536 Solutions Manual
Problem 3.84
The package handling system is designed to launch the small pack-
age of mass
m
from
A
, 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,
making any friction between the packages and the track negligible.
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, and determine the corresponding speed
with which the package reaches the conveyor at B.
Solution
We must determine the speed of the mass as it leaves the spring so that it reaches the top
Dynamics 2e 537
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.
538 Solutions Manual
Problem 3.85
A particle moves over the inner surface of an inverted cone. Assuming that the cone’s surface is frictionless
and using the cylindrical coordinate system shown, show that the particle’s equations of motion are
R
R1Ccot2RP
✓2Dgcot ;
RR
✓C2P
RP
✓D0:
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.
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.
540 Solutions Manual
Problem 3.86
Continue Prob. 3.85 by integrating the particle’s equations of motion and plotting its trajectory for
0t25
s. Use the following parameter values and initial conditions:
gD9:81 m=s2
,
D30ı
,
✓.0/ D0ı,P
✓.0/ D1:00 rad=s, R.0/ D5m, P
R.0/ D0m=s.
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.
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.
542 Solutions Manual
Problem 3.87
The disk shown, weighing
3lb
, rotates about
O
by sliding without friction over the horizontal surface
shown. The spring is linear with constant
k
and unstretched length
L0D0:75 ft
. The maximum distance
achieved by the disk from point
O
is
dmax D1:85 ft
while traveling at a speed
v0D20 ft=s
. Determine
the value of ksuch that the minimum distance between the disk and Ois dmin Ddmax=2.
Solution
We model the disk as a particle moving in the horizontal plane subject only to the spring
Dynamics 2e 543
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.
544 Solutions Manual
Problem 3.88
A pendulum with cord length
LD6ft
and a bob weighing
3lb
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 and 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 reached by the pendulum, measured from its
lowest point, if ✓iD20ı?
Solution
Dynamics 2e 545
where the minus sign in front of the square root is because, for
✓D0
, the pendulum is swinging to the left.
Now, the velocity of the pendulum bob has the form
EvDLP
✓Ou✓
(
PrD0
since
rDLDconstant
), for
✓D0