Problem 8.1 Use the method described in Active
Example 8.1 to determine IYand kyfor the rectangular
Solution: The height of the vertical strip of width dx is 0.6 m, so
the area is dA D(0.6 m) dx.
0.2m
D(0.6 m) x3
Problem 8.2 Use the method described in Active
Example 8.1 to determine Ixand kxfor the rectangular
0.6 m
Solution: It was shown in Active Example 8.1 that the moment
of inertia about the xaxis of a vertical strip of width dx and height
0.2m
3⊲0.6 m⊳3dx D1
3⊲0.6 m⊳3[x]0.4m
The radius of gyration about the xaxis is
Problem 8.3 In Active Example 8.1, suppose that the
triangular area is reoriented as shown. Use integration
to determine Iyand ky.
y
Solution: The height of a vertical strip of width dx is h⊲h/b⊳x,
Problem 8.4 (a) Determine the moment of inertia Iy
of the beam’s rectangular cross section about the yaxis.
(b) Determine the moment of inertia Iy0of the beam’s
Solution:
(a) IyD40 mm
x2dydx D1.28 ð106mm4
Problem 8.5 (a) Determine the polar moment of
inertia JOof the beam’s rectangular cross section about
the origin O.
Solution:
Problem 8.6 Determine Iyand ky.
y
Solution:
AD⊲0.3m⊳⊲1m⊳C1
602
Problem 8.7 Determine JOand kO.
Solution:
Problem 8.8 Determine Ixy.
00.3mC0.3x
0
Problem 8.9 Determine Iy.
y
y ⫽ 2 ⫺ x2
Solution: The height of a vertical strip of width dx is 2 x2,so
D0.467.
Problem 8.10 Determine Ix.
y
y ⫽ 2 ⫺ x2
Solution: It was shown in Active Example 8.1 that the moment
of inertia about the xaxis of a vertical strip of width dx and height
Problem 8.11 Determine JO.
x
y
1
Problem 8.12 Determine Ixy.
y
y ⫽ 2 ⫺ x2
Solution: It was shown in Active Example 8.1 that the product of
inertia of a vertical strip of width dx and height f(x)is
⊲Ixy ⊳strip D1
Problem 8.13 Determine Iyand ky.
y
y ⫽ ⫺ x2 ⫹ 4x ⫺ 7
1
4
604
Problem 8.14 Determine Ixand kx.
Solution: See Solution to Problem 8.13
IxD14
y2dydx D1333
Problem 8.15 Determine JOand kO.
Solution: See Solution to 8.13 and 8.14
Problem 8.16 Determine Ixy.
2x2/4C4x7
0
Problem 8.17 Determine Iyand ky.
y
y ⫽ ⫺ x2 ⫹ 4x ⫺ 7
1
4
Problem 8.18 Determine Ixand kx.
Solution: See Solution to Problem 8.17
IxD12
y2dydx D953
Problem 8.19 (a) Determine Iyand kyby letting dA
be a vertical strip of width dx.
(b) The polar moment of inertia of a circular area with
its center at the origin is JOD1
2R4. Explain how you
can use this information to confirm your answer to (a).
x
y
R
Solution: The equation of the circle is x2Cy2DR2, from which
yDš
pR2x2. The strip dx wide and ylong has the elemental area
dA D2pR2x2dx. The area of the semicircle is
(b) If the integration were done for a circular area with the center at
the origin, the limits of integration for the variable xwould be from
Rto R, doubling the result. Hence, doubling the answer above,
Problem 8.20 (a) Determine Ixand kxfor the area
in Problem 8.19 by letting dA be a horizontal strip of
height dy.
2. The equation for the circle is x2Cy2DR2, from which xD
dA DR2y2dy.
By symmetry IyDIx,
606
Problem 8.21 Use the procedure described in
Example 8.2 to determine the moment of inertia Ix
and Iyfor the annular ring.
x
y
Ri
Ro
Solution: We first determine the polar moment of inertia JOby
Problem 8.22 What are the values of Iyand kyfor the
elliptical area of the airplane’s wing?
y
2 m
⫹⫽ 1
x
x2
a2
y2
b2
Solution:
IyDA
x2dA Da
0y
y
x2dy dx
Evaluating, we get
IyD49.09 m4
The area of the ellipse (half ellipse) is
AD2a
0b1x2
a1/2
0
dy dx
kyD2.5m
608
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2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they
currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
Problem 8.23 What are the values of Ixand kxfor the
elliptical area of the airplane’s wing in Problem 8.22?
C3
a
IxD2b3
3a3a⊲0⊳
4C3a3p0
8C3
8a4
2
IxD3ab3
3.8Dab3
8
Evaluating (aD5, bD1)
8D1.96 m4
Problem 8.24 Determine Iyand ky.
y
y = x2 – 20
Solution: The straight line and curve intersect where xDx220.
Solving this equation for x, we obtain
xD1šp1C80
Problem 8.25 Determine Ixand kxfor the area in
Problem 8.24.
Solution: Let us determine the moment of inertia about the xaxis
of a vertical strip holding xand dx fixed:
Problem 8.24), we obtain Ixfor the entire area:
IxD5
1
Dx7
21 C4x5Cx4
12 400x3
3C8000x
35
4D10,900.
