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

28

ROTATION ABOUT A FIXED AXIS

Hence, 121
Hence, 121 v2 13. 32. 52,
2.13.32. 52
121
26.8
= 19 ft./sec. nearly.
6. The following example involves the use of the Calculus.
Find the energy communicated to a top when it is set in motion by a
string 1~ yds. in length, being pulled with a force which varies as the
length of the string already unwound.
Suppose that x ft. have been unwound at any moment. Then the pull
at that moment = x units of force. The whole work done or energy
communicated when 11 yds. are unwound
c2 4Jo
Si1
=81 units of energy.
8
Here neither the size of the spindle nor the radius of gyration make any
difference.
EXAMPLES FOR SOLUTION.
1. A spinning top weighing 6 oz. makes 300 revolutions a minute.
Taking the radius of gyration about the axis of revolution (supposed
vertical) as 1~ ins., calculate
(i) its angular momentum; (ii) its energy.
2. It is said that a Diabolo spool in full rotation spins 2000 turns a
minute. Taking the mass as 31 oz. and the radius of gyration as ~ in.,
find its angular momentum and how much energy has been expended on it,
assuming that half has been wasted.
3. The weights of the minute, hour, and seconds hands of a watch are as
15: 10:1. Compare their angular momenta, their lengths being as 3: 2:1,
assuming that the minute and hour hands revolve about one end, but the
seconds hand about its middle point.
(For the value of k, see Art. 18Q
4. A door a ft. wide is shutting with angular velocity o rad./sec. If
it comes to rest in -!6 sec., find the shock on the door post, the mass of the
door being M lbs. and its radius of gyration k ft.
5. A door weighing 40 lbs. is rotating about its hinges, when its edge,
3 ft. 6 ins. away from the hinges, and moving at 30 mi./hr., meets an
immovable object. What is the measure of the shock experienced by that
object during j of a second? Radius of gyration 21 ft.
6. Given that the top in Fig. 11 weighs 31 oz., that the string is 1 yds.
long, and the average radius of the conical body is 1 in., calculate the
angular momentum and the energy due to rotation when the top falls from
rest so as to unwind in 1~ secs.
Take the radius of gyration as 2 inch.
7. If the top in the preceding question is making 360 turns a minute just
after falling, how long did it take to fall?