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PROBLEM 16.93
Show that in the case of an unbalanced disk, the equation derived in Problem 16.92 is valid only when the
mass center G, the geometric center O, and the instantaneous center C happen to lie in a straight line.
PROBLEM 16.94
A wheel of radius r and centroidal radius of gyration
k
is released from rest on
the incline and rolls without sliding. Derive an expression for the acceleration of
the center of the wheel in terms of r,
,k
,
β
and g.
PROBLEM 16.95
A homogeneous sphere S, a uniform cylinder C, and a thin pipe P
are in contact when they are released from rest on the incline
shown. Knowing that all three objects roll without slipping,
determine, after 4 s of motion, the clear distance between (a) the
pipe and the cylinder, (b) the cylinder and the sphere.
PROBLEM 16.95 (Continued)
(b) Between sphere and cylinder.
52 1
/
2 21
SC
PROBLEM 16.96
A 40-kg flywheel of radius R = 0.5 m is rigidly attached to a shaft of radius
r = 0.05 m that can roll along parallel rails. A cord is attached as shown and
pulled with a force P of magnitude 150 N. Knowing the centroidal radius of
gyration is
0.4 m,k=
determine (a) the angular acceleration of the
flywheel, (b) the velocity of the center of gravity after 5 s.
SOLUTION
PROBLEM 16.96 (Continued)
(b) Velocity of center of gravity after 5 s.
22
PROBLEM 16.97
A 40-kg flywheel of radius R = 0.5 m is rigidly attached to a shaft of radius
r = 0.05 m that can roll along parallel rails. A cord is attached as shown and
pulled with a force P. Knowing the centroidal radius of gyration is
0.4 mk=
and the coefficient of static friction is
0.4,
s
µ
=
determine the
largest magnitude of force P for which no slipping will occur.
PROBLEM 16.97 (Continued)
Let C be the contact point between the flywheel and the rails.
PROBLEM 16.98
A drum of 60-mm radius is attached to a disk of 120-mm radius. The
disk and drum have a total mass of 6 kg and a combined radius of
gyration of 90 mm. A cord is attached as shown and pulled with a force
P of magnitude 20 N. Knowing that the disk rolls without sliding,
determine (a) the angular acceleration of the disk and the acceleration of
G, (b) the minimum value of the coefficient of static friction compatible
with this motion.
SOLUTION
2
2
32
(0.12 m)
(6 kg)(0.09 m)
48.6 10 kg m
ar
I mk
I
aa
−
= =
=
=
=×⋅
eff
( ) : (20 N)(0.12 m) ( )
CC
M M ma r I
a
Σ=Σ = +
2 32
3
2.4 N m (6 kg)(0.12 m) 48.6 10 kg m
2.4 135.0 10
a
a
−
−
⋅= + × ⋅
= ×
(a)
2
17.778 rad/s
a
=
2
17.78 rad/s=α
2
2
(0.12 m)(17.778 rad/s )
2.133 m/s
ar
a
= =
=
2
2.13 m/s=a
(b)
eff
( ):
yy
FFΣ=Σ
0N mg−=
2
(6 kg)(9.81 m/s )N=
58.86 N=N
PROBLEM 16.99
A drum of 60-mm radius is attached to a disk of 120-mm radius. The
disk and drum have a total mass of 6 kg and a combined radius of
gyration of 90 mm. A cord is attached as shown and pulled with a force P
of magnitude 20 N. Knowing that the disk rolls without sliding,
determine (a) the angular acceleration of the disk and the acceleration of
G, (b) the minimum value of the coefficient of static friction compatible
with this motion.
SOLUTION
2
2
32
(0.12 m)
(6 kg)(0.09 m)
48.6 10 kg m
ar
I mk
I
aa
−
= =
=
=
=×⋅
PROBLEM 16.100
A drum of 60-mm radius is attached to a disk of 120-mm radius. The
disk and drum have a total mass of 6 kg and a combined radius of
gyration of 90 mm. A cord is attached as shown and pulled with a force
P of magnitude 20 N. Knowing that the disk rolls without sliding,
determine (a) the angular acceleration of the disk and the acceleration
of G, (b) the minimum value of the coefficient of static friction
compatible with this motion.
SOLUTION
2
2
2
(0.12 m)
(6 kg)(0.09 m)
48.6 kg m
ar
I mk
I
aa
= =
=
=
= ⋅
PROBLEM 16.101
A drum of 60-mm radius is attached to a disk of 120-mm radius. The disk and
drum have a total mass of 6 kg and a combined radius of gyration of 90 mm. A
cord is attached as shown and pulled with a force P of magnitude 20 N.
Knowing that the disk rolls without sliding, determine (a) the angular
acceleration of the disk and the acceleration of G, (b) the minimum value of
the coefficient of static friction compatible with this motion.
