An elementary treatment of the theory of spinning tops and gyroscopic motion, by Harold Crabtree.

VELOCITIES IN TWO DIMENSIONS

111

And similarly,
v=velocity of P in direction 0 Y
=velocity of P relative to N in this direction+that of N
= +xO.
It will thus be seen that the total velocities of P in the
directions OX, OY are not merely dt and dc' but depend
also on 0.
Cor. If y=0, x and 0 become the polar coordinates of the
point P. Hence, with the more usual notation, we see that
the radial velocity is ir,
and the transversal velocity is rO.
115. Accelerations. We have above found u and v, the rates
of change, in definite directions, of x and y, subject to certain
conditions. To find the total accelerations in the same directions
we must find the rates of change of u and v in these directions,
subject to the same conditions.
Method I. The component of velocity in the direction OX
after time St is (u + 6u) cos 60 - (v + 6v) sin 60.
Hence the acceleration in direction OX
Lt (u + 86) cos 60-(v + 6v) sin 60-i
8t:0 et
= it - An -
=-u-v0.
Similarly, acceleration in direction O Y = v + i.
116. Method II. These expressions may also be obtained as
follows: The velocities of the moving point P at any instant in
the directions OX, O Y, considered fixed in space, can be
represented by = ON and v= QN,
Y Q
0
FIG. 66.
but it must be remembered that Q does not represent the
position of the moving point.