From the solution to Problem 8.24, AD122 so
kxDIx
AD10,900
122 D9.45.
y
xdx
Problem 8.26 A vertical plate of area Ais beneath the
surface of a stationary body of water. The pressure of
the water subjects each element dA of the surface of the
plate to a force ⊲p0Cy⊳ dA, where p0is the pressure
at the surface of the water and is the weight density
of the water. Show that the magnitude of the moment
about the xaxis due to the pressure on the front face of
the plate is
Mxaxis Dp0yA CIx,
where yis the ycoordinate of the centroid of Aand Ix
is the moment of inertia of Aabout the xaxis.
x
y
A
Solution: The moment about the xaxis is dM Dy⊲p0Cy⊳ dA
integrating over the surface of the plate:
610
Problem 8.27 Using the procedure described in Active
Example 8.3, determine Ixand kxfor the composite area
by dividing it into rectangles 1 and 2 as shown.
1
y
1 m
D21.0 m4
The moment of inertia about the xaxis for area 2 is
⊲Ix⊳2D1
4.
Problem 8.28 Determine Iyand kyfor the composite
area by dividing it into rectangles 1 and 2 as shown.
3 m
1 m
y
1 m
⊲Iy⊳2D1
4.
Problem 8.29 Determine Ixand kx.
x
0.6 m
0.2 m
Solution: Break into 3 rectangles
AD0.1653 m4
0.4m
2D0.643 m
)
IxD0.1653 m4
Problem 8.30 In Example 8.4, determine Ixand kxfor
the composite area.
x
y
20 mm
Solution: The area is divided into a rectangular area without the
cutout (part 1), a semicircular areas without the cutout (part 2), and
the circular cutout (part 3).
612
Problem 8.31 Determine Ixand kx.y
0.6 m
0.2 m
0.6 m
0.2 m
0.8 m
0.2 m
x
Solution: Break into 3 rectangles — See 8.29
First locate the centroid
AD0.0487 m4
0.4m
2D0.349 m
Problem 8.32 Determine Iyand ky.
AD0.01253 m4
0.4m
2D0.1770 m
Problem 8.33 Determine JOand kO.
Problem 8.34 If you design the beam cross section so
that IxD6.4ð105mm4, what are the resulting values
of Iyand JO?
y
h
Solution: The area moment of inertia for a triangle about the
614
Problem 8.35 Determine Iyand ky.
y
40
mm
40 mm
200
mm
160
mm
Solution: Divide the area into three parts:
Part (1): The top rectangle.
A2D⊲200 80⊳⊲40⊳D4.8ð103mm2,
dx2D20 mm,
From which,
Problem 8.36 Determine Ixand kx.
Solution: Use the solution to Problem 8.35. Divide the area into
A1D6.4ð103mm2,
dy1D200 20 D180 mm,
Ixx1D1
A2D4.8ð103mm2,
from which
A3D4.8ð103mm2,
dy3D20 mm,
kxDIx
Problems 8.35 and 8.36. Divide the area into three parts:
Part (1): A1D160⊲40⊳D6.4ð103mm2,
dx1D160
2D80 mm,
Ixy2Ddx2dy2A2D9.6ð106mm4.
Part (3): A3D120⊲40⊳D4.8ð103mm2,
616
Problem 8.38 Determine Ixand kx.
y
160
mm
40 mm
Solution: The strategy is to use the relationship IxDd2ACIxc,
where Ixc is the area moment of inertia about the centroid. From this
dx3D120
2D60 mm,d
y3D20 mm.
The total area is
D7.788 ð107mm4
AD69.77 mm
Problem 8.39 Determine Iyand ky.
Solution: The strategy is to use the relationship IyDd2ACIyc,
AD43.33 mm
Problem 8.40 Determine Ixy.
Solution: Use the solution to Problem 8.37. The centroid
Problem 8.41 Determine Ixand kx.
y
3 ft
4 ft
Solution: Divide the area into two parts:
Part (1) Ix1D1
12 4⊲33⊳D9ft
2.
AD1.549 ft.
Problem 8.41.
Iyc D1
where A1D6ft
2.
Part (2): The area moment of inertia about a centroid parallel to the
base for a rectangle is
Iyc D1
12 bh3D1
12 3⊲33⊳D6.75 ft4,
Iy2D⊲5.5⊳2A2CIyc D279 ft4,
where A2D9ft
The composite: IyDIy1CIy2D327 ft4, from which, using a result
AD4.92 ft
618
c
2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they
currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
Problem 8.43 Determine Ixy.
Solution: (See Figure in Problem 8.41.) Use the results of the
solutions to Problems 8.41 and 8.42. The area cross product of the
Problem 8.44 Determine Ixand kx.
y
Solution: Use the results of Problems 8.41, 8.42, and 8.43. The
AD1.3ft.
Problem 8.45 Determine JOand kO.
Solution: Use the results of Problems 8.41, 8.42, and 8.43. The
xD
AD4.3667 ft,
from which
AD1.855 ft
Problem 8.46 Determine Ixy.
Solution: Use the results of Problems 8.41–8.45. The strategy is
Problem 8.47 Determine Ixand kx.
40 mm
x
y
80
mm
120
mm 20
mm
620