PROBLEM 16.27
The 8–in.–radius brake drum is attached to a larger flywheel that is not
shown. The total mass moment of inertia of the drum and the flywheel 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 angular velocity of the flywheel
is 360 rpm counterclockwise when a force P of magnitude 75 lb is
applied to the pedal C, determine the number of revolutions executed by
the flywheel before it comes to rest.
SOLUTION
Lever ABC: Static equilibrium (friction force )
PROBLEM 16.28
Solve Problem 16.27, assuming that the initial angular velocity of the
flywheel is 360 rpm clockwise.
PROBLEM 16.27 The 8–in.–radius brake drum is attached to a larger
flywheel that is not shown. The total mass moment of inertia of the drum
and the flywheel 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 angular
velocity of the flywheel is 360 rpm counterclockwise when a force P of
magnitude 75 lb is applied to the pedal C, determine the number of
revolutions executed by the flywheel before it comes to rest.
SOLUTION
Lever ABC: Static equilibrium (friction force )
PROBLEM 16.29
The 100-mm-radius brake drum is attached to a flywheel which is not
shown. The drum and flywheel together have a mass of 300 kg and a
radius of gyration of 600 mm. The coefficient of kinetic friction between
the brake band and the drum is 0.30. Knowing that a force P of magnitude
50 N is applied at A when the angular velocity is 180 rpm counterclockwise,
determine the time required to stop the flywheel when a = 200 mm and
b = 160 mm.
SOLUTION
21
21
PROBLEM 16.29 (Continued)
0
= 74.5 st
PROBLEM 16.30
The 180-mm radius disk is at rest when it is placed in contact with a belt
moving at a constant speed. Neglecting the weight of the link AB and
knowing that the coefficient of kinetic friction between the disk and the
belt is 0.40, determine the angular acceleration of the disk while slipping
occurs.
SOLUTION
tan
tan
k
k
k
k
mg
FN
θµ
µ
µθµ
= = +
PROBLEM 16.30 (Continued)
0.18 m tan 60 0.40
PROBLEM 16.31
Solve Problem 16.30, assuming that the direction of motion of the belt is
reversed.
PROBLEM 16.30 The 180-mm disk is at rest when it is placed in contact
with a belt moving at a constant speed. Neglecting the weight of the link
AB and knowing that the coefficient of kinetic friction between the disk
and the belt is 0.40, determine the angular acceleration of the disk while
slipping occurs.
PROBLEM 16.32
In order to determine the mass moment of inertia of a flywheel of radius
600 mm, a 12–kg block is attached to a wire that is wrapped around the
flywheel. The block is released and is observed to fall 3 m in 4.6 s. To
eliminate bearing friction from the computation, a second block of mass 24 kg
is used and is observed to fall 3 m in 3.1 s. Assuming that the moment of the
couple due to friction remains constant, determine the mass moment of inertia
of the flywheel.
SOLUTION
f
PROBLEM 16.32 (Continued)
PROBLEM 16.33
The flywheel shown has a radius of 20 in. a weight of 250 lbs, and a radius of
gyration of 15 in. A 30-lb block A is attached to a wire that is wrapped around
the flywheel, and the system is released from rest. Neglecting the effect of
friction, determine (a) the acceleration of block A, (b) the speed of block A after
it has moved 5 ft.
SOLUTION
PROBLEM 16.34
Each of the double pulleys shown has a mass moment of inertia of
2
15 lb ft s⋅⋅
and is initially at rest. The
outside radius is 18 in., and the inner radius is 9 in. Determine (a) the angular acceleration of each pulley,
(b) the angular velocity of each pulley after Point A on the cord has moved 10 ft.
SOLUTION
Case 1:
2
9
Case 3:
PROBLEM 16.34 (Continued)
PROBLEM 16.35
Each of the gears A and B has a mass of 9 kg and has a radius of
gyration of 200 mm; gear C has a mass of 3 kg and has a radius of
gyration of 75 mm. If a couple M of constant magnitude 5 N-m is
applied to gear C, determine (a) the angular acceleration of gear A,
(b) the tangential force which gear C exerts on gear A.
0.25
0.25
0.1
2.5
tA
B
C
BA
CA
a
a
a
a
aa
aa
=
=
=
=
=
(1)
Gear A:
22
2
9(0.20)
0.36 kg m
A AA
I mk= =
= ⋅
eff
AA A
1.44
A
F
a
=
(2)
Because of symmetry, gear C exerts an equal force F on gear B.
2
PROBLEM 16.35 Each of the gears A and B has a mass of 9 kg and
has a radius of gyration of 200 mm; gear C has a mass of 3 kg and has
a radius of gyration of 75 mm. If a couple M of constant magnitude 5
N-m is applied to gear C, determine (a) the angular acceleration of
gear A, (b) the tangential force which gear C exerts on gear A.
