Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 378

The formula (A.139) is then written in the form

I 0 Area fð

I Area
centroid

¼ d d Þ 1 ð d d Þg A ;

where 2 A ¼ b .

□

The vector d is a vector from the centroid to the corner,

ð

1
3

d ¼ be 1 þ he 2 Þ:

Substituting I 0 and the formula for d into the equation for I centroid above, it
follows that the mass moment of inertia of the rectangular prism relative to its
centroid is given by

bh
72

2 h 2

bh

I Area

centroid ¼

:

hb

2 b 2

□

Problems

A.9.1 Find the center of mass of a set of four masses. The form masses and their
locations are mass 1 (2 kg) at (3, 1), mass 2 (4 kg) at (4, 4), mass 3 (5 kg) at

1).
A.9.2 Under what conditions does the center of mass of an object coincide with the
centroid?
A.9.3 Find the centroid of a cylinder of length L with a semicircular cross-section
of radius R .
A.9.4 Find the center of mass of a cylinder of length L with a semicircular cross-
section of radius R ( R

( 4, 4), mass 4 (1 kg) at ( 3,

<

2 L ) if the density varies in according to the rule

c ( x 2 ) 2 ). The coordinate system for the cylinder has been
selected so that x 3 is along its length L , x 2 is across its smallest dimension
(0

r ¼ r o ( þ

1

x 2 R ) and x 1 is along its intermediate dimension ( R x 1 R ).

A.9.5 Show that the moment of inertia matrix I is symmetric.
A.9.6 Develop the formulas for the mass moment of inertia of a thin plate of
thickness t and a homogeneous material of density . Illustrate these

specialized formulas by determining the mass moment of inertia of a thin
rectangular plate of thickness t , height h , and a width of base b , and a
homogeneous material of density

r

. Specify precisely where the origin of
the coordinate system that you are using is located and how the base vectors
of that coordinate system are located relative to the sides of the rectangular
plate.
A.9.7 In Example A.9.2 the occurrence of a multiple eigenvalue (7

r

a 3 /12) made

r

any vector perpendicular to the first eigenvector e 1 ¼ (1/ 3)[1, 1, 1] an

√

a 3 /12. In Example A.9.2 eigenvector associated the multiple eigenvalue 7 r

the two perpendicular unit vectors e 2 ¼ (1/ 2)[ 1, 0, 1] and e 3 ¼ (1/ 6)

√ √

[1, 2, 1] were selected as the eigenvectors associated with the multiple