PROBLEM 14.27
Derive the relation
OG
m HrvH
between the angular momenta O
H and G
H defined in Eqs. (14.7) and (14.24), respectively. The vectors r
and v define, respectively, the position and velocity of the mass center G of the system of particles relative to
the newtonian frame of reference Oxyz, and m represents the total mass of the system.
G
m
rvH
PROBLEM 14.28
Show that Eq. (14.23) may be derived directly from Eq. (14.11) by substituting for O
H the expression given
in Problem 14.27.
GG
MH
PROBLEM 14.29
Consider the frame of reference
A
xyz
in translation with respect to the
newtonian frame of reference Oxyz. We define the angular momentum H
A of a
system of n particles about A as the sum
1
n
A
iii
i
m
Hrv
(1)
of the moments about A of the momenta ii
m
v of the particles in their
motion relative to the frame .
A
xyz
Denoting by A
H the sum
1
n
A
iii
i
m
Hrv
(2)
of the moments about A of the momenta ii
mv of the particles in their
motion relative to the newtonian frame Oxyz, show that AA
HH
at a
given instant if, and only if, one of the following conditions is satisfied at
that instant: (a) A has zero velocity with respect to the frame Oxyz,
(b) A coincides with the mass center G of the system, (c) the velocity vA
relative to Oxyz is directed along the line AG.
or ( ) is parallel to . Velocity is directed along line .
A
AA A
cAG
vrr v
PROBLEM 14.30
AA
where A
Problem 14.29 and where A
M represents the sum of the moments about A of
the external forces acting on the system of particles, is valid if, and only if,
one of the following conditions is satisfied: (a) the frame
A
xyz
is itself a
newtonian frame of reference, (b) A coincides with the mass center G,
(c) the acceleration A
a of A relative to Oxyz is directed along the line AG.
or ( ) is parallel to . Acceleration is directed along line .
A
AA A
cAG
arr a
Problem 14.1.
PROBLEM 14.1 A 30-g bullet is fired with a horizontal
velocity of 450 m/s and becomes embedded in block B which
has a mass of 3 kg. After the impact, block B slides on 30-kg
carrier C until it impacts the end of the carrier. Knowing the
impact between B and C is perfectly plastic and the coefficient
of kinetic friction between B and C is 0.2, determine (a) the
velocity of the bullet and B after the first impact, (b) the final
velocity of the carrier.
SOLUTION
From the solution to Problem 4.1 the velocity of A and B after the first impact is 4.4554 m/sv and the
velocity common to A, B, and C after the sliding of block B and bullet A relative to the carrier C has ceased in
PROBLEM 14.32
Assuming that the airline employee of Prob. 14.3 first tosses
the 30-lb suitcase on the baggage carrier, determine the energy
lost (a) as the first suitcase hits the carrier, (b) as the second
suitcase hits the carrier.
PROBLEM 14.3 An airline employee tosses two suitcases, of
weight 30 lb and 40 lb, respectively, onto a 50-lb baggage
carrier in rapid succession. Knowing that the carrier is initially
at rest and that the employee imparts a 9-ft/s horizontal
velocity to the 30-lb suitcase and a 6-ft/s horizontal velocity to
the 40-lb suitcase, determine the final velocity of the baggage
carrier if the first suitcase tossed onto the carrier is (a) the 30-lb
suitcase,(b) the 40-lb suitcase.
SOLUTION
ABC
PROBLEM 14.32 (Continued)
PROBLEM 14.33
In Problem 14.6, determine the work done by the woman and
by the man as each dives from the boat, assuming that the
woman dives first.
PROBLEM 14.6
A 180-lb man and a 120-lb woman stand
side by side at the same end of a 300-lb boat, ready to dive,
each with a 16-ft/s velocity relative to the boat. Determine
the velocity of the boat after they have both dived, if (a) the
woman dives first, (b) the man dives first.
PROBLEM 14.34
Determine the energy lost as a result of the series of collisions described in Problem 14.8.
PROBLEM 14.8 Two identical cars A and B are at rest on a loading dock with brakes released. Car C, of a
slightly different style but of the same weight, has been pushed by dockworkers and hits car B with a velocity
of 1.5 m/s. Knowing that the coefficient of restitution is 0.8 between B and C and 0.5 between A and B,
determine the velocity of each car after all collisions have taken place.
