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					Fluid Mechanics                    Worksheet on First and Second Moments            page 1 of 7


Part 1. The "centroid" of an object is the geometric center of an object

The "centroid" of an object is independent of the density of the object. For a two dimensional
body, the centroid (xc, yc) can be calculated as:

                                  1                          1
                                  A                         A
                           xc       x dA             yc       y dA



Part 2. Gaining an intuition for first and second moments

The center of mass of a system of particles is a specific point about which the sum of all the
linear moments of the particles is zero. It is the point at which all of the system’s mass could be
concentrated without affecting its response to gravity. When the density is uniform, the center of
mass can be computed only from geometry – it is identical to the centroid. If the density is not
uniform, then the center of mass for a 2-dimensional body is given by:

                       1                 My                  1                Mx
                  x
                       M     x dA     M
                                                      y
                                                             M    y  dA    M

where M is the total mass of the body and Mx and My are called the “first moments” of the body.

        Mx is the first moment with respect to the x axis.

        My is the first moment with respect to the y axis.

              M x   y dA and M y   x dA and M    dA
                       A                          A                           A




The first moments of a body tell us the torque that it will exert about different axes in a
gravitational field.


Question/Demonstration 1: Something seems kind of backwards here. Why do you need to
sum up all of the “x” distances to get the moment with respect to the y-axis, and sum up all of the
“y” distances to get the moment with respect to the x-axis?
Fluid Mechanics              Worksheet on First and Second Moments                             page 2 of 7


We are often interested in the energy that is stored in a rotating body, or the amount of energy it
would take to get a body to rotate. In this case, we need to calculate the second moments
(moments of inertia) of the body, given by: I  y 2 dA
                                                        x      A
                                                                        I  x 2  dA   y   
                                                                                           A



The energy to rotate an object depends a lot on whether we're rotating it about the x-axis or the
y-axis.

Demonstration 2: Compare I x  A y dA with
                                   2
                                                                    I y   x 2  dA
                                                                          A




Demonstration 3: Shifting the axis has a big effect on the moment of inertia

To account for this, we can use the parallel axis theorem, which states that the moment of inertia
about any axis parallel to the axis through the center of mass is given by

                       Iparallel axis = Icenter of mass + Md2




Some important points :

       Moments are only meaningful with respect to some axis
       The axis doesn’t have to be the x or y axis, it can be some arbitrary axis
Fluid Mechanics                Worksheet on First and Second Moments                  page 3 of 7


Part 3: The term “moment of inertia” means different things to different people.


Quantity                  Engineers                            Physicists and mathematicians
                          tend to call the quantity            tend to call the quantity

                          First moment                         First area moment OR
M x   y dA              with respect to the x axis           First moment of the area
          A
                                                               with respect to the x axis
                          First moment                         First area moment OR
M y   x dA              with respect to the y axis           First moment of the area
          A
                                                               with respect to the y axis
                          First mass moment                    First moment
M x   y dA             with respect to the x axis           with respect to the x axis
          A


                          First mass moment                    First moment
M y   x dA             with respect to the y axis           with respect to the y axis
          A


                          Moment of inertia OR                 Area moment of inertia OR
I x   y 2 dA            Second moment                        Second moment of the area
      A
                          with respect to the x axis           with respect to the x axis
                          Moment of inertia OR                 Area moment of inertia OR
I y   x 2 dA            Second moment                        Second moment of the area
      A
                          with respect to the y axis           with respect to the y axis
                          Mass moment of inertia OR            Moment of inertia OR
I x   y 2  dA          Second mass moment                   Second moment
      A
                          with respect to the x axis           with respect to the x axis
                          Mass moment of inertia OR            Moment of inertia OR
I y   x 2  dA          Second mass moment                   Second moment
      A
                          with respect to the y axis           with respect to the y axis


Part 4: What terms will we use, in order to avoid confusion?

Answer: To avoid ambiguity, in this class we will always include the word “mass” when the
integral includes density, and we will always include the word “area” when the integral does not
include density. This means that our terms will be:

                   First area moment             and     First mass moment

                   Area moment of inertia        and     Mass moment of inertia

When you go out in the world – to industry, to graduate school, to a non-profit, or to consulting –
be sure that you are sharing the same definitions as the other people you are working with.
Fluid Mechanics            Worksheet on First and Second Moments   page 4 of 7


Part 5: Quiz time.    You need to know this for the exam

Write down the integrals that express:

A) First area moment with respect to the x axis




B) Mass moment of inertia with respect to the y axis




C) First mass moment with respect to the x axis




D) First area moment with respect to the y axis




E) Area moment of inertia with respect to the x axis




F) Mass moment of inertia with respect to the x axis
Fluid Mechanics             Worksheet on First and Second Moments                 page 5 of 7


Part 6: Problems. You need to know this for the exam

You will be provided with the table of integrals that appears on the last page of this handout.

Problem 1: Find the area moment of inertia with respect to the y-axis that goes through the
centroid of a rectangle of length A and height B.




 Problem 2: Find the area moment of inertia with respect to the x-axis that passes through the
centroid of a circle of radius R.
Fluid Mechanics             Worksheet on First and Second Moments                page 6 of 7


Problem 3: Find the mass moment of inertia with respect to the y-axis that passes through the
centroid of a circle of radius R. Assume constant density over the region.




Problem 4: Find the mass moment of inertia with respect to the x-axis for a half circle of radius
R that lies in the upper half plane (y>0). Assume constant density over the region.




Problem 5: Find the mass moment of inertia about the line parallel to the x-axis that passes
through the centroid of a half circle of radius R. Assume constant density over the region.




Homework: Practice finding the area moments of inertia on page 61 of Munson et al.
Fluid Mechanics              Worksheet on First and Second Moments   page 7 of 7


Possibly useful integrals:

				
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posted:11/27/2011
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