Problem 18.80 The mass of the object is 10 kg. Its
moment of inertia about L1is 10 kg-m2. What is its
moment of inertia about L2? (The three axes are in the
same plane.)
LL
1L2
0.6 m 0.6 m
Problem 18.81 An engineer gathering data for the
design of a maneuvering unit determines that the
astronaut’s center of mass is at x=1.01 m, y=0.16 m
and that her moment of inertia about the zaxis is
105.6 kg-m2. The astronaut’s mass is 81.6 kg. What is
her moment of inertia about the z′axis through her center
of mass?
y⬘
x⬘
x
y
Problem 18.82 Two homogenous slender bars, each of
mass mand length l, are welded together to form the
T-shaped object. Use the parallel-axis theorem to deter-
mine the moment of inertia of the object about the axis
through point Othat is perpendicular to the bars.
l
Ol
Problem 18.83 Use the parallel-axis theorem to deter-
mine the moment of inertia of the T-shaped object in
Problem 18.98 about the axis through the center of mass
of the object that is perpendicular to the two bars.
Solution: The location of the center of mass of the object is
x=
ml
2+lm
Problem 18.84 The mass of the homogeneous slender
x
x⬘
0.6 m 2 m
0.8 m
Solution: The density is ρ=30 kg
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Problem 18.85 The mass of the homogeneous slender
bar is 30 kg. Determine the moment of inertia of the bar
about the z′axis through its center of mass.
Problem 18.86 The homogeneous slender bar weighs
5 lb. Determine its moment of inertia about the zaxis.
4 in
y⬘
x⬘
x
y
Problem 18.87 Determine the moment of inertia of the
bar in Problem 18.86 about the z′axis through its center
of mass.
Solution: In the solution of Problem 18.86, it is shown that the
y=y1L1+y2L2+y3L3
Problem 18.88 The rocket is used for atmospheric
research. Its weight and its moment of inertia about
the zaxis through its center of mass (including its
fuel) are 10,1000 lb and 10,200 slug-ft2, respectively.
The rocket’s fuel weighs 6000 lb, its center of mass is
located at x=−3 ft, y=0, and z=0, and the moment
of inertia of the fuel about the axis through the fuel’s
center of mass parallel to zaxis is 2200 slug-ft2. When
the fuel is exhausted, what is the rocket’s moment of
inertia about the axis through its new center of mass
parallel to zaxis?
y
x
about a center of mass xE, and the moment of inertia of the fuel as IF
about a mass center xF. Using the parallel axis theorem, the moment
of inertia of the filled rocket is
124.224 (−3)=4.5ft
is the new location of the center of mass.
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Problem 18.89 The mass of the homogeneous thin plate
x
y
0.3 m
Solution: Divide the plate into two areas: the rectangle 0.4 m by
0.6 m on the left, and the rectangle 0.4 m by 0.3 m on the right. The
mass density is ρ=m
0.36 =100 kg/m2.
Problem 18.90 Determine the moment of inertia of the
of inertia about the y-axis using the same divisions as in Problem 8.89
and the parallel axis theorem is
Problem 18.91 The mass of the homogeneous thin plate
is 20 kg. Determine its moment of inertia about the
xaxis.
y
2(0.4 m)(0.6 m)=0.36 m2
0.36 m2=55.6 kg/m2
Using the integral tables we have
200
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Problem 18.92 The mass of the homogeneous thin plate
is 20 kg. Determine its moment of inertia about the
Problem 18.93 The thermal radiator (used to eliminate
excess heat from a satellite) can be modeled as a homo-
geneous thin rectangular plate. The mass of the radiator
is 5 slugs. Determine its moments of inertia about the
y
6 ft
3 ft
Problem 18.94 The mass of the homogeneous thin plate
is 2 kg. Determine the moment of inertia of the plate
about the axis through point Othat is perpendicular to
the plate.
30 mm
80 mm
10 mm
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Problem 18.95 The homogeneous cone is of mass m.
Determine its moment of inertia about the zaxis, and
compare your result with the value given in Appendix C.
(See Example 18.10.)
