Mathematics of Physics and Engineering

Uniform Rotation of Frames 51

circle of radius R with uniform angular velocity u (Figure 2.1.4). Denote by

r(t) the position vector of the point at time t and assume that the origin O

of the frame is chosen so that ||r|| does not change in time. How to express

r(t) in terms of r(t) and w?

Fig. 2.1.4 Rotating Point

To solve this problem, consider the plane that contains the circle of
rotation and define the rotation vector u> as follows (see Figure 2.1.4).
The vector u> is perpendicular to the plane of the rotation; the direction
of the vector u> is such that the rotation is counterclockwise as seen from
the tip of the vector (alternatively, the rotation is clockwise as seen in the
direction of the vector); the length of the vector u: is w, the angular speed
of the rotation. As seen from Figure 2.1.4, r(t) = r(t) + r and the vector
r does not change in time, so that r(t) = r'(t). Consider a cartesian
coordinate system (?, j) with the origin at the center O of the circle in the
plane of rotation so that r(t) = Rcoscjti + Rsinwtj. Direct computations
show that

r'(t) = (ixj)x («f(<)) = iv x r{t) = w x r(t), (2.1.17)

r{t) = LJ x r[t). (2.1.18)

EXERCISE 2.1.6.c Verify all equalities in (2.1.17).
We will now use (2.1.18) to derive the relation between the velocities
and accelerations of a point mass m relative to two frames O and Oi,
when the frame 0\ is rotating with respect to O. We assume that the
two frames have the same origins: O = 0. We also choose cartesian
coordinate systems (i, J, k) and (£i, jlt ki) in the frames; see Figure 2.1.9