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

ROTATION ABOUT A FIXED AXIS

25

(i) Angular momentum =Io.
Here o= 5. 27= 10r radians a second.
I:-(122)~ =~ l6lb.-ft.2.
Angular momentum =1o
1 1
236 10
=5 lb.-ft.2/sec.
(ii) Kinetic energy = I IW2
2
11
=1 1 1 10072 [2= =10 approx.]
250 125
=-6-= 1-8 = 7 ft.-pdls. approx.
2. Round the spindle of a top I in. in diameter, is wound a thin string
3 ft. in length which is pulled off by a uniform force in 2 sees.
(i) Find the angular velocity after that time.
(ii) If the angular velocity is 157r rad./sec., how long did it take to pull
off the string?
(i) The point where the string is held passes over 3 ft. in 2 sees..'. its average velocity =- ft./sec.
since the pulling force is uniform its final velocity = 3 ft./sec. (Art. 7),
and this is the final velocity of any point on the spindle's rim.
Hence, since the spindle is t in. in radius or AL ft., the final angular
velocity of the spindle or of the top=3 - = 144 rad./sec.
(ii) Let t sees. be the unknown time.
Then as before, the final velocity of a point on the string or on the rim of
the spindle
=2. ft./sec.
The final angular velocity of top therefore
2.3 1
t 48 rad./sec.
But this also =157r rad./sec.
6.48
-157r;
t -
6.48 96.. t= = — sees.
157r 5vr
3. A heavy top weighing 1~ lbs., whose radius of gyration may be taken
as 4 ins., is revolving at the rate of 450 turns a minute.
(i) What energy is expended in stopping it?
(ii) If a frictional force equal to r bs. wt. be applied to the rim at a
distance of 6 ins. from the axis, how soon will the top be brought to rest,
supposing the axis to be fixed vertically all the time?
(iii) How many turns will it have made since the force was applied?
(i) Here I= 4)2 or lb -ft.
450 27
= 6 --- rad./sec. = 15r.
60