Strategy: Use the same approach we used in
Example 18.10 to obtain the moments of inertia of a
homogeneous cylinder.
x
y
z
R
h
Solution: The differential mass
Problem 18.96 Determine the moments of inertia of
the homogeneous cone in Problem 18.95 about the x
and yaxes, and compare your results with the values
given in Appendix C. (See Example 18.10.)
Solution: The mass density is ρ=m
Problem 18.97 The homogeneous object has the shape
of a truncated cone and consists of bronze with mass
density ρ=8200 kg/m3. Determine the moment of
inertia of the object about the zaxis.
x
z
60 mm
180 mm
180 mm
y
Solution: Consider an element of the cone consisting of a disk
y
Problem 18.98 Determine the moment of inertia of the
object in Problem 18.97 about the xaxis.
Solution: Consider the disk element described in the solution
to Problem 18.97. The radius of the laminate is r=0.167z. Using
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Problem 18.99 The homogeneous rectangular paral-
lelepiped is of mass m. Determine its moments of inertia
about the x,y, and zaxes and compare your results with
the values given in Appendix C.
y
z
a
b
Solution: Consider a rectangular slice normal to the x-axis of
Since the labeling of the x–y– and z-axes is arbitrary,
dIx=dIz+dIy,
where the x-axis is normal to the area of the plate. Thus
dIx=1
12 b2+c2dm,
from which
Ix=1
12 (b2+c2)mdm =m
12 (b2+c2).
Problem 18.100 The sphere-capped cone consists of
material with density 7800 kg/m3. The radius R=
80 mm. Determine its moment of inertia about the
xaxis.
y
x
z
4R
R
Problem 18.101 Determine the moment of inertia of
the sphere-capped cone described in Problem 18.100
about the yaxis.
Problem 18.102 The circular cylinder is made of
aluminum (Al) with density 2700 kg/m3and iron (Fe)
with density 7860 kg/m3. Determine its moment of
inertia about the x′axis.
200 mm
y
z
Al
Fe
600 mm
y⬘
Solution:
Problem 18.103 Determine the moment of inertia of
the composite cylinder in Problem 18.102 about the y′
x=0.747 m
Iy=[(2700 kg/m3)π(0.1 m)2(0.6 m)]1
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Problem 18.104 The homogeneous machine part is
made of aluminum alloy with mass density ρ=
2800 kg/m3. Determine the moment of inertia of the part
about the zaxis.
120 mm
40
mm
20 mm
40 mm
xz
yy
Problem 18.105 Determine the moment of inertia of
the machine part in Problem 18.104 about the xaxis.
Problem 18.106 The object shown consists of steel of
density ρ=7800 kg/m3of width w=40 mm. Deter-
mine the moment of inertia about the axis L0.
100 mm
O
20 mm
Solution: Divide the object into four parts:
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Problem 18.107 Determine the moment of inertia of
the object in Problem 18.106 about the axis through the
Problem 18.108 The thick plate consists of steel of
density ρ=15 slug/ft3. Determine the moment of iner-
tia of the plate about the zaxis.
2 in 2 in
x
yy
z
4 in
Problem 18.109 Determine the moment of inertia of
the object in Problem 18.108 about the xaxis.
=0.1916 slug ft2
Problem 18.110 The airplane is at the beginning of its
takeoff run. Its weight is 1000 lb. and the initial thrust T
exerted by its engine is 300 lb. Assume that the thrust is
horizontal, and neglect the tangential forces exerted on
its wheels.
(a) If the acceleration of the airplane remains constant,
how long will it take to reach its takeoff speed of
80 mi/hr?
(b) Determine the normal force exerted on the forward
landing gear at the beginning of the takeoff run.
T
6 in
1 ft 7 ft
Solution: The acceleration under constant thrust is
W
0.002 kg-m2,IB=0.036 kg-m2, and IC=0.032 kg-m2.
They are initially stationary, and at t=0 a constant
M=2 N-m is applied at pulley A. What is the angular
velocity of pulley Cand how many revolutions has it
turned at t=2s?
200 mm
200 mm
A
BC
RB2
0.1αC=2αC.
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