6.3
O. A differential element dD of the body is shown.
Translated Coordinates
Figure 6.3 – 1: A photo of a hand (time-lapse) of a hand
rotating a ruler about its center and about its end.
It takes four times the moment to rotate the ruler about
one of its ends than to rotate it about its mass center (See Fig.
6.3 – 1). The difference is a direct result of the ruler’s mass
moment. The ruler’s mass moment about one of its ends is four
times larger than its mass moment about one of its ends. The
question arises how to calculate the space and mass integrals of
a body about different points.
As stated earlier, space and mass integral tables list
integrals of simple shapes. The tables only consider bodies that
have uniform mass densities. Furthermore, the origin of the
coordinate system for each body is located at the bodies’
geometric center. The area moments, polar moment, area
products, mass moments and mass products are said to be about
the geometric center C of the body. The integrals about other
points are not found in the space and mass integral tables except
in some special cases. The question arises how to find the
integrals of a body about a point O that is different than the
geometric center C, given the integrals of the body about the
geometric center. To find the integrals of the body about point
O, the coordinate system needs to be translated from point C to
point O.
Transformation between Translated
Coordinates
Referring to Fig. 6.3 – 2, the origin of coordinate
system x’-y’-z’ is located at the geometric center C of a body
(not shown) and the origin of coordinate system x-y-z is at point
Sub-Section
Transformation between Translated
Coordinates
The Dependence of the Integrals on
the Coordinate System
The Area Moments, Polar Moment,
and Area Product
The Mass Moments and Mass
Products
Figure 6.3 – 2: Translating Coordinates
The differential element is located at (x’ y’ z’) measured in the
x’-y’-z’ coordinates and it’s located at (x y z) measured in the xy-z coordinates. The coordinates of the two systems are related
to each other by the transformation
(6.3 – 1)
x  x' xC , y  y' yC , z  z ' zC .
where (xC yC zC) is the location of the geometric center in the xy-z coordinate system.
The Dependence of the Integrals on the
Coordinates
As an illustration, consider a surface body. Using the
x’-y’-z’ coordinates, the x’ coordinate of the geometric center of
1
the body is xC '   x' dA . Using the x-y-z coordinates, the x
A
1
coordinate of the geometric center is located at xC   xdA .
A
Let’s see how xC and xC’ are related. From Eq. (6.3 – 1)
(6.3
–
2)
1
1
1
1
xC '   x' dA   ( x  xC )dA   xdA   xC dA
A
A
A
A
1
1
1
  xdA  xC  dA  xC  xC A  0.
A
A
A
Section Objectives
Objective
To give the transformation between translated coordinate systems.
To point out that the space integrals and the mass integrals of a body depend on
the coordinate system.
To show how to calculate the space integrals of a body about a point O given the
space integrals of the body about the geometric center C.
To show how to calculate the mass integrals of a body about a point O given the
mass integrals of the body about the mass center M.
Equation (6.3 – 2) states that the geometric center of the surface
body is located at the origin O of the x’-y’-z’ coordinates. This
is an obvious result, considering the fact that the origin of the
coordinates x’-y’-z’ was originally taken to be located at the
geometric center of the body. Of course, the y’ and z’
coordinates of the geometric center are located at the origin C,
as well, and these results apply to the other types of bodies, not
just surface bodies.
(6.3 – 4)
I Oy  I Cy  xC2 A.
Polar Moment
Referring again to Table 6.2 – 3, the polar moment
about point O is J O   r 2 dA and the polar moment about
point C is J C   r '2 dA . From Eqs. (6.3 – 1), (6.3 – 3) and (6.3
– 4)
The dependence of a space integral on the location of
the origin of the coordinate system is obvious in the case of the
geometric center. The dependence of other space integrals on
the location of the origin of the coordinate system is obvious,
too. For example, it’s obvious that length, area, volume, and
mass do not depend on the location of the origin of the
coordinate system (nor the orientation of the coordinate
system). On the other hand, the area moments, polar moments,
area products, mass moments, and mass products, do depend on
the location of the origin of the coordinate system.
The Area Moments, Polar Moment and
Area Product
The following considers the section integrals, namely,
the area moments, polar moments, and area products. The mass
integrals are considered in the following sub-section.
Area Moments
Referring to Table 6.2 – 3, the area moment about the x
axis (through point O) is I Ox   y 2 dA and the area moment
about the x’ axis (through point C) is I Cx   y'2 dA . From Eq.
J O   r 2 dA   ( x 2  y 2 )dA   x 2 dA   y 2 dA  I Ox  I Oy
 ( I Cx  yC2 A)  ( I Cy  xC2 A)  ( I Cx  I Cy )  ( xC2  yC2 ) A.
But I Cx  I Cy  J C and so
(6.3 – 5)
Equation (6.3 – 5) is used to calculate the polar moment JO
given the polar moment JC.
Area Product
The area product about point O is I Oxy   xydA and
the area product about point C is I Cxy   x' y ' dA . From Eqs.
(6.3 – 1)
I Oxy   xydA   ( x' xC )( y' yC )dA
  x' y' dA   x' yC dA   xC y' dA   xC yC dA
1

