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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
where z
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.
PROBLEM 18.121
Show that the angular velocity vector 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
I
I
nI
where
z is the rectangular component of along the axis of symmetry of the body.
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.
PROBLEM 18.122 (Continued)
211
tan tan tan( ) tan tan( )
22
11
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 .
I
I
(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.)
PROBLEM 18.123 (Continued)
Using 22
and yields
z
I
I
nn
I
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 of the
coin forms with GD, (b) the rate of precession of the coin about GD.
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.
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
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.
z 20.7 rev/h
