PROBLEM 18.114
A homogeneous cone of height h = 12 in. and with a base diameter
d = 6 in. is attached as shown to a cord AB. Knowing that the angles that cord
AB and the axis BC of the cone form with the vertical are, respectively,
= 45° and
= 30°, and that the cone precesses at the constant rate
= 8 rad/s in the sense indicated, determine (a) the rate of spin
of the cone
about its axis BC, (b) the length of cord AB.
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PROBLEM 18.114 (Continued)
Data: 3 in. 0.25 ft, 12 in. 1 ft,rh
2
30.75 ft, 32.2 ft/s
4
ch g
22 2
222
33 3 3
0.25 1 0.75
20 5 20 5
Irhc
m
(a) Dividing Eq. (3) by m and substituting numerical data,
2
32.2 0.75 sin15 0.01875 8 sin30 0.028125 8 sin30 cos30
(b) Substituting data into Eq. (2),
2
PROBLEM 18.115
A solid cube of side
80 mm
c
is attached as shown to cord AB. It is
observed to spin at the rate
40 rad/s
about its diagonal BC and to
precess at the constant rate
5 rad/s
about the vertical axis AD.
Knowing that
30 ,
determine the angle
that the diagonal BC
forms with the vertical. (Hint: The moment of inertia of a cube about an
axis through its center is independent of the orientation of that axis.)
PROBLEM 18.115 (Continued)
Data: 3
80 mm 0.08 m (0.08) 0.069282 m
2
ce
11.06667 10 m 0
II cII
PROBLEM 18.116
A solid cube of side
120 mmc
is attached as shown to a cord AB of
length 240 mm. The cube spins about its diagonal BC and precesses about
the vertical axis AD. Knowing that
25
and
40 ,
determine (a) the
rate of spin of the cube, (b) its rate of precession. (See hint of Problem
18.115.)
PROBLEM 18.116 (Continued)
Data: 120 mm 0.12 m
3(0.12) 0.103923 m
c
From Eq. (3), 3
cos40
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PROBLEM 18.117
A high-speed photographic record shows that a certain projectile was fired
with a horizontal velocity
v
of 2000 ft/s and with its axis of symmetry
forming an angle
3
with the horizontal. The rate of spin
of the
projectile was 6000 rpm, and the atmospheric drag was equivalent to a
force
D
of 25 lb acting at the center of pressure
P
C
located at a distance
c
6in.
from G. (a) Knowing that the projectile has a weight of 45 lb and
a radius of gyration of 2 in. with respect to its axis of symmetry, determine
its approximate rate of steady precession. (b) If it is further known that the
radius of gyration of the projectile with respect to a transverse axis through
G is 8 in., determine the exact values of the two possible rates of precession.
PROBLEM 18.117 (Continued)
22
8
PROBLEM 18.118
If the earth were a sphere, the gravitational attraction of the sun, moon, and
planets would at all times be equivalent to a single force R acting at the mass
center of the earth. However, the earth is actually an oblate spheroid and the
gravitational system acting on the earth is equivalent to a force R and a
couple M. Knowing that the effect of the couple M is to cause the axis of the
earth to precess about the axis GA at the rate of one revolution in 25,800
years, determine the average magnitude of the couple M applied to the earth.
Assume that the average density of the earth is 3
5.51 g/cm , that the average
radius of the earth is 6370 km, and that 2
2
5.ImR (Note: This forced
precession is known as the precession of the equinoxes and is not to be
confused with the free precession discussed in Problem 18.123.)
PROBLEM 18.119
Show that for an axisymmetrical body under no force, the rates of precession and spin can be expressed,
respectively, as
G
H
I
and
cos ( )
G
HII
II
where G
H is the constant value of the angular momentum of the body.
I