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PROBLEM 17.27
Greek engineers had the unenviable task of moving large
columns from the quarries to the city. One engineer,
Chersiphron, tried several different techniques to do this. One
method was to cut pivot holes into the ends of the stone and
then use oxen to pull the column. The 4-ft diameter column
weighs 12,000 lbs, and the team of oxen generates a constant
pull force of 1500 lbs on the center of the cylinder G. Knowing
that the column starts from rest and rolls without slipping,
determine (a) the velocity of its center G after it has moved
5 ft, (b) the minimum static coefficient of friction that will keep
it from slipping.
SOLUTION
12000 lb =372.67 slugs
2
gg
r
2
PROBLEM 17.27 (Continued)
Free Body Diagram:
s
N
s
PROBLEM 17.28
A small sphere of mass m and radius r is released from rest at A
and rolls without sliding on the curved surface to Point B where
it leaves the surface with a horizontal velocity. Knowing that
a 1.5 m and b 1.2 m, determine (a) the speed of the sphere as
it strikes the ground at C, (b) the corresponding distance c.
SOLUTION
Work:
12
Umga
Kinetic energy:
1
0T
v
2
( ) (9.81 m/s )(0.49462 s) 4.8522 m/s
yC C
vgt
PROBLEM 17.28 (Continued)
Horizontal motion: Let the x coordinate point to the left with origin below B.
PROBLEM 17.29
The mass center G of a 3-kg wheel of radius
180 mmR
is located at a distance
r 60 mm from its geometric center C. The centroidal radius of gyration of the wheel
is 90 mm.k As the wheel rolls without sliding, its angular velocity is observed to
vary. Knowing that 8 rad/s
in the position shown, determine (a) the angular
velocity of the wheel when the mass center G is directly above the geometric center C,
(b) the reaction at the horizontal surface at the same instant.
SOLUTION
22
22 2
222
22
2
2
(3)(9.81)(0.06)
1.7658 J
11
22
11
(3)(0.24 ) (3)(0.09)
22
0.09855
mgh
Tmv I
PROBLEM 17.29 (Continued)
(a) Conservation of energy.
11 2 2
2
2
2
2
2
4.2336 J 0 0.09855 1.7658 J
25.041
TV T V
2
(b) Reaction at B.
2
PROBLEM 17.30
A half-cylinder of mass m and radius r is released from rest in the position shown.
Knowing that the half-cylinder rolls without sliding, determine (a) its angular velocity
after it has rolled through 90 , (b) the reaction at the horizontal surface at the same
instant. [Hint: Note that 4/3GO r
and that, by the parallel-axis theorem,
22
1().
2
I
mr m GO
]
r
PROBLEM 17.30 (Continued)
Kinematics:
Translation Rotation about O
Rolling Motion
(b) Acceleration of C. x component. 00
0 ar a r
Acceleration of G. x component.
0x
aaOG b
22
4
1.553 mgR
PROBLEM 17.31
A sphere of mass m and radius r rolls without slipping inside a curved
surface of radius R. Knowing that the sphere is released from rest in the
position shown, derive an expression (a) for the linear velocity of the sphere
as it passes through B, (b) for the magnitude of the vertical reaction at that
instant.
G
PROBLEM 17.31 (Continued)
The sphere rolls so that its mass center moves on a circle of radius .Rr
2
2
v
7
PROBLEM 17.32
Two uniform cylinders, each of weight W 14 lb and radius r 5 in., are connected by
a belt as shown. Knowing that at the instant shown the angular velocity of cylinder B is
30 rad/s clockwise, determine (a) the distance through which cylinder A will rise before
the angular velocity of cylinder B is reduced to 5 rad/s, (b) the tension in the portion of
belt connecting the two cylinders.
PROBLEM 17.32 (Continued)
Principle of work and energy.
22 22
1122 1 2
77
:() ()
BB
TU T mr mgh mr
2
22
12
7[( ) ( ) ]
16 BB
r
2
ft [(30 rad/s) (5 rad/s) ] 2.064 ft
16 12 32.2 ft/s
h
2
22
Q I mr mg W
7
PROBLEM 17.33
Two uniform cylinders, each of weight W 14 lb and radius r 5 in., are connected by
a belt as shown. If the system is released from rest, determine (a) the velocity of the
center of cylinder A after it has moved through 3 ft, (b) the tension in the portion of belt
connecting the two cylinders.
PROBLEM 17.33 (Continued)
Principle of work and energy:
2
2
714lb 5
2212
7.43 ft/s
(b) Tension in cord .
D
E Let Q be its value.
Recall that
2
DA
vv thus D moves twice the distance that A moves, i.e 2h
1
0
T
PROBLEM 17.34
A bar of mass m 5 kg is held as shown between
four disks each of mass
m
2 kg and radius
r 75 mm. Knowing that the forces exerted on
the disks are sufficient to prevent slipping and
that the bar is released from rest, for each of the
cases shown determine the velocity of the bar
after it has moved through the distance h.
2
a
PROBLEM 17.34 (Continued)
Kinematics and kinetic energy for case (b).
The instantaneous center C of a typical disk lies at its point of contact
with the fixed wall.
2
v
r
1,
1
1122
(a)
05 04.5gh v
1.054vgh
2
PROBLEM 17.35
The 1.5-kg uniform slender bar AB is connected to the 3-kg gear B
which meshes with the stationary outer gear C. The centroidal radius of
gyration of gear B is 30 m. Knowing that the system is released from
rest in the position shown, determine (a) the angular velocity of the bar
as it passes through the vertical position, (b) the corresponding angular
velocity of gear B.
2
Conservation of energy. 11 2 2
:TV T V
22
0.050
PROBLEM 17.36
The motion of the uniform rod AB is guided by small wheels of
negligible mass that roll on the surface shown. If the rod is released from
rest when
0,
determine the velocities of A and B when
30 .
0.6
L
PROBLEM 17.37
A 5-m long ladder has a mass of 15 kg and is placed against a house at an
angle 20 .
Knowing that the ladder is released from rest, determine the
angular velocity of the ladder and the velocity of A when 45 .
Assume
the ladder can slide freely on the horizontal ground and on the vertical
wall.
/2.GA GB GC L The velocity of the mass center G is
/2
G
vv L
Kinetic energy: 22
11
Tmv I
2
PROBLEM 17.37 (Continued)
Position 1. 20 ;
rest (T1 0)
Position 2. 45 ; ?
11
LL
