104 6. SYSTEMS OF PARTICLES
is quadratic equation gives v
A
D 83:49 m/s or 63:49 m/s.
e negative value is inadmissible since v
A
is pointing upward. us,
*
v
A
D 83:49
*
| m/s:
Substituting this into Equation (6.39e),
.
v
B
/
y
D 22:5 1:25
.
83:49
/
m/s D 81:86 m/s:
erefore, the velocity of satellite B is
*
v
B
D
.
v
B
/
x
*
{ C
.
v
B
/
y
*
| D 112:5
*
{ 81:86
*
| m/s:
(b) Distance h
A
from y-axis
Applying the principle of conservation of angular momentum about O,
*
H
O
o
D
*
H
O
) 1:11
10
3
*
k kg.m
2
=s D h
A
*
{ m
A
*
v
A
C h
B
*
{ m
B
*
v
B
D h
A
*
{
.
100
/
83:49
*
|
C
.
80
/
*
{
.
80
/
112:5
*
{ 81:86
*
|
D h
A
8349
*
k
523;904
*
k
kg.m
2
=s:
is gives the required distance from y-axis to satellite A,
h
A
D 62:6176 m:
6.10 EXERCISES
6.1. A system of three particles A; B; and C , has the following masses and position vectors:
m
A
D 1:0 kg; m
B
D 2:0 kg; m
C
D 5:0 kg;
*
r
A
D 3
*
{ C 3
*
| C 6
*
k m;
*
r
B
D 3
*
{ C 4
*
| C 2
*
k m;
and
*
r
C
D 3
*
{ 3
*
| 5
*
k m:
e velocities of these particles are given as,
*
v
A
D 5
*
{ C 4
*
| C 3
*
k m/s;
*
v
B
D 2
*
{ C
2
*
| C 5
*
k
m
s
, and
*
v
C
D 6
*
{ C 5
*
| C 4
*
k m/s. Determine
(a) the position vector
*
r of the mass center G of the system,
(b) the linear momentum m
*
v of the system, and
6.10. EXERCISES 105
(c) the angular momentum
*
H
G
of the system.
6.2. An object of 10 kg is falling vertically along y-axis, as shown in Figure 6.7. At point E
it explodes into three pieces A; B, and C whose masses are, respectively 3 kg, 2 kg, and
5 kg. Immediately after the explosion the velocity of each piece is directed as indicated
in the figure and the velocity of A has been detected to be 100 m/s. Find the velocity of
the 10 kg object just before the explosion.
⃑
A
⃑
C
⃑
B
E
C
B
A
z
y
x
O
5
5
6
2
2
Figure 6.7: Explosion of an object (dimensions in m).
6.3. ree identical billiard balls are on a smooth horizontal table, as shown in Figure 6.8.
Balls B and C are at rest and in contact while ball A moves with a velocity
*
v
A
to the right
of the table. Given that the coefficient of restitution e D 1:0, determine the velocity of
ball A immediately after impact if (a) its path is perfectly centered such that balls B
and C are struck by ball A simultaneously and (b) ball A strikes ball C slightly before it
strikes ball B.
⃑
A
A
B
C
Figure 6.8: ree identical billiard balls on a smooth horizontal table.
106 6. SYSTEMS OF PARTICLES
6.4. ree identical billiard balls in a game of pool are on a smooth horizontal table. Ball
A moves with a velocity magnitude v
A
D 20 m/s, as indicated in Figure 6.9. If after
collision the three balls move in the directions shown in the figure and the coefficient
of restitution e D 1:0; determine the magnitudes of the velocities
*
v
0
A
;
*
v
0
B
; and
*
v
0
C
:
A
B
C
⃑
A
⃑
B
'
⃑
A
'
⃑
C
'
45°
45°
45°
Figure 6.9: ree identical billiard balls in a game of pool.
⃑
A
A D
B
C
Figure 6.10: Four identical billiard balls on a smooth horizontal table.
6.5. Four identical billiard balls are on a smooth horizontal table, as shown in Figure 6.10.
Balls B; C; and D are at rest and in contact while ball A moves with a velocity
*
v
A
to the
right of the table. Given that the coefficient of restitution e D 1; determine the velocity
of ball A immediately after impact if its path is perfectly centered such that balls B and
C are struck by ball A simultaneously while ball D moves parallel to the longer side of
the table and its velocity immediately after impact is
*
v
0
D
D 0:5
*
v
A
:
6.6. ree identical satellites A; B; and C in space are connected by high strength inexten-
sible and inelastic cables to a ring R located at the mass center of the three satellites,
shown in Figure 6.11, such that `
c
D ` cos : e satellites are initially rotating in a
plane about ring R which is at rest (stationary with respect to planet earth). e rotat-
ing speeds of the satellites are proportional to their distances from the ring R. When
6.10. EXERCISES 107
the original speeds of A and B are v
A
D v
B
D v
o
and the angle D 20
ı
, cable CR sud-
denly breaks. is causes satellite C to fly away. One is interested in the motion of the
satellites A and B, and of the ring R after cables AR and BR have again become taut
(at is, after C flies away and when cables AR and BR have become taut again R is the
mass center of A and B). Determine (a) the speed of the ring R, and (b) the relative
speed at which satellites A and B rotate about R.
2θ
180° – θ
180° – θ
A
B
C
R
⃑
C
⃑
B
⃑
A
ℓ
C
ℓ
ℓ
Figure 6.11: ree identical satellites in space.
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