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

OF THE INTRODUCTORY CHAPTER

49

But at the start w is very big, though it gradually diminishes
under the action of the friction, and also the resistance of the
air.
Hence, at the start we have
aw > AQ,..............................(i)
i.e. the top is skidding and P is moving into the paper; friction
accordingly acts in a direction which hurries the precession,
and the top therefore rises.
In the meantime w is diminishing (and also a as the top gets
more vertical), and after a time we have
ca = A,..............................(ii)
i.e. P has no motion relative to the table, and steady rolling
motion becomes possible.
And, later, we have (a < A2,..............................(iii)
i.e. friction is retarding" the precession and the top begins to
fall.
Were it not for the resistance of the air, the state of steady
motion, when once established, would, theoretically, continue
indefinitely: for as soon as aw becomes = A2 rolling commences,
which can only be disturbed by air resistance. Immediately
aw becomes < A2 friction acts so as to increase w, but never
beyond the point necessary to maintain steady motion, since
it is a self-adjusting force. Hence, the position of steady
motion is a stable one, for friction produces an oscillating
tendency to return to it; but eventually, the resistance of the
air, tending to destroy this state, overcomes the tendency of
friction to restore it.
[13.] Even in vacuo the top would not really spin for ever. What is the
explanation of this?
[14.] In Fig. xvII. why must the gyroscope be placed on a smooth
surface?
[15.] The rougher the table the more quickly the top will rise when first
spun. Why?
[16.] Given a rough table, why does a top with a blunt peg rise more
quickly than one with a fine peg?
[17.] Explain the apparent contradiction of the tight and loose screw in
the top of Figs. iv. (a) and iv. (b)..)l4.
[18.] Supposing our object in spinning a top is to make it "go for as long
as possible," how would
(i) a blunt peg, (ii) a fine peg,
contribute to length of time?
[19.] How would the centre of gravity behave if the table were perfectly
smooth?
*It will be shown in Chapter IV. that, as the top dies, though friction is
resisting precession, yet owing to other causes, Q increases. Amongst these
is the fact that the gravity couple acts on a longer arm.
D