978-0077687342 Chapter 18 Part 11

subject Type Homework Help
subject Pages 14
subject Words 3160
subject Authors Brian Self, E. Johnston, Ferdinand Beer, Phillip Cornwell

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page-pf1
PROBLEM 18.120
(a) Show that for an axisymmetrical body under no force, the rate of precession can be expressed as
cos
z
I
I
ω
φθ
=
where
z
ω
is the rectangular component of
ω
along the axis of symmetry of the body. (b) Use this result to
check that the condition (18.44) for steady precession is satisfied by an axisymmetrical body under no force.
0
page-pf2
PROBLEM 18.121
Show that the angular velocity vector
ω
of an axisymmetrical body under no force is observed from the body
itself to rotate about the axis of symmetry at the constant rate
z
II
nI
ω
=
where
ω
z is the rectangular component of
ω
along the axis of symmetry of the body.
2
z
z
page-pf3
PROBLEM 18.122
For an axisymmetrical body under no force, prove (a) that the rate of retrograde precession can never be less
than twice the rate of spin of the body about its axis of symmetry, (b) that in Figure 18.24 the axis of symmetry
of the body can never lie within the space cone.
22
page-pf4
PROBLEM 18.122 (Continued)
2
11
tan tan tan( ) tan tan( )
22
θ θ γθ θ γθ
+ −> +
2
11
page-pf5
PROBLEM 18.123
Using the relation given in Problem 18.121, determine the period of precession of the north pole of the earth
about the axis of symmetry of the earth. The earth may be approximated by an oblate spheroid of axial
moment of inertia I and of transverse moment of inertia
0.9967 .II
=
(Note: Actual observations show a
period of precession of the north pole of about 432.5 mean solar days; the difference between the observed
and computed periods is due to the fact that the earth is not a perfectly rigid body. The free precession
considered here should not be confused with the much slower precession of the equinoxes, which is a forced
precession. See Problem 18.118.)
page-pf6
PROBLEM 18.123 (Continued)
II
page-pf7
PROBLEM 18.124
A coin is tossed into the air. It is observed to spin at the rate of 600 rpm about an axis GC
perpendicular to the coin and to precess about the vertical direction GD. Knowing that GC
forms an angle of 15° with GD, determine (a) the angle that the angular velocity
ω
of the
coin forms with GD, (b) the rate of precession of the coin about GD.
z

from which
sin sin 28.187 2.0705
sin( ) sin13.187
ψγ ψ
ϕψ
θγ
°
= = = −
−− °


With
600 rpm, ( 2.0705)(600)
ψϕ
= = −

| | 1242 rpm
ϕ
=
(retrograde)
page-pf8
PROBLEM 18.125
The angular velocity vector of a football which has just been kicked is
horizontal, and its axis of symmetry OC is oriented as shown. Knowing
that the magnitude of the angular velocity is 200 rpm and that the ratio
of the axis and transverse moments of inertia is
1
3
/,II
=
determine
(a) the orientation of the axis of precession OA, (b) the rates of
precession and spin.
page-pf9
PROBLEM 18.126
A space station consists of two sections A and B of equal masses, which are rigidly
connected. Each section is dynamically equivalent to a homogeneous cylinder of
length 15 m and radius 3 m. Knowing that the station is precessing about the fixed
direction GD at the constant rate of 2 rev/h, determine the rate of spin of the station
about its axis of symmetry
.CC
x
1.28557 26.3005 rev/h
tan tan
ω
ωγγ
=−= =
x
z
page-pfa
PROBLEM 18.127
If the connection between sections A and B of the space station of Prob. 18.126 is
severed when the station is oriented as shown and if the two sections are gently pushed
apart along their common axis of symmetry, determine (a) the angle between the spin
axis and the new precession axis of section A, (b) the rate of precession of section A,
(c) its rate of spin.
SOLUTION
22 2
11
x
(c)
cos 26.3005 5.7806cos12.8497
ψω ϕ θ
=−= − °
z
20.7 rev/h
ψ
=
page-pfb
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
0
60 rpm.
ω
=
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.
page-pfc
PROBLEM 18.128 (Continued)
2
tan 10.0515
11.2832
y
z
ω
γγ
ω
= = = °
90 , 100.05 , 10.05
xy z
γγ γ
=°= °=°
page-pfd
PROBLEM 18.129
An 800-lb geostationary satellite is spinning with an angular velocity
ω
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)=−+ +v ijk
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.
SOLUTION
Mass of satellite:
2
800 24.845 lb s /ft
32.2
W
mg
= = =
2
2
28.8

0 00 0
xz y
page-pfe
PROBLEM 18.129 (Continued)
Angular momentum of satellite-meteorite system before impact:
0 0/ 0
() ( )
G y BG
Im
ω
= +×
H jr v
i jk
xz
The motion is a steady precession
φ
about the precession axis together with a steady spin
ψ
about the spin or
symmetry axis. Since
,II
>
the precession is retrograde.
page-pff
PROBLEM 18.129 (Continued)
Precession axis. The precession axis is directed along the angular momentum vector
,
G
H
which remains fixed.
Immediately after impact, its direction cosines relative to the body axes
,,xyz
are:
() 77.64
135.319
Gx
x
G
H
H
x
() 108.637
cos 0.80282
135.319
Gy
y
G
H
H
θ
= = =
36.6
y
θ
= °
() 21.935
135.319
Gz
z
G
H
H
z
The angle
θ
between the spin axis (y axis) and the precession axis remains constant.
36.600
y
θθ
= = °
0.82317
ω
The angle
γ
could also have been calculated from
181.118
I
page-pf10
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)
ω
=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.
SOLUTION
2
800 24.845 lb s /ft
32.2
W
Principal moments of inertia:
2
22
2
22
2
28.8
(24.845) 143.106 lb s ft
12
32.4
(24.845) 181.118 lb s ft
12
143.106 lb s ft
xx
yy
zx
I mk
I mk
II

= = = ⋅⋅



= = = ⋅⋅


= = ⋅⋅
2
60.011649 lb s /ft
page-pf11
PROBLEM 18.130 (Continued)
Angular momentum of satellite-meteorite system before impact:
0 0/ 0
() ( )
(181.118)(1.5) 3.5 0 0
18.633 15.140 46.584
(108.637 lb s ft) (52.99 lb s ft)
G y AG
Im
ω
= +×
= +
= ⋅⋅ + ⋅⋅
H jr v
i jk
j
jk
Principle of impulse and momentum for satellite-meteorite system. Moments about G:
page-pf12
PROBLEM 18.130 (Continued)
Precession axis. The precession axis is directed along the angular momentum vector
,
G
H
which remains
fixed. Immediately after impact, its direction cosines relative to the body axes
,,xyz
are:
()
Gx
H
page-pf13
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,
0
40 ,
θ
= °
0
0,
θ
=
and
0
10
φ
=
rad/s, determine for the ensuring motion (a) the range
of values of
,
θ
(b) the minimum value of
,
φ
(c) the maximum
value of
.
θ
22 2 2
24
1[4 (1 4 sin )]
24
mc
θφ θ
= ++

page-pf14
PROBLEM 18.131 (Continued)
Using the initial conditions, including
0
0,
θ
=
Eq. (3) yields
22222
00
4 (1 4 sin ) (1 4 sin )
θφθφθ
++ =+
 
(4)
(a) With
0
40
θ
= °
and
010 rad/s
φ
=
in. Eqs. (2) and (4),
2
max
max

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