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PROBLEM 17.64 (Continued)
Moments about A:
0
AAB AA A A
rT t rTt I
6
0.9 0.9
0 ( ) (0.75)(0.24) (169.837 10 )(133.333)
12 12
0.48193 lb s
AB
AB
Tt
Tt
PROBLEM 17.65
Show that the system of momenta for a rigid body in plane motion reduces to a single vector, and express the
distance from the mass center G to the line of action of this vector in terms of the centroidal radius of gyration
k of the body, the magnitude
v
of the velocity of G, and the angular velocity
.
PROBLEM 17.66
Show that, when a rigid body rotates about a fixed axis through O perpendicular
to the body, the system of the momenta of its particles is equivalent to a single
vector of magnitude
,mr
perpendicular to the line OG, and applied to a Point
P on this line, called the center of percussion, at a distance
2
/GP k r
from the
mass center of the body.
PROBLEM 17.67
Show that the sum
A
H
of the moments about a Point A of the momenta of the particles of a rigid body in
plane motion is equal to
,
A
I
where
is the angular velocity of the body at the instant considered and
A
I
the
moment of inertia of the body about A, if and only if one of the following conditions is satisfied: (a) A is
the mass center of the body, (b) A is the instantaneous center of rotation, (c) the velocity of A is directed along
a line joining Point A and the mass center G.
PROBLEM 17.68
Consider a rigid body initially at rest and subjected to an impulsive force
F
contained in the plane of the body. We define the center of percussion P as the point
of intersection of the line of action of
F
with the perpendicular drawn from G.
(a) Show that the instantaneous center of rotation C of the body is located on line GP
at a distance
2
/GC k GP
on the opposite side of G. (b) Show that if the center of
percussion were located at C the instantaneous center of rotation would be located
at P.
1
12
2
PROBLEM 17.68 (Continued)
Components parallel to :t
F
0
Ft mv
Moments about G:
0()
P
dFt I
2
vIk
md d
PROBLEM 17.69
A flywheel is rigidly attached to a 1.5-in.-radius shaft that rolls without sliding
along parallel rails. Knowing that after being released from rest the system
attains a speed of 6 in./s in 30 s, determine the centroidal radius of gyration of
the system.
SOLUTION
PROBLEM 17.70
A wheel of radius r and centroidal radius of gyration k is released from rest on
the incline shown at time
0.
t
Assuming that the wheel rolls without sliding,
determine (a) the velocity of its center at time t, (b) the coefficient of static
friction required to prevent slipping.
22
rk
PROBLEM 17.70 (Continued)
components normal to incline:
0cos0
Nt mgt
PROBLEM 17.71
The double pulley shown has a mass of 3 kg and a radius of gyration of 100 mm.
Knowing that when the pulley is at rest, a force
P
of magnitude 24 N is applied
to cord B, determine (a) the velocity of the center of the pulley after 1.5 s,
(b) the tension in cord C.
SOLUTION
For the double pulley,
0.150 m
0.080 m
0.100 m
C
B
r
r
k
Principle of impulse and momentum.
22
3(0.100 0.150 )
17.0077 rad/s
PROBLEM 17.71 (Continued)
PROBLEM 17.72
A 9-in.-radius cylinder of weight 18 lb rests on a 6-lb carriage. The
system is at rest when a force
P
of magnitude 2.5 lb is applied as shown
for 1.2 s. Knowing that the cylinder rolls without sliding on the carriage
and neglecting the mass of the wheels of the carriage, determine the
resulting velocity of (a) the carriage, (b) the center of the cylinder.
SOLUTION
2
2
2
1
2
118lb 9in.
2 32.2 12
0.15722 slug ft
A
Cylinder alone:
Moments about C:
00
Imvr
A
Cylinder and carriage:
AB
PROBLEM 17.72 (Continued)
Kinematics.
AB
vvr
9
PROBLEM 17.73
A 9-in.-radius cylinder of weight 18 lb rests on a 6-lb carriage. The system is
at rest when a force
P
of magnitude 2.5 lb is applied as shown for 1.2 s.
Knowing that the cylinder rolls without sliding on the carriage and
neglecting the mass of the wheels of the carriage, determine the resulting
velocity of (a) the carriage, (b) the center of the cylinder.
SOLUTION
2
2
2
1
2
118lb 9in.
2 32.2 12
0.15722 slug ft
A
Cylinder alone:
1
Syst. Momenta
12
Syst. Ext. Imp.
2
Syst. Momenta
Moments about C:
0
Ptr I m v r
PROBLEM 17.73 (Continued)
Kinematics.
AB
vvr
9
A
PROBLEM 17.74
Two uniform cylinders, each of mass
6
m
kg and radius
125
r
mm,
are connected by a belt as shown. If the system is released from rest
when
0,t
determine (a) the velocity of the center of cylinder B at
3s,t (b) the tension in the portion of belt connecting the two
cylinders.
1
12
2
PROBLEM 17.74 (Continued)
Moments about C:
02
11
02 22
22 2
AA
BBB
B
Ptr mgtr mv r I
ImgtrmrrI
7
4
4
B
rgt
g
t
(2)
277
(b) Tension in the belt.
From Eqs. (1) and (2), 4
7
g
t
Ptr I r
11 4 2 2
2777
gt
tr r
16.82 NP
PROBLEM 17.75
Two uniform cylinders, each of mass
6m
kg and radius r125 mm,
are connected by a belt as shown. Knowing that at the instant shown the
angular velocity of cylinder A is 30 rad/s counterclockwise, determine
(a) the time required for the angular velocity of cylinder A to be reduced
to 5 rad/s, (b) the tension in the portion of belt connecting the two
cylinders.
1
12
2
PROBLEM 17.75 (Continued)
Moments about C: 11 2 2
() () 2 () ()
AA A A
I
m v r Ptr mgtr I m v r
2
1[( ) ( ) [( ) ( ) ] 2 0
mr mr r Ptr mgtr
(4)(9.81)
(b) Tension in belt between cylinders.
1(6)(0.125) (30 5)
(0.55747)(0.125)
PROBLEM 17.76
In the gear arrangement shown, gears A and C are attached to rod ABC,
which is free to rotate about B, while the inner gear B is fixed. Knowing
that the system is at rest, determine the magnitude of the couple
M
which
must be applied to rod ABC, if 2.5 s later the angular velocity of the rod is
to be 240 rpm clockwise. Gears A and C weigh 2.5 lb each and may be
considered as disks of radius 2 in.; rod ABC weighs 4 lb.
1
12
2
