PROBLEM 18.128
Solve Sample Problem 18.6, assuming that the meteorite strikes the satellite at C with a
velocity 0(2000 m/s) .vi
PROBLEM 18.6 A space satellite of mass m is known to be dynamically equivalent to
two thin disks of equal mass. The disks are of radius a 800 mm and are rigidly
connected by a light rod of length 2a. Initially the satellite is spinning freely about its axis
of symmetry at the rate 060 rpm.
A meteorite, of mass 0/1000mm and traveling
with a velocity v0 of 2000 m/s relative to the satellite, strikes the satellite and becomes
embedded at C. Determine (a) the angular velocity of the satellite immediately after
impact, (b) the precession axis of the ensuing motion, (c) the rates of precession and spin
of the ensuing motion.
PROBLEM 18.128 (Continued)
2
tan 10.0515
y
90 , 100.05 , 10.05
xy z
4 11.2832
sin sin( ) sin
sin sin 23.9
sin( ) 109.4sin13.85
sin sin 23.9
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PROBLEM 18.129
An 800-lb geostationary satellite is spinning with an angular velocity
0(1.5 rad/s)
j when it is hit at B by a 6-oz meteorite traveling with a
velocity 0(1600 ft/s) (1300 ft/s) (4000 ft/s) vijk
relative to the
satellite. Knowing that 20 in.b
and that the radii of gyration of the
satellite are 28.8 in.
xz
kk and 32.4 in.
y
k, determine the
precession axis and the rates of precession and spin of the satellite after
the impact.
PROBLEM 18.129 (Continued)
Angular momentum of satellite-meteorite system before impact:
00/ 0
() ( )
Gy BG
Im
Hjrv
ijk
symmetry axis. Since ,II
the precession is retrograde.
PROBLEM 18.129 (Continued)
Precession axis. The precession axis is directed along the angular momentum vector
,
G
H
which remains fixed.
135.319
G
H
The angle
between the spin axis (y axis) and the precession axis remains constant.
0.82317
The angle
could also have been calculated from
181.118
I
6.626
Rates of precession and spin.
Set up the triangle of vector addition for the components of angular velocity. Apply the law of sines.
sin36.600 sin 6.626 sin 43.226
PROBLEM 18.130
Solve Problem 18.129, assuming that the meteorite hits the satellite
at A instead of B.
PROBLEM 18.129 An 800-lb geostationary satellite is spinning
with an angular velocity 0(1.5 rad/s)
j
when it is hit at B by a 6-
oz meteorite traveling with a velocity 0(1600 ft/s)vi
(1300 ft/s)j (4000 ft/s)k relative to the satellite. Knowing that
20 in.b and that the radii of gyration of the satellite are
28.8 in.
xz
kk and 32.4 in.,
y
k determine the precession axis
and the rates of precession and spin of the satellite after the impact.
PROBLEM 18.130 (Continued)
Angular momentum of satellite-meteorite system before impact:
(181.118)(1.5) 3.5 0 0
18.633 15.140 46.584
(108.637 lb s ft) (52.99 lb s ft)
Gy AG
ijk
j
jk
Principle of impulse and momentum for satellite-meteorite system. Moments about G:
10
()()
GGG
HHH
Angular velocity immediately after impact.
x
yz
ωijk
22
22
(0.59981) (0.37028) 0.70490 rad/s
(108.637) (52.99) 120.872 lb s ft
G
H
Motion after impact. Since the moments of inertia
x
I and z
I are equal, the body moves as an axisymmetrical
body with the y axis as the symmetry axis.
Moment of inertia about the symmetry axis:
2
181.118 lb s ft
y
II
symmetry axis. Since ,II
the precession is retrograde.
PROBLEM 18.130 (Continued)
Precession axis. The precession axis is directed along the angular momentum vector
,
G
H
which remains
x
G
H
x
cos 0.89878
120.872
y
G
H
26.0
y
() 52.99
120.872
Gz
z
G
H
H
z
The angle
between the spin axis (y axis) and the precession axis remains constant.
26.002
y
The angle
between the angular velocity vector and the spin axis (y axis) is
0.59981
0.70490
y
The angle
could also have been calculated from
181.118
tan tan tan 26.002
I
PROBLEM 18.131
A homogeneous rectangular plate of mass m and sides c and 2c
is held at A and B by a fork-ended shaft of negligible mass,
which is supported by a bearing at C. The plate is free to rotate
about AB, and the shaft is free to rotate about a horizontal axis
through C. Knowing that, initially, 040 ,
00,
and
010
rad/s, determine for the ensuring motion (a) the range
of values of ,
(b) the minimum value of ,
(c) the maximum
value of .
PROBLEM 18.131 (Continued)
Using the initial conditions, including 00,
Eq. (3) yields
(4)
(a) With
040
and 010 rad/s
2
(1 4sin ) 26.527
26.527
4 (1 4sin ) 265.27
Eliminate
and solve for 2:
22
(26.527)
(5)
min 5.3054
min 5.31rad/s
(c) From Eq. (5),
175.92