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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 mmc=
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 .
θ
= °
PROBLEM 18.109 (Continued)
Substituting into Eq. (1),
66
−−
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
I I Wc
ω φ θφ
′
−=
where
φ
is the rate of precession and
z
ω
is the rectangular component of the
angular velocity along the axis of symmetry of the top. (b) Show that if the rate
of spin
ψ
of the top is very large compared with its rate of precession
,
φ
the
condition for steady precession is
.I Wc
ψφ
≈
(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)
4.807
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
β
= °
with the
vertical, determine the rate of spin of the sphere about line AB.
PROBLEM 18.111 (Continued)
2 22
22
Ir
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
β
for which the sphere will precess about a vertical axis at the constant rate
of 60 rpm in the sense indicated.
SOLUTION
Use principal axes x, y, z with origin at A.
2
3
10 4 14 in.
14 7 ft,
12 6
32.2 ft/s
c
g
= +=
= =
=
PROBLEM 18.112 (Continued)
2
22
2 21 0.04444 ft
Ir
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
ψ
and precesses about the vertical through A at the constant
rate
.
φ
Determine the angle
β
for which the axis BC of the cone is aligned
with the cord AB (
θ
=
β
).
PROBLEM 18.11
3 (Continued)
( )
22
dh
( )
22
dh
+
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,
β
= 45° and
θ
= 30°, and that the cone precesses at the constant rate
φ
= 8 rad/s in the sense indicated, determine (a) the rate of spin
ψ
of the cone
about its axis BC, (b) the length of cord AB.
PROBLEM 18.114
(Continued)
( ) ( ) ( )
22 2
2 22
33 3 3
0.25 1 0.75
20 5 20 5
Ir hc
m
′= + −= + −
2
0.046875 ft=
2
0.028125 ft
II
m
′−=
45 , 30 15
β θ βθ
= ° = ° −=°
(a) Dividing Eq. (3) by m and substituting numerical data,
( )( ) ( )
( )
( )( )
2
32.2 0.75 sin15 0.01875 8 sin 30 0.028125 8 sin 30 cos30
cos 45
ψ
°= °− ° °
°
8.8395 0.075 0.77942
ψ
= −
128.252 rad/s
ψ
=
128.3 rad/s
ψ
=
(b) Substituting data into Eq. (2),
( )( )
2
32.2 tan 45 sin 45 0.75sin 30 8l°= °+ °
32.2 45.255 24, 0.1812 ftll=+=
2.17 in.l=
PROBLEM 18.115
A solid cube of side
80 mmc=
is attached as shown to cord AB. It is
observed to spin at the rate
40 rad/s
ψ
=
about its diagonal BC and to
precess at the constant rate
5 rad/s
φ
=
about the vertical axis AD.
Knowing that
30 ,
β
= °
determine the angle
φ
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)
3
80 mm 0.08 m (0.08) 0.069282 m
ce= = = =
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
θ
25= °
and
40 ,
β
= °
determine (a) the
rate of spin of the cube, (b) its rate of precession. (See hint of Problem
18.115.)
PROBLEM 18.116 (Continued)
120 mm 0.12 m
c
= =
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
β
= °
with the horizontal. The rate of spin
ψ
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
6 in.=
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.
SOLUTION
Choose principal centroidal axes x, y, z as shown.
The symmetry (spin) axis is the z axis.
Reduce the drag force to a force-couple system at the mass center.
( sin cos ) ( cos sin )
DW D W
ββ ββ
=− −−
Fik
(0.0388205)(628.32)
PROBLEM 18.117 (Continued)
2
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
.I mR=
(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.)
SOLUTION
9
12
25,800 years (25,800 yr)(365.24 day/yr)(24 h/day)(3600 s/h)
814.16 10 s
27.7173 10 rad/s
π
−
=
= ×
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
ϕ
=′
and
cos ( )
G
H II
II
θ
ψ
′−
=′
where
G
H
is the constant value of the angular momentum of the body.
cos cos cos ( )cos
cos
G GGG
I
H H H HI I
I I I II
θ θθ θ
ψ ϕθ
′−
=−=−=
′′
