PROBLEM 18.109
The 85-g top shown is supported at the fixed Point O. The radii of gyration of
the top with respect to its axis of symmetry and with respect to a transverse axis
through O are 21 mm and 45 mm, respectively. Knowing that
37.5 mm
c
and
that the rate of spin of the top about its axis of symmetry is 1800 rpm,
determine the two possible rates of steady precession corresponding to
30 .
PROBLEM 18.109 (Continued)
Substituting into Eq. (1),
PROBLEM 18.110
The top shown is supported at the fixed Point O and its moments of inertia
about its axis of symmetry and about a transverse axis through O are denoted,
respectively, by
I
and
.
I
(a) Show that the condition for steady precession
of the top is
(cos)
z
II Wc
where
is the rate of precession and
z
is the rectangular component of the
angular velocity along the axis of symmetry of the top. (b) Show that if the rate
of spin
of the top is very large compared with its rate of precession
,
the
condition for steady precession is
.IWc
(c) Determine the percentage
error introduced when this last relation is used to approximate the slower of
the two rates of precession obtained for the top of Problem 18.109.
PROBLEM 18.110 (Continued)
(c) Data:
85 g 0.085 kg
(0.085)(9.81) 0.83385 N
m
Wmg
c
PROBLEM 18.111
A solid aluminum sphere of radius 4 in. is welded to the end of a10-in.-long rod
AB of negligible mass which is supported by a ball-and-socket joint at A. Knowing
that the sphere is observed to precess about a vertical axis at the constant rate of 60
rpm in the sense indicated and that rod AB forms an angle
20
with the
vertical, determine the rate of spin of the sphere about line AB.
PROBLEM 18.111 (Continued)
222
22
(0.3333) 0.04444 ft
55
Ir
m
PROBLEM 18.112
A solid aluminum sphere of radius 4 in. is welded to the end of a 10-in.-long rod
AB of negligible mass which is supported by a ball-and-socket joint at A. Knowing
that the sphere spins as shown about line AB at the rate of 600 rpm, determine the
angle
for which the sphere will precess about a vertical axis at the constant rate
of 60 rpm in the sense indicated.
PROBLEM 18.112 (Continued)
2
PROBLEM 18.113
A homogeneous cone of height h and with a base of diameter d < h is
attached as shown to a cord AB. The cone spins about its axis BC at the
constant rate
and precesses about the vertical through A at the constant
rate
.
Determine the angle
for which the axis BC of the cone is aligned
with the cord AB (
=
).
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PROBLEM 18.113 (Continued)
:
sin
GG
MTc H
(3)
20 5 80 80
22
33
80 80
I I md mh
2
22
2II d
Idh
2
dh
2
dh