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PROBLEM 17.132 (Continued)
Add Equations (1) and (2) to eliminate .Pdt
11
or
AB BA
mv mv mv v v v
(3)
Condition of rolling without slipping:
AA
vr
Moments about C:
0
AAA
IImvr
1
7
A
vv
Motion after impact. Sphere B.
PROBLEM 17.132 (Continued)
Condition of rolling without slipping: BB
vr
Moments about C:
2
0
2
BBB
mv r I mv r
2
7
Av
5
Bv
PROBLEM 17.133
In a game of pool, ball A is rolling without slipping with a velocity
0
v
as it hits
obliquely ball B, which is at rest. Denoting by r the radius of each ball and by
k
the coefficient of kinetic friction between a ball and the table and assuming
perfectly elastic impact, determine (a) the linear and angular velocity of each ball
immediately after the impact, (b) the velocity of ball B after it has started rolling
uniformly.
By
PROBLEM 17.133 (Continued)
Moments about y axis:
00 cos
B
I
(7)
Moments about x axis:
00 sin
B
I
(8)
Adding Equations (1) and (5) to eliminate
,Pdt
0
Ay By
0
(cos)
B
vv
i
From Equations (3) and (4) simultaneously,
0
,
v
0
(sin cos )
v
ij
1
5
cos
55
BB
B
mr m r r
vv
1
7
B
0
7
B
PROBLEM 17.134
Each of the bars AB and BC is of length
400 mmL
and mass 1.2 kg.m Determine the
angular velocity of each bar immediately after the impulse Q (1.5 N s)
i
t is applied at C.
PROBLEM 17.134 (Continued)
Moments about A:
2
0( ) 2
1
xABAB
L
BtLI mv
LL
Subtract Eq. (1) from Eq. (4): 51
066
A
BBC
mL mL
5
BC AB
(5)
Substitute for BC
in Eq. (1): 11
(5 )
23
A
BAB
Qt mL mL
Qt
7(7)(1.2)(0.4)
mL
PROBLEM 17.135
A uniform disk of constant thickness and initially at rest is placed in
contact with the belt shown, which moves at a constant speed v
80 ft/s. Knowing that the coefficient of kinetic friction between the
disk and the belt is 0.15, determine (a) the number of revolutions
executed by the disk before it reaches a constant angular velocity,
(b) the time required for the disk to reach that constant angular velocity.
SOLUTION
Kinetic friction.
0.15 N
fk
FN
cos 25 sin 25 0
yf
FN F mg
cos 25 0.15sin 25
1.18636
(0.15)(1.18636)
0.177954
k
f
mg
N
mg
Fmg
mg
Final angular velocity.
2
v
r
2
1
2
0 0.177954 4
mgr mv
v
PROBLEM 17.135 (Continued)
(b) Principle of impulse-momentum.
0.177954
32.2
mgr
v
PROBLEM 17.136
The 8-in.-radius brake drum is attached to a larger flywheel that is not
shown. The total mass moment of inertia of the flywheel and drum is
2
14 lb ft s
and the coefficient of kinetic friction between the drum and
the brake shoe is 0.35. Knowing that the initial angular velocity of the
flywheel is 360 rpm counterclockwise, determine the vertical force
P
that
must be applied to the pedal C if the system is to stop in 100 revolutions.
2
3
2
1(14)(12 )
2
9.9486 10 ft lb
PROBLEM 17.137
Charpy impact test pendulums are used to determine the amount of
energy a test specimen absorbs during an impact (see ASTM
Standard E23). The hammer weighs 71.2 lbs and has a mass moment
of inertia about its center of gravity G
H
of 20.9 slug·in
2
. The arm
weighs 19.5 lbs and has a mass moment of inertia about its own
center of gravity G
A
of 47.1 slug·in
2
. The pendulum is released from
rest from an initial position of
θ
= 39°. Knowing that the friction at
pin O is negligible, determine (a) the impact speed when the hammer
hits the test specimen, (b) the force on the pin O just before the
hammer hits the test specimen, (c) the amount of energy that the test
specimen absorbs if the hammer swings up to a maximum of
= 70°
after the impact.
