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PROBLEM 18.34
The coordinate axes shown represent the principal centroidal axes of inertia of a 3000-lb space probe whose
radii of gyration are 1.375 ft,
x
k 1.425 ft,
y
k and 1.250 ft.
z
k The probe has no angular velocity when a
5-oz meteorite strikes one of its solar panels at Point A and emerges on the other side of the panel with no
change in the direction of its velocity, but with a speed reduced by 25 percent. Knowing that the final angular
velocity of the probe is (0.05 rad/s) (0.12 rad/s) z
ijk and that the x component of the resulting
change in the velocity of the mass center of the probe is 0.675 in./s,
determine (a) the component z
of the
final angular velocity of the probe, (b) the relative velocity 0
v with which the meteorite strikes the panel.
PROBLEM 18.34 (Continued)
PROBLEM 18.35
A 1200-kg satellite designed to study the sun has an angular velocity
0
= (0.050 rad/s)i + (0.075 rad/s)k when two small jets are activated
at A and B in a direction parallel to the y axis. Knowing that the
coordinate axes are principal centroidal axes, that the radii of gyration
of the satellite are 1.120 m,
x
k 1.200 m,
y
k0.900 m,
z
kand that
each jet produces a 50-N thrust, determine (a) the required operating
time of each jet if the angular velocity of the satellite is to be reduced
to zero, (b) the resulting change in the velocity of the mass center G.
Copyright © McGraw-Hill Education. Permission required for reproduction or display.
PROBLEM 18.35 (Continued)
Solving (1) and (2) simultaneously,
6.4518 N s,
A
Ft
54.298 N s
B
Ft
Since both impulse components are negative, the 50-N thrusts of jets A and B act in the negative y direction.
50 NF
(a) Operating times of jets.
6.4518
50
A
A
Ft
tF 0.1290 s
A
t
54.298
50
B
B
Ft
tF
1.086 s
B
t
Principle of impulse and momentum. Linear momentum:
AB
Ft Ft m
jj
v
6.4518 54.298 1200 jjv
3
10 m/s
vj
(b) Change in velocity of mass center.
50.6 mm/svj
PROBLEM 18.36
If jet A in Prob. 18.35 is inoperative, determine (a) the required operating
time of jet B to reduce to zero the x component of the angular velocity of
the satellite, (b) the resulting final angular velocity, (c) the resulting
change in the velocity of the mass center G.
PROBLEM 18.36 (Continued)
PROBLEM 18.37
Denoting, respectively, by ,
,
O
H
and T the angular velocity, the angular momentum, and the kinetic energy
of a rigid body with a fixed Point O, (a) prove that 2;
OTH (b) show that the angle
between
and
HO will always be acute.
PROBLEM 18.38
Show that the kinetic energy of a rigid body with a fixed Point O can be
expressed as 2
1
2,
OL
TI
where
is the instantaneous angular velocity of
the body and OL
I is its moment of inertia about the line of action OL of .
Derive this expression (a) from Eqs. (9.46) (or Eq. B.19 in the Appendix) and
(18.19), (b) by considering T as the sum of the kinetic energies of particles i
P
describing circles of radius i
about line OL.
PROBLEM 18.39
Determine the kinetic energy of the disk of Problem 18.1.
PROBLEM 18.1 A thin, homogeneous disk of mass m and radius r
spins at the constant rate 1
about an axle held by a fork-ended vertical
rod, which rotates at the constant rate 2.
Determine the angular
momentum G
H of the disk about its mass center G.
PROBLEM 18.40
Determine the kinetic energy of the plate of Problem 18.2.
PROBLEM 18.2 A thin rectangular plate of weight 15 lb rotates about
its vertical diagonal AB with an angular velocity
. Knowing that the
z axis is perpendicular to the plate and that
is constant and equal to
5 rad/s, determine the angular momentum of the plate about its mass
center G.
