Dynamics 2e 727
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
728 Solutions Manual
Problem 4.48
Packages for transporting delicate items (e.g., a laptop or glass) are designed
to “absorb” some of the energy of the impact in order to protect their contents.
These energy absorbers can get very complicated (e.g., the mechanics of
Styrofoam peanuts can be complex), but we can begin to understand how
they work by modeling them as a linear elastic spring of constant
k
that is
placed between the contents (an expensive vase) of mass
m
and the package
P
. Assume that the vase’s mass is
3kg
and that the box is dropped from
rest from a height of
1:5
m. Treating the vase as a particle and neglecting
all forces except for gravity and the spring force, determine the value of the
spring constant
k
so that the maximum displacement of the vase relative to
the box is
0:15
m. Assume that the spring relaxes after the box is dropped and
that it does not oscillate.
Solution
We denote by
¿
and
¡
the positions of the vase at which it is dropped (
h
in
the figure is equal to
3
m) and at which the maximum displacement is achieved,
Dynamics 2e 729
Problem 4.49
The pendulum is released from rest when
✓D0ı
. If the string holding the
pendulum bob breaks when the tension is twice the weight of the bob, at
what angle does the string break? Treat the pendulum as a particle, ignore
air resistance, and let the string be inextensible and massless.
Solution
To find the desired value of
✓
, we need the tension in the cord
Fc
as a function
730 Solutions Manual
Problem 4.50
The force acting on a stationary electric charge
qA
interacting with a charge
qB
is described by Coulomb’s
law and takes the form
E
FDkqAqB
r2Our;
where
kD8:9875 ⇥109
N
m
2=
C
2
(C is the symbol for coulomb, the unit used to measure
electric charge) is a constant and
r
is the distance between
A
and
B
. This force law is mathemati-
cally very similar to Newton’s universal gravitation law. With this in mind, determine an expression of the
electrostatic potential energy, choosing the datum at infinity, i.e., such that the potential energy is equal to
zero when the two charges are separated by an infinite distance.
Solution
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 731
Problem 4.51
The force-compression profile of a rubber bumper
B
is given by
FBDˇx3
, where
ˇD3:5⇥106lb=ft3
and
x
is the bumper’s compression measured in the horizontal direction. Determine the expression for the
potential energy of the bumper
B
. In addition, if the cruiser
C
weighs
70;000 lb
and impacts
B
with a
speed of
5ft=s
, determine the compression required to bring
C
to a stop. Model
C
as a particle and neglect
C’s vertical motion as well as the drag force between the water and the cruiser C.
Solution
We model
C
as a particle subject only to its own weight
mg
, the force
of the bumper
FB
, and the force of the water
FW
. We neglect vertical
732 Solutions Manual
A satellite orbits the Earth along the orbit shown. The minimum
and maximum distances from the center of the Earth are
RPD
4:5 ⇥107
m and
RAD6:163 ⇥107
m, respectively, where the
subscripts
P
and
A
stand for perigee (the point on the orbit closest
to Earth) and apogee (the point on the orbit farthest from Earth),
respectively. Modeling the satellite as a particle and assuming
that the center of the Earth can be chosen as the origin of an
inertial frame of reference, if the satellite’s speed at
P
is
jEvjPD
3:2⇥103m=s, determine the satellite’s speed at A.
Solution
We model the satellite as a particle subject only to the gravitational
attraction
FG
. We use a polar coordinate system with origin at
O
,
Dynamics 2e 733
Problem 4.53
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 system is released from rest when
✓D30ı
, determine the
maximum angle ✓reached by the arm AB.
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
734 Solutions Manual
Next, we determine
`0
by considering the equilibrium of the system
when
✓D0
. Referring to the FBD at the right, setting equal to zero the
Dynamics 2e 735
Problem 4.54
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 system is released from rest when
✓D30ı
, determine the
maximum speed achieved by the ball
B
, and determine the angle at which
it occurs.
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
736 Solutions Manual
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.