SOLUTION
PROBLEM 17.137 (Continued)
13
PROBLEM 17.138
The gear shown has a radius R 150 mm and a radius of gyration
125 mm.k
The gear is rolling without sliding with a velocity
1
v
of magnitude 3 m/s when it strikes a step of height h 75 mm.
Because the edge of the step engages the gear teeth, no slipping
occurs between the gear and the step. Assuming perfectly plastic
impact, determine (a) the angular velocity of the gear immediately
after the impact, (b) the angular velocity of the gear after it has
rotated to the top of the step.
1
0.150 m
R
Angular velocity.
222
(150)(75)
(150 125 )
2
PROBLEM 17.138 (Continued)
Part (b) Conditions at the top of the step.
The gear pivots about the edge of the step. Use the principle of conservation of energy.
Position (2): The gear has just broken contact with the floor.
Position (3): The center of the gear is above the edge of the step.
3
()
VmgRh
Principle of conservation of energy:
2233
TVTV
222 222
11
PROBLEM 17.139
A uniform slender rod is placed at corner B and is given a slight
clockwise motion. Assuming that the corner is sharp and becomes
slightly embedded in the end of the rod, so that the coefficient of
static friction at B is very large, determine (a) the angle
through
which the rod will have rotated when it loses contact with the corner,
(b) the corresponding velocity of end A.
A
PROBLEM 17.140
The motion of the slender 250-mm rod AB is guided by pins at A and B that
slide freely in slots cut in a vertical plate as shown. Knowing that the rod has
a mass of 2 kg and is released from rest when
0,
determine the reactions
at A and B when
90 .
0.125 m
BG
Point E is the instantaneous center of rotation of bar AB.
0.125
2
G
L
v
1
1
0
(2)(9.81)(0.125) 2.4525 J
2
T
L
Vmg
PROBLEM 17.140 (Continued)
Position 2 90
11
2.1848 m/s
(0.125)(10.091)
1.2614 m/s
G
v
More kinematics: For Point A moving in the curved slot,
2
2
()
(2.1847)
( ) 38.1833
A
ACx
Cx
Cx
v
aR
a
aij
ij
For the rod AB, ,BB
v
kv j
2
(10.091) ( 0.125 0.21651 )
0.125 0.21651 12.7285 22.0468
B
a
ij
jj i i j
PROBLEM 17.140 (Continued)
Matching vertical components of
A
a
38.1833 0.125 22.0468
0.125 60.2301
B
B
a
a
Kinetics: Use rod AB as a free body.
eff
():
EE
MM
(2)(9.81)(0.125)sin 30
L
k
2
22
0.0104667 (0.0625 0.10825 ) ( )
1.22625 0.0104667 0.03125 4.7730
0.0417167 5.99925
143.808 rad/s
( 21.933 m/s ) (40.2189 m/s )
G
G
m
ija
aij
PROBLEM 17.141
A baseball attachment that helps people with mobility
impairments play T-ball and baseball is powered by a
spring that is unstretched at position 2. The spring is
attached to a cord, which is fastened to point B on the
75-mm radius pulley. As the pulley, which is fixed at
point O, rotates backwards to the cocked position at
,
the rope wraps around the pulley and stretches the
spring of stiffness k = 2000 N/m. The combined mass
moment of inertia of all rotating components about
point O is 0.40 kg·m2. The swing is timed perfectly to
strike a 145 gram baseball travelling with a speed of
v0= 10 m/s at a distance of h = 0.7 m away from point
O. Knowing that the coefficient of restitution between
the bat and ball is 0.59, determine the velocity of the
baseball immediately after the impact. Assume that the
ball is travelling primarily in the horizontal plane and
that its spin is negligible.
2
2
11.107 rad/s
O
I
PROBLEM 17.141 (Continued)
Conservation of Momentum about point O during impact:
,2 ,3
OO
Bat
HH
v
PROBLEM 17.142
Two panels A and B are attached with hinges to a rectangular
plate and held by a wire as shown. The plate and the panels are
made of the same material and have the same thickness. The
entire assembly is rotating with an angular velocity
0
when
the wire breaks. Determine the angular velocity of the
assembly after the panels have come to rest against the plate.
panel 0
44
12
82
12 3
tb tb
