1818 Solutions Manual
a Cartesian coordinate system with origin at the center of pulley D and with axes x and y.
We have labeled
P
the point on the spool at which the cord comes off the spool. We recall that we model the
cord as inextensible and we assume that the cord does not slip relative to the spool or the pulley
C
. Since the
spool rolls without slipping, using rigid body kinematics, we have
mAgt3Zt2
t1
TdtD1
3mA!s2.R C2⇢/;
Zt2
Dynamics 2e 1819
Problem 8.97
A spool of mass
msD150 kg
and inner and outer radii
⇢D0:8
m
and
RD1:2
m, respectively, is connected to a counterweight
A
of
mass
mAD50 kg
by a pulley system whose cord, at one end, is
wound around the inner hub of the spool. The center
G
and the
center of mass of the spool coincide, and the radius of gyration of
the spool is
kGD1
m. The system is at rest when the counterweight
is released, causing the spool to move to the right. The spool rolls
without slip, and the cord unwinds from the spool without slip.
Assume that the inertia of the cord and of pulleys
B
and
D
can be neglected, but model pulley
C
as a uni-
form disk mass
mCD15 kg
and radius
rCD0:3
m.
If the cord does not slip relative to pulley
C
, use the impulse-
momentum principles to determine the angular speed of the spool
3
s
after release.
Solution
where
IG
is the mass moment of inertia of the spool about the spool’s center of mass
G
,
E!sD!sO
k
is the
angular velocity of the spool,
IH
is the mass moment of inertia of pulley
C
about
H
, and
E!CD!CO
k
is the
angular velocity of pulley C. For IGand IH, we have
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 1821
Problem 8.98
An
0:8 lb
collar with center of mass at
G
and a uniform cylindrical
horizontal arm
A
of length
LD1ft
, radius
riD0:022 ft
, and
weight
WAD1:5 lb
are rotating as shown with
!0D1:5 rad=s
while the collar’s mass center is at a distance
dD0:44 ft
from the
´
axis. The vertical shaft has radius eD0:03 ft and negligible mass.
After the cord restraining the collar is cut, the collar slides with
no friction relative to the arm. Assuming that no external forces
and moments are applied to the system, determine the collar’s
impact speed with the end of
A
if (a) the collar is modeled as a
particle coinciding with its own mass center (in this case, neglect
the collar’s dimensions), and (b) the collar is modeled as a uniform
hollow cylinder with length
`D0:15 ft
, inner radius
ri
, and outer
radius roD0:048 ft.
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.
1822 Solutions Manual
IOD.IO/ACIGDI⇤CmCr2;(4)
where
I⇤D.IO/ACIGD(0:01697 slugft2for the collar as a particle;
Dynamics 2e 1823
Problem 8.99
The uniform disk
A
, of mass
mAD1:2 kg
and radius
rAD0:25
m, is
mounted on a vertical shaft that can translate along the horizontal guide
C
. The uniform disk
B
, of mass
mBD0:85 kg
and radius
rBD0:38
m,
is mounted on a fixed vertical shaft. Both disks
A
and
B
can rotate about
their own axes, namely,
`A
and
`B
, respectively. Disk
A
is initially spun
with
!AD1000 rpm
and then brought into contact with
B
, which is
initially stationary. The contact is maintained by a spring, and due to
friction between
A
and
B
, disk
B
starts spinning and eventually
A
and
B
will stop slipping relative to one another. Neglecting any friction except
at the contact between the two disks, determine the angular velocities of
Aand Bwhen slipping stops.
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.
1824 Solutions Manual
where we have obtained the last equation by eliminating the term
Rt2
t1Fdt
from the first two equations. The
last of Eqs. (5) and the last of Eqs. (4) form a system of two equations in the two unknowns
!A2
and
!B2
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 1825
Problem 8.100
The uniform disk
A
, of mass
mAD1:2 kg
and radius
rAD0:25
m, is
mounted on a vertical shaft that can translate along the horizontal arm
E
.
The uniform disk
B
, of mass
mBD0:85 kg
and radius
rBD0:18
m, is
mounted on a vertical shaft that is rigidly attached to arm
E
. Disk
A
can
rotate about axis
`A
, disk
B
can rotate about axis
`B
, and the arm
E
, along
with disk
C
, can rotate about the fixed axis
`C
. Disk
C
has negligible mass
and is rigidly attached to
E
so that they rotate together. While keeping
both
B
and
C
stationary, disk
A
is spun to
!AD1200 rpm
. Disk
A
is
then brought in contact with disk
C
(contact is maintained by a spring),
and
B
and
C
(and the arm
E
) are then allowed to freely rotate. Due to
friction between
A
and
C
, disks
C
(and arm
E
) and
B
start spinning.
Eventually, Aand Cstop slipping relative to one another. Disk Balways
rotates without slip over
C
. Let
dD0:27
m and
wD0:95
m. If the only
elements of the system that have mass are
A
and
B
, and if all friction in the
system can be neglected except for that between
A
and
C
and between
C
and
B
, determine the angular speeds of
A
and
C
when they stop slipping
relative to one another.
Solution
We let the subscripts
1
and
2
denote the time instants at which the wheels
are first brought into contact (and the system starts spinning) and when slip
stops between the wheels, respectively. We use subscripts
1
and
2
to denote
quantities at times
t1
and
t2
, respectively. The figure at the right is a top view of
Dynamics 2e 1827
Problem 8.101
The double pulley
D
has mass of
mDD15 kg
, center of mass
G
coinciding with
its geometric center, radius of gyration
kGD10 cm
, outer radius
roD15 cm
, and
inner radius
riD7:5 cm
. It is connected to the pulley
P
with radius
RD.rori/=2
by a cord of negligible mass that unwinds from the inner and outer spools of the
double pulley
D
. The crate
C
, which has a mass
mCD20 kg
, is released from
rest. The cord does not slip relative to the pulleys, and the inner and outer pulleys
rotate as a single unit.
Neglecting the mass of the pulley
P
, use the impulse-momentum principles to
determine the speed of the crate 4s after release.
Solution