Chapter 9 Homework Cc That Perpendicular The Plate B The

PROBLEM 9.115

A piece of thin, uniform sheet metal is cut to form the machine

component shown. Denoting the mass of the component by

m

,

determine its mass moment of inertia with respect to (

a

) the

x

axis,

(

b

) the

y

axis.

SOLUTION

First note

2

mass

1

(2 )( ) (2 )

22

3

2

mV tA

a

taa a

ta





 

















PROBLEM 9.115 (Continued)

Finally, ,mass ,mass ,mass

yxz

III



y

PROBLEM 9.116

A piece of thin, uniform sheet metal is cut to form the machine

component shown. Denoting the mass of the component by m,

determine its mass moment of inertia with respect to (a) the axis

AA, (b) the axis BB, where the AA and BB axes are parallel to the

x axis and lie in a plane parallel to and at a distance a above the xz

plane.

SOLUTION

First note that the x axis is a centroidal axis so that

2

,massx

II md

and that from the solution to Problem 9.115,

PROBLEM 9.117

A thin plate of mass m has the trapezoidal shape shown. Determine

the mass moment of inertia of the plate with respect to (a) the x

axis, (b) the y axis.

SOLUTION

First note

2

mass

1

(2 )( ) (2 )( ) 3

2

mV tA

taa aa ta



 









Also

mass area area

2

3

m

ItI I

a





(a) Now

,area 1,area 2,area

33

() ()

11

(2 )( ) (2 )( )

312

xx x

II I

aa aa



PROBLEM 9.118

A thin plate of mass m has the trapezoidal shape shown. Determine

the mass moment of inertia of the plate with respect to (a) the

centroidal axis CC that is perpendicular to the plate, (b) the axis

AA that is parallel to the x axis and is located at a distance 1.5a

from the plate.

SOLUTION

First locate the centroid C.

22 2 2

1

:(2 )(2)2 2()

3

XA xA Xa a aa a aa



    





or 14

9

Xa

PROBLEM 9.118 (Continued)

From the solution to Problem 9.193:

2

,mass

5

18

x

Ima

PROBLEM 9.119

Determine by direct integration the mass moment of inertia

with respect to the z axis of the right circular cylinder shown,

assuming that it has a uniform density and a mass m.

SOLUTION

For the cylinder: 2

mV aL





For the element shown: 2

dm a dx

mdx

L





PROBLEM 9.120

The area shown is revolved about the x axis to form a

homogeneous solid of revolution of mass m. Using direct

integration, express the mass moment of inertia of the solid

with respect to the x axis in terms of m and h.

SOLUTION

We have

2



hh

yxh

a

so that ()

h

rxa

a

For the element shown:

PROBLEM 9.121

The area shown is revolved about the x axis to form a

homogeneous solid of revolution of mass m. Determine by

direct integration the mass moment of inertia of the solid with

respect to (a) the x axis, (b) the y axis. Express your answers in

terms of m and the dimensions of the solid.

SOLUTION

We have at (, ): k

ah h a

or

kah

For the element shown:

4r



PROBLEM 9.121 (Continued)

(b) For the element: 2

22

1

4

yy

dI dI x dm

rdm xdm



PROBLEM 9.122

Determine by direct integration the mass moment of inertia with

respect to the x axis of the tetrahedron shown, assuming that it has

a uniform density and a mass m.

SOLUTION

We have

1









ay

xyaa

hh

and

1









by

zybb

hh

PROBLEM 9.122 (Continued)

Now

2

,mass

4

3

22

2

434

32

2

1

3

11

36

11

32

11

12

12 2

xAA

dI dI y z dm

y

ab dy

h

yy

yb ab dy

hh

yyy

ab dy ab y dy

hh

h





































 

  







 

 

















 

 

PROBLEM 9.123

Determine by direct integration the mass moment of inertia with

respect to the y axis of the tetrahedron shown, assuming that it has

a uniform density and a mass m.

SOLUTION

We have

1









ay

xyaa

hh

and

1









by

zybb

hh

PROBLEM 9.123 (Continued)

Now

,mass ,mass

22

1()

12

11()1

12











 







 

 









yBB DD

dI dI dI

xz x z dy

yy

ab a b dy

hh

PROBLEM 9.124

Determine by direct integration the mass moment of inertia and

the radius of gyration with respect to the x axis of the paraboloid

shown, assuming that it has a uniform density and a mass m.

SOLUTION

222 :ryzkx at

2

,; ; a

xhra a kh k h

  

Thus:

2

2a

rx

h



2

a

dm r dx xdy

h

 



PROBLEM 9.125

A thin rectangular plate of mass m is welded to a vertical shaft AB as

shown. Knowing that the plate forms an angle



with the y axis,

determine by direct integration the mass moment of inertia of the

plate with respect to (a) the y axis, (b) the z axis.

SOLUTION

Projection on yz plane

PROBLEM 9.125 (Continued)

222

1(4sin)

12

y

Imba



 

(b) 2222



1

(cos ) cos

zz

dI dI v dm b dm v dm





  

PROBLEM 9.126*

A thin steel wire is bent into the shape shown. Denoting the mass per

unit length of the wire by m, determine by direct integration the mass

moment of inertia of the wire with respect to each of the coordinate

axes.

SOLUTION

First note 1/ 3 2 /3 2/ 3 1/ 2

()

dy xa x

dx



 

Then

2

2/3 2/3 2/3

2/3

11()

dy xa x

dx

a

x





 











PROBLEM 9.126* (Continued)

Alternative solution:

1/3

aa

PROBLEM 9.127

Shown is the cross section of an idler roller. Determine its

mass moment of inertia and its radius of gyration with respect

to the axis AA. (The specific weight of bronze is 0.310 lb/in

3

;

of aluminum, 0.100 lb/in

3

; and of neoprene, 0.0452 lb/in

3

.)

SOLUTION

First note for the cylindrical ring shown that



22 22

21 21

44

mV t dd tdd



 

  

and, using Figure 9.28, that

22

21

11

2222

AA

dd

Im m



 



 

 