1718 Solutions Manual
A payload
B
of mass
mBD50 kg
is lifted via the pulley system shown by the application
of a constant force
FD300
N. The pulleys are identical and can be modeled as uniform
disks of radius
rpD10 cm
and mass
mpD8kg
. The cord does not slip relative to the
pulleys. Modeling the cord as inextensible and neglecting friction at the pulley bearings,
determine the speed of
B
after
B
has been lifted a height
hD1
m from its initial rest
position. Treat all cable segments as being purely vertical.
Solution
We model the payload as a particle subject to its own weight
mBg
and, following
Dynamics 2e 1719
Kinematic Equations. Denoting by Lthe length of the cord, we have
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permission of McGraw-Hill, is prohibited.
1720 Solutions Manual
A winch drawing
9hp
powers a pulley system lifting a
600 lb
crate
C
with a constant
speed
vc
. The pulleys are all identical and have a radius
rpD1:25 ft
. The cord is
inextensible and does not slip relative to the pulleys. For each pulley, the friction
at the pulley bearings produces a moment about the pulley’s center with magnitude
j!pj
opposing the rotation of the pulley, where
D1:5 lbfts
and
j!pj
is the angular
speed of the pulley. Neglecting the inertia of the pulleys and of the cord, and treating
segments of cord that do not touch the pulleys as being vertical, determine
vc
if the
motor’s efficiency is
✏D0:87
.Hint: Adapt to this problem the solution in Part (a) of
Example 4.16 on p. 300, observing that friction at the pulley bearings causes the tension
in the cord on the two sides of a pulley to be different.
Solution
We model the crate as a particle subject only to its own weight
mcg
and to the tension
F1
in the cord connecting
C
to pulley
B
. Pulley
B
Dynamics 2e 1721
Kinematic Equations.
Points on the portion of the cord immediately to the right of
A
move with the same
speed as the crate, i.e.,
vc
. Pulley
A
is in a fixed axis rotation about its center and the cord does not slip
relative to it. Hence,
!ADvc=r
p
. Points on the cord immediately to the left of
A
also travel with a speed
vc
.
Hence, enforcing again rolling without slipping, we conclude that
!BD2vc=r
p
. Proceeding in a similar
way, we also conclude that
!DD3vp=r
p
. Finally, points on the cord immediately to the right of pulleys
D
travel with a speed equal to 3vc. In summary, we have
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permission of McGraw-Hill, is prohibited.
1722 Solutions Manual
A
10 lb
uniform thin bar
BC
of length
LD10 ft
is pinned at
B
to
the edge of a
20 lb
uniform disk of radius
RD3:5 ft
. The system
is initially at rest in the position shown when a constant horizontal
force
PD60 lb
is applied to the end
C
. Assume that the disk
rolls without slip. In addition, neglect the friction between the
end
C
of the bar and the ground, as well as the friction at the pin.
Determine the angular speed of the disk when point
B
is directly
above the center of the disk. Hint: To determine the displacement
of point
C
, keep in mind that the overall displacement of point
A
is ⇡R=2.
Solution
Dynamics 2e 1723
Kinematic Equations.
Referring to the bottom two figures, denoting by
xA
and
xC
the
x
coordinates of
A
and C, respectively,
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permission of McGraw-Hill, is prohibited.
1724 Solutions Manual
The two blocks
A
and
B
weighing
30 lb
and
25 lb
, respectively, are released from
rest. The pulleys are identical and can be modeled as uniform disks of radius
rpD0:75 ft
and weight
WpD8lb
. Modeling the cord as inextensible and
neglecting friction at the pulley bearings, determine the speed of
A
after
B
has
dropped a height hD3ft.
Solution
Dynamics 2e 1725
Kinematic Equations. Denoting by Lthe length of the cord,
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.
1726 Solutions Manual
An eccentric wheel with mass
mD150 kg
, mass center
G
, and radius of
gyration
kGD0:4
m is placed on the incline shown, such that the wheel’s
center of mass
G
is vertically aligned with
P
, which is the point of contact
with the incline. If the wheel rolls without slip once it is gently nudged away
from its initial placement, letting
RD0:55
m,
hD0:25
m,
✓D25ı
, and
dD0:5
m, determine whether the wheel arrives at
B
and, if yes, determine the
corresponding speed of the center
O
. Note that the angle
POG
is not equal to
90ıat release.
Solution
We model the wheel as a rigid body subject to its own weight
mg
and the components
N
and
F
of the
Dynamics 2e 1727
where we have used the data
RD0:55
m,
hD0:25
m,
✓D25ı
, and
dD0:5
m. As far as speeds are
concerned, recalling that the system starts from rest, due to rolling without slip we have
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permission of McGraw-Hill, is prohibited.