0.25
0.1
t AA CC
A
CA A
C
ar r
r
r
aa
aa a
= =
= =
0.1 0.25
0.25 0.1
t BB CC
C
B C AA
A
ar r
r
r
aa
a a aa
= =
==⋅=
Kinetics:
0.25 0.1
AC B C A
C
r
Gear A:
eff
( ):
AA
MM= Σ
A A AB A A
M rF I
a
−=
2
0.1 0.25
0.25 0.1
0.25
0.1
A
A B C A AA
C
A AB CA
r
M I II
r
M II I
aa
a
−+ =
= ++
(2)
PROBLEM 16.36 (Continued)
Data: From Eq. (2)
22
2
9 kg
0.2 m
9(0.2)
0.36 kg m
AB
AB
A B AA
mm
kk
I I mk
= =
= =
= = =
= ⋅
22
2
3 kg
0.075 m
3(0.075) 0.016875 kg m
5 N m
0.25
5 0.36 0.36 (0.016875)
0.1
C
C
C
A
A
m
k
I
M
a
=
=
= = ⋅
= ⋅
= ++
(a) Angular acceleration.
2
6.0572 rad/s
A
a
=
2
6.06 rad/s
A
=a
From Eq. (1)
1 0.1 0.25
(0.36) (0.016875) (6.0572)
0.1 0.25 0.1
11.278
AC
F
= +
=
AC
PROBLEM 16.37
Gear A weighs 1 lb and has a radius of gyration of 1.3 in; Gear B weighs
6 lb and has a radius of gyration of 3 in.; gear C weighs 9 lb and has a
radius of gyration of 4.3 in. Knowing a couple M of constant magnitude
of 40 lb ⋅ in is applied to gear A, determine (a) the angular acceleration
of gear C, (b) the tangential force which gear B exerts on gear C.
SOLUTION
Masses and moments of inertia.
2
2
2
2
2
2
2
22
32
22
1 lb 0.031056 lb s /ft
32.2 ft/s
6 lb 0.18634 lb s /ft
32.2 ft/s
9 lb 0.27950 lb s /ft
32.2 ft/s
1.3 in.
(0.031056 lb s /ft) 12 in./ft
3 in.
(0.18634 lb s /ft) 12 in./f
A
B
C
A AA
B BB
m
m
m
I mk
I mk
−
= = ⋅
= = ⋅
= = ⋅
= = ⋅
= = ⋅
2
32
2
22
32
t
11.646 10 lb s ft
4.3 in.
(0.27950 lb s /ft) 12 in./ft
35.889 10 lb s ft
C CC
I mk
−
−
= × ⋅⋅
= = ⋅
= × ⋅⋅
Kinematics. Gear A:
2 in.
A
r=
Gear B:
12
4 in., 2 in.rr= =
Gear C:
6 in.
C
r=
Point of contact between A and B.
PROBLEM 16.37 (Continued)
Point of contact between B and C.
2
2
6 in.
2 in.
t B CC
C
BC C
ar r
r
r
aa
aa a
= =
= =
Summary.
3
BC
aa
=
(1)
26
ABC
aaa
= =
(2)
Kinetics. Applied couple:
40 lb in. 3.3333 lb ftM= ⋅= ⋅
Gear A:
eff
( ):
AA
MMΣ=Σ
AB A A A
M Fr I
a
−=
3
(6 )
3.3333 lb ft (0.36448 10 )(6)
(2/12) ft (2/12)
20 lb 0.013121
AC
AB
AA
C
C
I
M
Frr
a
a
a
−
= −
⋅×
= −
= −
(3)
Gear B:
eff
( ):
BB
MMΣ=Σ
12AB BC B C
Fr Fr I
a
−=
1
22
2
3
3
2
(3)(11.646 10 )
2[20 lb 0.013121 ] (2/12)
BB
BC AB
B
AB C
CC
C
rI
FFrr
I
Fr
a
a
aa
−
= = −
= −
×
=−−
Gear C:
eff
( ):
CC
MMΣ=Σ
BC C C C
Fr I
a
=
3
6
(40 0.23587 ) (35.889 10 )
12
20 0.153824
CC
C
aa
a
−
−=×
=
(a) Angular acceleration of gear C.
2
130.0 rad/s
C
=a
(b) Tangential force which gear B exerts on gear C.
40 lb (0.23587)(130.0) 9.33 lb
BC
F=−=
9.33 lb
PROBLEM 16.38
The 25-lb double pulley shown is at rest and in equilibrium when a constant
3.5 lb ft⋅
couple M is applied. Neglecting the effect of friction and knowing
that the radius of gyration of the double pulley is 6 in., determine (a) the
angular acceleration of the double pulley, (b) the tension in each rope.
PROBLEM 16.38 (Continued)
Kinetics:
:A
10
10 32.2
AA
FT a
Σ= − =
:B
5
532.2
BB
FT a
Σ= −=
Elimination:
( )
2
10 4 5 8 25
3.5 10 5 0.5
12 12
AB
aa
g gg
a
+ − −+ =
3.5 3.333+1.111 3.333
g
a
−−
2.222 6.25
gg
aa
−=
(a)
2
11.76 rad/s=a
(b)
8.78 lb , 6.22 lb
AB
TT= =
PROBLEM 16.39
A belt of negligible mass passes between cylinders A and B and is pulled to
the right with a force P. Cylinders A and B weigh, respectively, 5 and 20 lb.
The shaft of cylinder A is free to slide in a vertical slot and the coefficients of
friction between the belt and each of the cylinders are
0.50
s
µ
=
and
µ
k
0.40.=
For
3.6 lb,P=
determine (a) whether slipping occurs between the
belt and either cylinder, (b) the angular acceleration of each cylinder.
SOLUTION
Assume that no slipping occurs.
1