164 8. DYNAMICS OF RIGID BODIES
Solving Equations (8.34a)–(8.34c), one finds
!
0
D 3:8462 rad/s;
*
!
0
D 3:8462 rad/s ;
v
0
b
D 2:3076 m/s;
*
v
0
b
D 2:3076 m/s :
8.9 EXERCISES
8.1. A high-frequency radar used in a frigate can be considered as a square plate of uniform
thickness of mass m and side length b. e plate is welded to a vertical rod AB, as shown
in Figure 8.16. e angle between the vertical rod and the plate is D 40
ı
: If the vertical
rod rotates at a constant angular velocity
*
! ; determine
(a) the force and couple system representing the dynamic reaction at A,
(b) the dynamic reaction at A when ! D 5 rad/s anti-clockwise, m D 1,000 kg, b D
1:6 m, and
(c) repeat (a) and (b) when now D 90
ı
:
x
B
y
ω⃑
z
θ
b
b
A
B
Figure 8.16: Rotating high-frequency square radar.
8.2. A slender uniform rigid rod DA of length L D 2 m and mass m D 15 kg is pinned at A
to a vertical axle BAE which rotates with a uniform angular velocity ! D 10 rad/s. e
rod is maintained in the horizontal position, shown in Figure 8.17, by means of a wire
DC attached at D of the rod and C of the axle. Given that D 45
ı
, find the tension in
the wire and the reaction at the pin A.
8.3. A uniform rigid beam AB of mass m is suspended from two identical extensible cables,
C
1
and C
2
as shown in Figure 8.18. If for some unknown reasons cable C
2
breaks, find
8.9. EXERCISES 165
B
ω⃑
L
θ
A
D
C
E
B
Figure 8.17: Rotating vertical axle with a horizontal rod.
C
2
C
1
A B
ℓ
G•
Figure 8.18: Rigid beam suspended from two extensible cables.
(a) the angular acceleration
*
˛ of the rigid beam at this instant,
(b) the acceleration at point A, and
(c) the acceleration at point B.
8.4. In a shooting target practice a uniform square plate has mass m hanging by two strip
cables C
1
and C
2
that are attached to the ceiling, as shown in Figure 8.19. e square
plate is hit at E by a bullet in a direction perpendicular to the plane of the plate. Suppose
the impulse imparted at E is
*
F t, find immediately after the impact (a) the velocity of
the mass center G and (b) the angular velocity of the plate.
8.5. A uniform circular disk of radius r and mass m is mounted on an axle AG of length `, as
shown in Figure 8.20. e axle rotates at a constant angular velocity !
2
with respect to
the vertical shaft AO which rotates at a constant angular velocity !
1
with respect to the
fixed vertical axis. Assume the length of the shaft AO is equal to the radius of the circular
166 8. DYNAMICS OF RIGID BODIES
C
2
C
1
A
B
ℓ
ℓ/4
ℓ/
4
F
⃑
Δt
•G
E
Ceiling Supports
Figure 8.19: Square plate in target practice.
A
G
A
O
r
ω⃑
1
ω⃑
2
ℓ
Figure 8.20: Circular disk rolling on a horizontal floor.
disk, and the masses of the axle and shaft can be disregarded, determine (a) the force
(for simplicity, it is assumed that this force is vertical) exerted by the floor on the disk
(assuming the disk rolls on the horizontal floor without slipping) and (b) the dynamic
reaction at point O.
8.6. A uniform thin circular disk of mass m and radius r rolls with a constant spin
P
on the
horizontal floor, as shown in Figure 8.21. e center of mass G of the disk describes a
circular path of radius
r
G
. If the disk maintains a constant angle of inclination during
its motion, derive the equation of motion for this particular event. Describe the motion
of the disk if the precession
P
D 0 and D 90
ı
(that is, when the axis of
P
in Figure 8.21
is horizontal).
8.7. e angular velocity of a 1,500 kg space capsule is
*
!
c
D 0:02
*
{ C 0:10
*
| rad/s when
two small jets are activated at A and B, each in a direction parallel to the z-axis. is
8.9. EXERCISES 167
ψ˙
θ
G
r
r
G
ϕ
˙
Figure 8.21: in disk rolling on a horizontal floor.
G
z
y
B
x
A
1.20 m
2.50 m
1.20 m
2.0 m
Figure 8.22: Space capsule with two small jets.
capsule is shown in Figure 8.22. Knowing that the radii of gyration of the capsule are
x
D
z
D 1:00 m and
y
D 1:20 m, and that each of the jets generates a thrust of 50 N,
determine (a) the required operation time of each jet if the angular velocity of the capsule
is to be reduced to zero and (b) the resulting change in the velocity of the mass center
G.
8.8. e conceptual model of a type of aircraft turn indicator is shown in Figure 8.23. Springs
AC and BD are initially stretched and exert equal vertical forces at A and B when the
path of the airplane is straight. If the rotating disk has a mass of 0.20 kg and spins
168 8. DYNAMICS OF RIGID BODIES
100 mm
40
mm
ω
A
C
B
D
Figure 8.23: Conceptual model of an aircraft turn indicator.
θ
Z
z
O
z
G
G
•
Figure 8.24: A top supported at fixed point O.
at 5,000 rpm, determine the angle through which the yoke rotates when the aircraft
executes a horizontal turn of radius 800 m at a speed of 500 km/h. Given that each
spring constant is 2,000 N/m.
8.9. A top is supported at the fixed point O, as shown in Figure 8.24. Suppose the moments
of inertia of the top about its axis of symmetry and about a transverse axis through O
are, respectively, I and I
x
, show that the condition for steady precession is
I!
z
I
x
cos
d
dt
d
dt
D mgz
G
;
where
d
dt
is the precession and !
z
is the component of the angular velocity along the
axis of symmetry of the top.
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