SOLUTION
2
2
2
(0.12 m)
(6 kg)(0.09 m)
48.6 kg m
ar
I mk
I
aa
= =
=
=
= ⋅
PROBLEM 16.102
A drum of 4-in. radius is attached to a disk of 8-in. radius. The disk and
drum have a combined weight of 10 lb and a combined radius of
gyration of 6 in. A cord is attached as shown and pulled with a force P of
magnitude 5 lb. Knowing that the coefficients of static and kinetic
friction are
0.25
s
µ
=
and
0.20,
k
µ
=
respectively, determine (a)
whether or not the disk slides, (b) the angular acceleration of the disk
and the acceleration of G.
SOLUTION
Assume disk rolls:
8ft
12
ar
aa
= =
2
2
10 lb 6 ft
32.2 12
2
0.07764 lb ft sI= ⋅⋅
eff
8
( ) : (5 lb) ft ( )
12
CC
M M ma r I
a
Σ=Σ = +
2
10 lb 8
3.333 lb ft ft 0.07764
32.2 12
aa
⋅= +
3.333 0.21566
a
=
2
15.456 rad/s
a
=
2
15.46 rad/s=a
2
8ft (15.456 rad/s )
12
ar
a
= =
2
10.30 ft/s=a
eff
( ) : 5 lb
xx
F F F maΣ =Σ −+ =
2
10 lb
5 lb (10.30 ft/s )
32.2
1.80 lb
F
F
−+ =
=
eff
( ):
yy
FFΣ=Σ
10 lb 0N−=
10 lbN=
0.25(10 lb) 2.5 lb
ms
FN
µ
= = =
(a) Since
,
m
FF<
Disk rolls without sliding
(b) Angular acceleration of the disk.
2
15.46 rad/s=a
Acceleration of G.
2
10.30 ft/s=a
PROBLEM 16.103
A drum of 4-in. radius is attached to a disk of 8-in. radius. The disk and
drum have a combined weight of 10 lb and a combined radius of
gyration of 6 in. A cord is attached as shown and pulled with a force P
of magnitude 5 lb. Knowing that the coefficients of static and kinetic
friction are
0.25
s
µ
=
and
0.20,
k
µ
=
respectively, determine
(a) whether or not the disk slides, (b) the angular acceleration of the disk
and the acceleration of G.
SOLUTION
Assume disk rolls:
8ft
12
ar
aa
= =
2
I mk=
2
10 lb 6 ft
PROBLEM 16.104
A drum of 4-in. radius is attached to a disk of 8-in. radius. The disk and
drum have a combined weight of 10 lb and a combined radius of
gyration of 6 in. A cord is attached as shown and pulled with a force P
of magnitude 5 lb. Knowing that the coefficients of static and kinetic
friction are
0.25
s
µ
=
and
0.20,
k
µ
=
respectively, determine (a) whether
or not the disk slides, (b) the angular acceleration of the disk and the
acceleration of G.
32.2 12
2
0.07764 lb ft sI= ⋅⋅
4
PROBLEM 16.104 (Continued)
(b) Angular acceleration.
eff
84
( ) : ft (5 lb) ft
12 12
GG
MM F I
a
Σ=Σ − =
2
8
(2 lb) ft 1.6667 lb ft (0.07764 lb ft s )
a
− ⋅= ⋅⋅
PROBLEM 16.105
A drum of 4-in. radius is attached to a disk of 8-in. radius. The disk and drum
have a combined weight of 10 lb and a combined radius of gyration of 6 in. A
cord is attached as shown and pulled with a force P of magnitude 5 lb.
Knowing that the coefficients of static and kinetic friction are
0.25
s
µ
=
and
0.20,
k
µ
=
respectively, determine (a) whether or not the disk slides, (b) the
angular acceleration of the disk and the acceleration of G.
32.2 12
2
0.07764 lb ft sI= ⋅⋅
4
12 ma r I
2
10 lb 8
1.6667 lb ft ft 0.07764
32.2 12
aa
⋅= +
1.6667 0.21566
a
=
2
7.728 rad/s
a
=
2
7.73 rad/s=a
2
8ft (7.728 rad/s )
12
ar
a
= =
2
5.153 ft/s=a
eff
( ):
xx
FFΣ=Σ
F ma=
2
10 lb (5.153 ft/s ); 1.60 lb
32.2
FF= =
eff
( ):
yy
FFΣ=Σ
10 lb 5 lb 0N− +=
5 lbN=
PROBLEM 16.105 (Continued)
(b) Angular acceleration.
48
( ) : (5 lb) ft ft
MM F I
a
Σ=Σ − =
PROBLEM 16.106
A 12-in.-radius cylinder of weight 16 lb rests on a 6-lb carriage. The
system is at rest when a force P of magnitude 4 lb is applied. Knowing
that the cylinder rolls without sliding on the carriage and neglecting the
mass of the wheels of the carriage, determine (a) the acceleration of the
carriage, (b) the acceleration of Point A, (c) the distance the cylinder
has rolled with respect to the carriage after 0.5 s.
SOLUTION
Masses and moments of inertia.
22
16 lb 6 lb
AB
Cylinder alone. Point C is contact point.
PROBLEM 16.106 (Continued)
eff
( ):0
C C A AA
M M I mar
a
Σ=Σ = +