PROBLEM 14.35
Two automobiles A and B, of mass A
m and ,
B
m respectively, are traveling in opposite directions when they
collide head on. The impact is assumed perfectly plastic, and it is further assumed that the energy absorbed by
each automobile is equal to its loss of kinetic energy with respect to a moving frame of reference attached to
the mass center of the two-vehicle system. Denoting by EA and EB, respectively, the energy absorbed by
automobile A and by automobile B, (a) show that //,
AB BA
E
Emm that is, the amount of energy absorbed by
each vehicle is inversely proportional to its mass, (b) compute A
E
and ,
B
E
knowing that 1600
A
m kg and
mB900 kg and that the speeds of A and B are, respectively, 90 km/h and 60 km/h.
SOLUTION
2
(2)(2500)
B
B
PROBLEM 14.36
It is assumed that each of the two automobiles involved in the collision described in Problem 14.35 had been
designed to safely withstand a test in which it crashed into a solid, immovable wall at the speed v0. The
severity of the collision of Problem 14.35 may then be measured for each vehicle by the ratio of the energy it
absorbed in the collision to the energy it absorbed in the test. On that basis, show that the collision described
in Problem 14.35 is 2
(/)
AB
mm times more severe for automobile B than for automobile A.
()
A
A
mm m
vv vv
B
A
Sm
PROBLEM 14.37
Solve Sample Problem 14.5, assuming that cart A is given an initial horizontal velocity
v
0
while ball B is at rest.
2
BAB
mg m m
2
AB
PROBLEM 14.38
Two hemispheres are held together by a cord which maintains a spring under
compression (the spring is not attached to the hemispheres). The potential
energy of the compressed spring is
120 J
and the assembly has an initial
velocity
0
v
of magnitude
08m/s.v
Knowing that the cord is severed when
30 ,
causing the hemispheres to fly apart, determine the resulting velocity
of each hemisphere.
SOLUTION
PROBLEM 14.39
A 15-lb block B starts from rest and slides on the 25-lb wedge A, which is
supported by a horizontal surface. Neglecting friction, determine (a) the velocity
of B relative to A after it has slid 3 ft down the inclined surface of the wedge,
(b) the corresponding velocity of A.
SOLUTION
PROBLEM 14.39 (Continued)
125 115
PROBLEM 14.40
A 40-lb block B is suspended from a 6-ft cord attached to a 60-lb cart A,
which may roll freely on a frictionless, horizontal track. If the system is
released from rest in the position shown, determine the velocities of A and B
as B passes directly under A.
B
AB
mm
PROBLEM 14.40 (Continued)
60
A
w
PROBLEM 14.41
In a game of pool, ball A is moving with a velocity 0
v of magnitude
0
v 15 ft/s when it strikes balls B and C, which are at rest and aligned
as shown. Knowing that after the collision the three balls move in the
directions indicated and assuming frictionless surfaces and perfectly
elastic impact (that is, conservation of energy), determine the
magnitudes of the velocities ,
A
v ,
B
v and .
C
v
SOLUTION
A
A
2
C
C
PROBLEM 14.42
In a game of pool, ball A is moving with a velocity 0
v of magnitude
v0 15 ft/s when it strikes balls B and C, which are at rest and aligned
as shown. Knowing that after the collision the three balls move in the
directions indicated and assuming frictionless surfaces and perfectly
elastic impact (that is, conservation of energy), determine the
magnitudes of the velocities vA, vB and vC.
SOLUTION
C
C
PROBLEM 14.43
Three spheres, each of mass m, can slide freely on a frictionless, horizontal
surface. Spheres A and B are attached to an inextensible, inelastic cord of
length l and are at rest in the position shown when sphere B is struck
squarely by sphere C which is moving with a velocity 0.v Knowing that the
cord is taut when sphere B is struck by sphere C and assuming perfectly
elastic impact between B and C, and thus conservation of energy for the
entire system, determine the velocity of each sphere immediately after
impact.
SOLUTION
B
A
A
B
ABA
Bv