Previous:  Inertial Forces

In the previous sections, we have concentrated on coordinate systems and frames of reference that are separated, but parallel.  We saw that when one frame is accelerating relative to the other, so-called inertial forces appear in the coordinate system that is accelerating.  In this section, we will consider two coordinate systems that have coincident origins, but one of the coordinate systems is free to rotate relative to the other.

The following figure shows two such systems, with the primed system rotated relative to the unprimed system.  Throughout it will be assumed that the axis of rotation passes through the common origin of the two systems.

Now, consider the line segment OA in this figure.  This represents a vector in either coordinate system, and since the origins are coincident, the vector OA will be the same in both coordinate systems.  Hence we can write

as the representation of the vector OA in the unprimed and primed systems.

Now we can differentiate both sides of this equation with respect to time.  We need to bear in mind that the primed unit vectors on the right hand side of this equation may vary with time, under a continuous rotation of the primed system.  Differentiating and using the product rule on the right hand side of this equation gives

where

Respectively, these two expressions represent velocity as seen by observers in the fixed and rotating coordinate systems.

The problem now is to obtain the rate of change of the primed unit vectors with respect to time.  To do this, we first consider how to define rotation and angular velocity as a vector.  This is done by making the following definition:

That is, the angular velocity vector ω is defined to be of magnitude ω, and n is a unit vector that points along the axis of rotation.  For example, in the figure above, if the axis of rotation were along the z-axis, then the angular velocity vector would be ω = ωk.

The following figure illustrates how to compute the rate of change of a primed unit vector, noting that the derivative of a vector is itself a vector.

On the left, it can be seen that as a unit vector rotates about some axis of rotation, it generates a circle around the axis of rotation, and the plane of that circle is normal to the axis of rotation.  The rate of change of the unit vector is therefore normal to the angular velocity vector ω.

Now, suppose that in a time dt, a rotation angle dφ takes place, as shown in the figure.  The magnitude of the change in the unit vector is then approximately the length of the arc AB in the figure on the right, where the radius A represents the unit vector at some time, and the radius B represents the unit vector at some later time.  This length is easily computed in terms of the angle dφ and the angle of the unit vector to the angular velocity vector.  The answer is

The figure on the right also shows that the increment in the unit vector (the vector AB) is perpendicular to the unit vector itself, in the limit of a small rotation angle dφ.

Noting that dφ/dt = ω, we conclude that for some unit vector n, we have

since the magnitude of the right-hand side of the above, by definition, is

and the increment dn is perpendicular both to ω and to n.

Our earlier expression for velocity can now be written in the form

Writing out the vector products and rearranging leads to the following:

This is an equation of the utmost importance, as it relates the velocity of a particle in the unprimed coordinate system to the velocity as seen by an observer in the rotating system.  The two are clearly not equivalent, and the difference depends on the angular velocity of the rotating system.  In the next section, we will see how this expression leads to the appearance of the Coriolis and centrifugal forces in the study of rotating systems.

Next:  Coriolis and Centrifugal Forces