466 Solutions Manual
Problem 3.40
A
6lb
collar is constrained to travel along a rectilinear and friction-
less bar of length
LD5ft
. The springs attached to the collar are
identical and are unstretched when the collar is at
B
. Treating the
collar as a particle, neglecting air resistance, and knowing that at
A
the collar is moving to the right with a speed of
11 ft=s
, determine
the linear spring constant
k
so that the collar reaches
D
with zero
speed. Points Eand Fare fixed.
Solution
Referring to the FBD at the right, we model the collar as a particle subject
Dynamics 2e 467
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permission of McGraw-Hill, is prohibited.
468 Solutions Manual
Problem 3.41
A
10 lb
collar is constrained to travel along a rectilinear and fric-
tionless bar of length
LD5ft
. The springs attached to the collar
are identical, they have a spring constant
kD4lb=ft
, and they
are unstretched when the collar is at
B
. Treating the collar as a
particle, neglecting air resistance, and knowing that at
A
the collar
is moving to the right with a speed of
14 ft=s
, determine the speed
with which the collar arrives at D. Points Eand Fare fixed.
Solution
Referring to the FBD at the right, we model the collar as a particle subject
only to its own weight
W
, the normal reaction
N
due to the contact with the
Dynamics 2e 469
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.
470 Solutions Manual
Problem 3.42
An
11 kg
collar is constrained to travel along a rectilinear and frictionless bar
of length
LD2
m. The springs attached to the collar are identical, and they are
unstretched when the collar is at
B
. Treating the collar as a particle, neglecting
air resistance, and knowing that at
A
the collar is moving upward with a speed
of
23 m=s
, determine the linear spring constant
k
so that the collar reaches
D
with zero speed. Points Eand Fare fixed.
Solution
Referring to the FBD at the right, we model the collar as a particle subject only to
its own weight
mg
, the normal reaction
N
due to the contact with the bar, and the
Dynamics 2e 471
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472 Solutions Manual
Problem 3.43
Derive the equation of motion of the mass
m
released from rest at
xDx0
from the slingshot-like device (the mass
m
is attached to
the elastic cords). Assume that the cords connecting the mass to
the device are linear springs with spring constant
k
and unstretched
length
L0
. In addition, assume that the mass is equidistant from
the two supports and that the mass and both springs lie in the
xy
plane. Ignore gravity and assume L>L
0.
Solution
Referring to the FBD at the right, ignoring gravity, and since the cords on either
Dynamics 2e 473
Problem 3.44
Determine the speed of the mass
m
when it reaches
xD0
if it
is released from rest at
xDx0
from the device in Prob. 3.43.
Assume that the cords connecting the mass to the device are linear
springs with spring constant kand unstretched length L0. Ignore
gravity and assume L>L
0.
Solution
Referring to the FBD at the right, ignoring gravity and since the cords on either
side of the mass are identical, we model the mass
m
as a particle moving only in
474 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.
Dynamics 2e 475
Problem 3.45
Given the approximation
1
p.x=L/2C1⇡1.x=L/2
2;(1)
show that the equation of motion for the mass
m
when it is released
from rest at
xDx0
from the device in Prob. 3.43 can be written
as
RxC!2
0x1C⇤x2D0; (2)
where
!2
0D2k
m LL0
L!and ⇤DL0
2L2.L L0/:(3)
Assume that the cords connecting the mass to the device are linear
springs with spring constant
k
, length
L
, and unstretched length
L0
. Ignore gravity and assume
L>L
0
. Equation (2) is a famous
equation in mechanics called Duffing’s equation.
Solution
Referring to the FBD at the right, ignoring gravity and due to the fact that the cords
on either side of the mass are identical, we model the mass
m
as a particle moving