Dynamics 2e 737
Problem 4.55
The arm
AB
can rotate freely about the pin at
A
. The spring with stiffness
kD500 N=m
is designed so that the system is in static equilibrium when
✓D0ı
. Let
LD18:2 cm
,
hD24:6 cm
, and the mass of the ball
B
be
5kg
. Neglect the mass of arm
AB
.Hint: Sketch an FBD of the ball and
the arm together. For Probs. 4.53 and 4.54, let
`
be the distance between
Cand Dand choose `as the primary unknown.
If the arm
AB
is released from rest when
✓D90ı
such that it swings
to the left, determine the speed of the ball
B
when the arm reaches
✓D0ı
.
Solution
Following the hint, we sketch an FBD of
B
and the arm
AB
together.
We model the system as being subject to the weight of
Bmg
, the
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Dynamics 2e 739
Problem 4.56
The package handling system is designed to launch the small package
of mass
m
from
A
by using a compressed linear spring of constant
k
. After launch, the package slides along the track until it lands on
the conveyor belt at
B
. The track has small, well-oiled rollers so that
you can neglect any energy loss due to the movement of the package
along the track. Modeling the package as a particle, determine the
minimum initial compression of the spring so that the package gets
to
B
without separating from the track at
C
. Finally, determine the
speed with which the package reaches the conveyor at B.
Solution
The figure at the right shows the FBD of
A
at two different
generic locations along with track. We model the package
A
as
740 Solutions Manual
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 741
Problem 4.57
Compressed gas is used in many circumstances to propel objects
within tubes. For example, you can still find pneumatic tubes
in use in many banks to receive and send items to drive-through
tellers,and it is compressed gas that propels a bullet out of a gun
barrel. Let the cross-sectional area of the tube be given by
A
and
the position of the cylinder by
s
, and assume that the compressed
gas is an ideal gas at constant temperature so that the pressure
P
times the volume
˝
is a constant, i.e.,
P˝ Dconstant
. Show
that the potential energy of this compressed gas is given by
VD
P0s0Aln.s=s0/
, where
P0
is the initial pressure and
s0
is the
initial value of
s
. Model the cylinder as a particle, and assume that
the forces resisting the motion of the cylinder are negligible.
Solution
742 Solutions Manual
Problem 4.58
When a gun fires a bullet, the gun barrel acts as the tube, and the
bullet acts as the cylinder in Prob. 4.57. Using the assumptions and
result of that problem, determine the velocity of a bullet at the end
of a
24 in:
gun barrel, given that the bore diameter is
0:458 in:
, the
bullet weight is
300 gr
(
7000 gr D1lb
), the initial firing pressure
is
27;000 psi
, and the initial distance between the back of the bullet
and the back wall of the firing chamber (i.e., s0in Prob. 4.57) is
(a) 1:855 in:(this distance is realistic and accurate),
(b) 1:5 in:, and
(c)
explain why the velocity of the bullet at the end of the gun
barrel is lower in Part (b) when compared to Part (a).
Solution
We model the bullet as subject only to the force of the gas
Fg
, which is conservative.
Hence, we apply the work-energy principle as the statement of conservation of
Dynamics 2e 743
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744 Solutions Manual
Problem 4.59
The resistance of a material to fracture is assessed with a fracture test. One
such test is the Charpy impact test, in which the fracture toughness is assessed
by measuring the energy required to break a specimen of a specified geometry.
This is done by releasing a heavy pendulum from rest at an angle
✓i
and by
measuring the maximum swing angle
✓
f
reached by the pendulum after the
specimen is broken. Suppose that in an experiment
✓iD45ı
,
✓
fD23ı
, the
weight of the pendulum’s bob is
3lb
, and the length of the pendulum is
3ft
.
Neglecting the mass of any other component of the testing apparatus, assuming
that the pendulum’s pivot is frictionless, and treating the pendulum’s bob as a
particle, determine the fracture energy of the specimen tested. Assume that the
fracture energy is the energy required to break the specimen.
Solution
We model the bob as a particle subject to its own weight
mg
, the tension in
the pendulum arm
Fa
, and the contact force with the specimen
Fc
. Clearly,
Dynamics 2e 745
Problem 4.60
A pendulum with mass
mD1:4 kg
and length
LD1:75
m is released from rest
at an angle
✓i
. Once the pendulum has swung to the vertical position (i.e.,
✓D0
),
its cord runs into a small fixed obstacle. In solving this problem, neglect the size
of the obstacle, model the pendulum’s bob as a particle, model the pendulum’s
cord as massless and inextensible, and let gravity and the tension in the cord be
the only relevant forces.
What is the maximum height, measured from its lowest point, reached by the
pendulum if ✓iD20ı?
Solution
We denote by
¿
the position of the pendulum at release, and by
¡
the position at which the pendulum bob achieves the maximum height
746 Solutions Manual
Problem 4.61
A pendulum with mass
mD1:4 kg
and length
LD1:75
m is released from rest
at an angle
✓i
. Once the pendulum has swung to the vertical position (i.e.,
✓D0
),
its cord runs into a small fixed obstacle. In solving this problem, neglect the size
of the obstacle, model the pendulum’s bob as a particle, model the pendulum’s
cord as massless and inextensible, and let gravity and the tension in the cord be
the only relevant forces.
If the bob is released from rest at
✓iD90ı
, at what angle
does the cord go
slack?
Solution
We denote by
¿
the position of the pendulum at release, and by
¡
the position at of the bob corresponding to a generic value of
.We