CHAPTER 18
PROBLEM 18.1
A thin, homogeneous disk of mass m and radius r spins at the constant
rate
1
ω
about an axle held by a fork-ended vertical rod, which rotates at
the constant rate
2
.
ω
Determine the angular momentum
G
H
of the disk
about its mass center G.
PROBLEM 18.2
A thin rectangular plate of weight 15 lb rotates about its vertical
diagonal AB with an angular velocity
ω
. Knowing that the z axis is
perpendicular to the plate and that
ω
is constant and equal to 5 rad/s,
determine the angular momentum of the plate about its mass center G.
12 0.8(5 rad/s) 4 rad/s
15
90.6(5 rad/s) 3 rad/s
15
0
x
y
z
ωω
ωω
ω
′
′
= = =
= = =
=
PROBLEM 18.2 (Continued)
Components along x and y axes:
34 3 4
(0.087345) (0.11646)
55 5 5
0.040761
43
55
43
(0.087345) (0.11646) 0.13975
55
xx y
yxy
HH H
HHH
′′
′′
=−= −
= −
= +
= +=
22
PROBLEM 18.3
Two uniform rods AB and CE, each of weight 3 lb and length
2 ft, are welded to each other at their midpoints. Knowing that
this assembly has an angular velocity of constant magnitude
ω
= 12 rad/s, determine the magnitude and direction of the
angular momentum HD of the assembly about D.
SOLUTION
2
30.093168 lb s /ft, 2 ft,
32.2
W
ml
g
=== ⋅=
(12 rad/s)=ωi
For rod
ADB,
0, since 0.
Dx x
II
ω
=≈≈Hi
For rod
CDE, use principal axes
, xy
′′
as shown.
9
cos , 41.410
12
θθ
= = °
2
cos 9 rad/s
x
ω ωθ
′
= =
2
sin 7.93725 rad/s
y
ω ωθ
′= =
0
z
ω
′=
0
′
≈
x
I
22
2
11
(0.093168)(2)
12 12
0.0310559 lb s ft
y
I ml
′= =
= ⋅⋅
0 (0.0310559)(7.93725) 0
D xx yy zz
II I
ωωω
′′ ′′ ′′
′′′
=++
′
=++
H i jk
j
0.246498 ′
=j
0.246 lb s ft
D
H= ⋅⋅
0.246498(sin cos ) 0.163045 0.184874
D
θθ
= += +H ij i j
cos 0
z
θ
=
90
θ
= °
z
PROBLEM 18.4
A homogeneous disk of weight
6 lbW=
rotates at the constant
rate
1
16
ω
=
rad/s with respect to arm ABC, which is welded to
a shaft DCE rotating at the constant rate
2
8 rad/s.
ω
=
Determine the angular momentum
A
H
of the disk about its
center A.
PROBLEM 18.5
A thin disk of mass
4 kgm=
rotates at the constant rate
2
15 rad/s
ω
=
with respect to arm ABC, which itself rotates at the constant rate
1
5 rad/s
ω
=
about the y axis. Determine the angular momentum of the
disk about its center C.
PROBLEM 18.6
A solid rectangular parallelepiped of mass m has a square base of side a
and a length 2a. Knowing that it rotates at the constant rate
ω
about its
diagonal
AC′
and that its rotation is observed from A as
counterclockwise, determine (a) the magnitude of the angular
momentum
G
H
of the parallelepiped about its mass center G, (b) the
angle that
G
H
forms with the diagonal
.AC′
PROBLEM 18.7
PROBLEM 18.6 A solid rectangular parallelepiped of mass m has a
square base of side a and a length 2a. Knowing that it rotates at the
constant rate
ω
about its diagonal
AC′
and that its rotation is observed
from A as counterclockwise, determine (a) the magnitude of the angular
momentum
G
H
of the parallelepiped about its mass center G, (b) the
angle that
G
H
forms with the diagonal
.AC′
2
2 22
12 2 3 15
1 77
(2 )
12 2 12 60
z
a
I m a m m a ma
′′
= +==
PROBLEM 18.7 (Continued)
For each plate parallel to the xy plane:
1
5
mm
′=
2
2 22
2
2 22
22 2 2
1 77
(2 )
12 2 12 60
1 11
12 2 3 15
1 51
[ (2 ) ]
12 12 12
x
y
z
a
I m a m m a ma
a
I ma m ma ma
I m a a m a ma
′′ ′
= +==
′′ ′
=+==
′′
= += =
22
22
22
13 1 7 37
2120 12 60 60
111 3
260 15 15 10
13 7 1 37
2120 60 12 60
x
y
z
I ma ma
I ma ma
I ma ma
= ++ =
= ++ =
= ++ =
2
ma
ω
PROBLEM 18.8
A thin homogeneous disk of mass m and radius r is mounted
on the horizontal axle AB. The plane of the disk forms an
angle
20
β
= °
with the vertical. Knowing that the axle rotates
with an angular velocity
,ω
determine the angle
θ
formed by
the axle and the angular momentum of the disk about G.
