can be regarded as just a specific application of a much more general principle.