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

100

CONDITIONS OF STABILITY

Relation (a) gives the upper limit of oscillation, i.e. the original
position at which the top was spun (see Art. 98).
Relation (f) gives the value of the azimuthal velocity f at
the lower limit of oscillation.
101. To determine the least value of w which will enable a
top to spin, if set down at an inclination 00 to the vertical.
We have seen that the top will first fall to a position 01 and
then rise to its original position; but there must be some angle
02 at which the top fails to spin, either through the peg slipping,
namely the constraint at 0 giving way, or through the side of
the top touching the ground. Hence the limiting value 01 must
be reached before this value 02, and the necessary condition is
02 > 01, and therefore (0)2 > (f)1; for writing
* Ow(l-cos0-7) Cof 1 k
A A(1-cos20) A il+cos0 1-cos20'
we see that since k is a positive constant ' increases with 0.
But ()2 C (cos 0-cos 02) from (iii) Art. 98,
A sin'202
and ()= 2ga from (/) Art. 100.
Ow (cos 0 -cos 02) 2Mgca
A sin202 > Co
2M 2iIgaA sin202
> C2(cos 00- cos 02)'
Example. Suppose the radii of gyration of the top are
respectively 8 in. and 2 ins. about the axle and perpendicular
axis through the point of support. Let the distance a between
the centre of gravity and the toe be 2 ins.; let 00=30~ and
02 = 60~.
If w is the least angular velocity required
2 > 2MgaA sin202
2 (cos 0o-cos 02)
2 3862sin260~ )
>2.386.2.4.642 ( sn260
252 3cos 30~- cos 60~
since = 386 inch/sec2;.'. c > /. 386 x /2049
> 289 rad./sec. approx.. number of revolutions per sec. must be
>47.
Thus if the top is thrown from a string coiled round it in
circles of average radius 1 in., the hand must be withdrawn
from the top at a relative velocity of about 16 mi./hr.