 1

  x' y' dA  yC A  x' dA  xC  A  y' dA  xC yC  dA
A

 A

(6.3 – 1)
I Ox   y 2 dA   ( y ' yC ) 2 dA   ( y'2 2 yC y' yC2 )dA 
  y '2 dA  2 yC  y ' dA  yC2  dA  I Cx  2 yC
yC '
 yC2  dA.
A
J O  J C  ( xC2  yC2 ) A.
But xC ' 
(6.3 – 6)
1
1
x' dA  0 and yC '   y' dA  0 so

A
A
I Oxy  I Cxy  xC yC A.
Equation (6.3 – 6) is used to calculate the area product
IOxy given the area product ICxy.
But yC’ = 0 so
I Ox  I Cx  yC2 A.
(6.3 – 3)
Equation (6.3 – 3) is used to calculate the area moment
IOx given the area moment ICx. Similarly, the area moment about
In summary, the equations used to calculate the area
integrals of a body about an arbitrary point O given the area
integrals of the body about the geometric center C are
the y axis is I Oy   x 2 dA and the area moment about the y’
axis is I Cy   x' dA . From Eq. (6.3 – 1)
I Ox  I Cx  yC2 A.
2
(6.3 – 7)
I Oy   x 2 dA   ( x' xC ) 2 dA   ( x'2 2 xC x' xC2 )dA 
  x'2 dA  2 xC  x' dA xC2  dA  I Oy  2 xC
Since xC’ = 0, it follows that
xC '
 xC2  dA.
A
I Oy  I Cy  xC2 A.
J O  J C  ( xC2  yC2 ) A.
I Oxy  I Cxy  xC yC A.
Equations (6.3 – 7) are called the parallel-axis equations for
the space integrals. Parallel axis equations exist for the mass
integrals, too, as the following shows.
The Mass Moments and Mass Products
The following develops parallel-axis equations for the
mass integrals, specifically the mass moments and the mass
products. The results apply to point bodies, line bodies, surface
bodies, and volume bodies. The results below also apply to
bodies that have non-uniform mass densities.
Mass Moments
Referring to Table 6.2 – 4, the mass moment of a body
about the x axis (through point O) is I Oxx   rx2dm and the
mass moment of the body about the x’ axis (through the mass
center M) is I Mxx   rx '2 dm . From Eq. (6.3 – 1)
I Oxx   rx2 dm   ( y 2  z 2 )dm   [( y ' yC ) 2  ( z ' zC ) 2 ]dm
I Oxy   xydm   ( x' xC )( y ' yC )dm   ( x' y ' xC x' yC y ' xC yC )dm
 1

 1

  x' y ' dm  xC M   x' dm   yC M   y ' dm   xC yC  dm
M

M

Since xM ' 
1
1
x' dm  0 and yM ' 
y' dm  0

M
M
that
(6.3 – 8)
it follows
I Oxy  I Mxy  xM yM M .
Similarly, the parallel-axis equations for the other mass
products are
I Oyz  I Myz  y M z M M ,
(6.3 – 9)
I Ozx  I Mzx  z M xM M .
  [( y '2  z '2 )  2 yC y '2 zC z '( yC2  zC2 )]dm
In summary, the equations used to calculate the mass
integrals of a body about an arbitrary point O given the mass
 1

 1

  ( y '2  z '2 )dm  2 yC M   y ' dm   2 zC M   z ' dm   ( yC2 integrals
zC2 )  dm of the body about the mass center M are
M

M

I Oxx  I Mxx  rx2 M ,
I Oyy  I Myy  ry2 M ,
Figure 6.3 – 3: rx, ry, and rz are distances
between dm and the x, y, and z axes,
respectively.
(6.3 – 10)
I Ozz  I Mzz  rz2 M ,
I Oxy  I Mxy  xM y M M ,
I Oyz  I Myz  y M z M M ,
I Ozx  I Mzx  z M xM M .
These are the parallel axis equations for the mass integrals.
1
1
But yM ' 
y' dm  0 and z M ' 
z ' dm  0 so

M
M
(6.3 – 8)
I Oxx  I Mxx  rx2 M .
Equation (6.3 – 8) is used to find the mass moment IOxx of a
body about the x axis given the mass moment IMxx of the body
about the x’ axis. The same steps that led to Eq. (6.3 – 8) can be
followed to obtain the parallel axis equations for the other two
mass moments. The parallel axis equations for the other two
mass moments are
(6.3 – 9)
I Oyy  I Myy  ry2 M ,
I Ozz  I Mzz  rz2 M .
Mass Products
From Table 6.2 – 4, the mass product IOxy of a body is
I Oxy   xydm and the mass product IMxy of the body
is I Mxy   x' y ' dm . From Eq. (6.3 – 1)
Key Terms
Geometric Center; Mass Center; Mass Integrals about a
Point; Parallel-Axis Equations; Space Integrals about a
Point; Translating Coordinates;
Review Questions
1. State whether or not the parallel-axis equation for the mass
integral of a body is valid when the body is non-uniform.
2. Explain why the space and mass integral tables do not
provide space integrals and mass integrals of bodies about
arbitrary points.
3. The parallel-axis equation for a space integral is used to
calculate the space integral of a body about an arbitrary point O
given the space integral about the geometric center C. Describe
how this equation could be used to calculate the space integral
of a body about point O given the space integral about a point B
that is not the geometric center C?