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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.
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.
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.
Copyright © McGraw-Hill Education. Permission required for reproduction or display.
PROBLEM 14.30
where A
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.
PROBLEM 14.31
Determine the energy lost due to friction and the impacts for
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.
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.
PROBLEM 14.32 (Continued)
Kinetic energies:
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.
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