Problem 8.132
A uniform thin ring
A
and a uniform disk
B
roll without slip as
shown. Letting
TA
and
TB
be the kinetic energies of
A
and
B
,
respectively, if the two objects have the same mass and radius and if
their centers are moving with the same speed
v0
, state which of the
following statements is true and why: (a)
TA<T
B
; (b)
TADTB
;
(c) TA>T
B.
Solution
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Dynamics 2e 1889
Problem 8.133
At the instant shown, the disk
D
, which has mass
m
and radius of
gyration
kG
, is rolling without slip down the flat incline with angular
velocity
!0
. The disk is attached at its center to a wall by a linear
elastic spring of constant
k
. If, at the instant shown, the spring is
unstretched, determine the distance
d
down the incline that the disk
rolls before coming to a stop. Use
kD65 N=m
,
RD0:3
m,
mD
10 kg, kGD0:25 m, !0D60 rpm, and ✓D30ı.
Solution
We model
D
a a rigid body rolling without slipping on a flat surface
under the action of its own weight mg, the spring force Fs, and the
components
F
and
N
of the reaction force with the incline. Because
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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.
Dynamics 2e 1891
Problem 8.134
A pendulum consists of a uniform disk
A
of diameter
dD5in:
and weight
WAD
0:25 lb
attached at the end of a uniform bar
B
of length
LD2:75 ft
and weight
WBD1:3 lb
. At the instant shown, the pendulum is swinging with an angular velocity
!D0:55 rad=s
clockwise. Determine the kinetic energy of the pendulum at this instant,
using Eq. (8.11) on p. 584.
Solution
Observe that the system is in a fixed axis rotation about point
O
, so the kinetic energy of the system is given
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permission of McGraw-Hill, is prohibited.
1892 Solutions Manual
A uniform disk
D
of radius
RDD7mm
and mass
mDD0:15 kg
is
connected to point
O
via the rotating arm OC and rolls without slip
over the stationary cylinder
S
of radius
RSD15 mm
. Assuming that
!DD25 rad=s
, and treating the arm OC as a uniform slender bar of
length
LDRDCRS
and mass
mOC D0:08 kg
, determine the kinetic
energy of the system.
Solution
The kinetic energy of the system is equal to the sum of the kinetic energy
of each part of the system. Referring to the figure at the right, we label
Q
the point on the disk
D
that is in contact with the stationary cylinder
S
. Observing that the arm is in a fixed axis rotation about
O
, that the disk
Dynamics 2e 1893
Problem 8.136
The figure shows the cross section of a garage door with length
LD9ft
and weight
WD175 lb
. At the ends
A
and
B
there are rollers of neg-
ligible mass constrained to move in a vertical and a horizontal guide,
respectively. The door’s motion is assisted by two springs (only one
spring is shown), each with constant
kD9:05 lb=ft
. If the door is
released from rest when horizontal and the spring is stretched
4in:
,
neglecting friction, and modeling the door as a uniform thin plate, de-
termine the speed with which
B
strikes the left end of the horizontal
guide.
Solution
We model the door
AB
as a rigid bar subject to its own weight
mAB g
,
the reactions
NA
and
NB
at
A
and
B
, respectively, and the spring forces
2Fs. Work is done only by gravity and the spring force, both of which
are conservative. Let
¿
be the position at release, and
¡
be the position
achieved by the door when
B
strikes the left end of the horizontal guide.
We use subscripts
1
and
2
to denote quantities at
¿
and
¡
, respectively.
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 1895
Problem 8.137
Body
B
has mass
m
and mass moment of inertia
IG
, where
G
is the mass center
of
B
. If
B
is in fixed-axis rotation about its center of mass
G
, determine which of
the following statements is true and why: (a)
ˇˇE
hEBˇˇ<ˇˇE
hPBˇˇ
, (b)
ˇˇE
hEBˇˇD
ˇˇE
hPBˇˇ, (c) ˇˇE
hEBˇˇ>ˇˇE
hPBˇˇ.
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.
Problem 8.138
The weights of the uniform thin pin-connected bars
AB
,BC,
and CD are
WAB D4lb
,
WBC D6:5 lb
, and
WCD D10 lb
,
respectively. Letting
D47ı
,
RD2ft
,
LD3:5 ft
, and
HD
4:5 ft
, and knowing that bar
AB
rotates at a constant angular
velocity
!AB D4rad=s
, compute the angular momentum of the
system about Dat the instant shown.
For the velocity of point Ewe can write
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.
Substituting the result in Eq. (7) along with Eqs. (9) into Eqs. (8), carrying out the cross products, and
enforcing equality between the two expressions for EvC, we have
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permission of McGraw-Hill, is prohibited.