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

30

ROTATION ABOUT A FIXED AXIS

(ii) the average pressure between the teeth of the cone and those of
the axle.
(iii) If the wheel continues spinning for 30 minutes, find the mean
resisting couple due to air and friction.11. A small top weighing 3 oz., is so constructed that on its head is a
series of small slanting flanges arranged like the sails of a windmill. The
object of these is that the spinning of the top may be prolonged by blowing
down on the flanges. Supposing that it is given an initial spin of
7 revolutions a sec., in the direction in which blowing assists, find the
value of the couple due to friction at the toe and the resistance of the
air when it is kept spinning uniformly under the following conditions:
There are n flanges each of area A sq. ins. inclined to the vertical at
an angle of 0~, the blowing pressure is P lbs. to the square in., and the
distance of the centre of pressure of each flange from the axis = 1 gsSupposing the initial spin is given in the wrong (i.e. opposite) direction,
and the same blowing force applied, how soon will it be brought to rest,
taking into account the resistance of air and friction, assuming their mean
value to be half of that in the previous part of the question, and that the
radius of gyration of the top is k ins.?
12. Fig. 12 represents a top which is spun in the following manner. The
spindle projecting from the head of the top works freely in a little cap to
be held in one hand, while the other pulls a short string of about 4 ins.
length, one end of which is fixed in the spindle. The string is pulled out,
allowed to slack and wind again on the spindle in the reverse direction.
The process is repeated, and after two or three pulls the top seems to have
its maximum rotation, for it starts humming. Why should the rotation be
greater after, say, three pulls than after one pull?
FIG. 12. FIG. 13.
13. In another top (Fig. 13) the spindle projects from its mounting, which
can be held without the top's rotation being affected; one end of the string
is looped once round the spindle and this end is pulled till the whole
string has been pulled over the spindle. Taking the tension in one part of
the string to be e20' x the tension in the other part, where /~ is the coefficient
of friction and 27r the angle through which the string is wrapped, how does
this method of spinning with a string compare with the more ordinary
one?
* Further information on the steering of a torpedo is given in Chapter V.
For the information there contained, and for the above approximate measurements,
the author is indebted to the Admiralty authorities and to the courtesy of the
Assistant Superintendent of H.M. Torpedo Factory, Woolwich Arsenal.