()
Gx
H
9.7
θ
= °
PROBLEM 18.9
Determine the angular momentum HD of the disk of Problem
18.4 about Point D.
PROBLEM 18.4 A homogeneous disk of weight
6 lbW=
rotates at the constant rate
1
16
ω
=
rad/s with respect to arm
ABC, which is welded to a shaft DCE rotating at the constant
rate
2
8 rad/s.
ω
=
Determine the angular momentum
A
H
of the
disk about its center A.
PROBLEM 18.10
Determine the angular momentum of the disk of Problem 18.5 about
Point A.
PROBLEM 18.5 A thin disk of mass
4 kgm=
rotates at the constant
rate
2
15 rad/s
ω
=
with respect to arm ABC, which itself rotates at the
constant rate
1
5 rad/s
ω
=
about the y axis. Determine the angular
momentum of the disk about its center C.
PROBLEM 18.11
Determine the angular momentum
O
H
of the disk of Sample Problem 18.2
from the expressions obtained for its linear momentum
mv
and its angular
momentum HG using Eq. (18.11). Verify that the result obtained is the same
as that obtained by direct computation.
PROBLEM 18.2 A homogeneous disk of radius r and mass m is mounted
on an axle OG of length L and negligible mass. The axle is pivoted at the
fixed Point O, and the disk is constrained to roll on a horizontal floor.
Knowing that the disk rotates counterclockwise at the rate
ω
1 about the axle
OG, determine (a) the angular velocity of the disk, (b) its angular
momentum about O, (c) its kinetic energy, (d) the vector and couple at G
equivalent to the momenta of the particles of the disk.
PROBLEM 18.12
The 100-kg projectile shown has a radius of gyration of 100 mm about
its axis of symmetry
Gx
and a radius of gyration of 250 mm about the
transverse axis
.Gy
Its angular velocity
ω
can be resolved into two
components; one component, directed along
,Gx
measures the rate of
spin of the projectile, while the other component, directed along GD,
measures its rate of precession. Knowing that
6
θ
= °
and that the
angular momentum of the projectile about its mass center G is
2
(500 g m /s)
G= ⋅Hi
2
(10 g m /s) ,−⋅j
determine (a) the rate of spin,
(b) the rate of precession.
G Gx Gy Gz x x y y z z
2
2
() 0.500 kg m /s 0.5 rad/s
1 kg m
Gx
x
x
H
I
ω
⋅
= = =
⋅
2
() 0.01 kg m /s 0.0016 rad/s
Gy
H
−⋅
PROBLEM 18.13
Determine the angular momentum HA of the projectile of Problem
18.12 about the center A of its base, knowing that its mass center G has
a velocity
v
of 750 m/s. Give your answer in terms of components
respectively parallel to the x and y axes shown and to a third axis z
pointing toward you.
PROBLEM 18.12 The 100-kg projectile shown has a radius of
gyration of 100 mm about its axis of symmetry
Gx
and a radius of
gyration of 250 mm about the transverse axis
.Gy
Its angular velocity
ω
can be resolved into two components; one component, directed along
,Gx
measures the rate of spin of the projectile, while the other
component, directed along GD, measures its rate of precession.
Knowing that
6
θ
= °
and that the angular momentum of the projectile
about its mass center G is
2
(500 g m /s)
G
= ⋅Hi
2
(10 g m /s) ,−⋅j
determine (a) the rate of spin, (b) the rate of precession.
PROBLEM 18.14
(a) Show that the angular momentum
B
H
of a rigid body about Point B can be obtained by adding to the
angular momentum
A
H
of that body about Point A the vector product of the vector
/AB
r
drawn from B to A and
the linear momentum
mv
of the body:
/B A AB
m=+×H Hr v
(b) Further show that when a rigid body rotates about a fixed axis, its angular momentum is the same about
any two Points A and B located on the fixed axis
()
AB
=HH
if, and only if, the mass center G of the body is
located on the fixed axis.
/
GA
Let
/GA
= ×urλ
, so that
0.×=uλ
Note that u must be either perpendicular to λ or equal to zero. But if u is perpendicular
/GA
PROBLEM 18.15
Two L–shaped arms, each of mass 5 kg, are welded at the one–
third points of the 600 mm shaft AB to form the assembly shown.
Knowing that the assembly rotates at the constant rate of
360 rpm, determine (a) the angular momentum
A
H
of the
assembly about point A, (b) the angle formed by
A
H
and AB.
G yz
y
( )
Gz
z
HI
ω
=
Segments 1, 2, 3, and 4, each of mass
2.5 kg,
′=m
contribute to
, and .
xz yz z
II I
Part
xz
I
yz
I
z
I
2
1
−ma
2
11
1
2
1
−ma
2
1
ma
2
11
1
′
++
ma
PROBLEM 18.15 (Continued)
( )
2
3
2
Gx
H ma
ω
′
=−−
2
32.5 0.2 12
2
2
5.6549 kg m /s= ⋅
2
1
2
2
1.8850 kg m /s=−⋅
2
10