Instructor's
Notes:
Centroid of
an Irregular Region
After we have discussed (and derived formulas
for and worked problems involving) the centroid of a planar region bounded
between two curves in our calculus, I ask my students how they would determine
the centroid of the state of Washington -- “Where would Washington
balance if was uniformly thick and dense?" Since many of my calculus
students plan to major in engineering and are “hands on" types, someone
usually suggests cutting out a map of the state and actually trying to
balance it. At that point I agree that is a fine idea, and I produce
the
state of “chaos" (see Figure 1) and ask someone in the front row to
find the centroid by trying to balance it on a finger. (Sometimes
I hand a rectangular piece of cardboard and scissors to someone and ask
them to create a “crazy" shape for me.)
Figure 1.
Then I pull out a pin and a piece of string,
and pin the shape (somewhere along its edge) to a classroom bulletin board
(using a loose enough hole so the shape can swing on the pin). I explain
that the centroid (center of mass) of the region is attracted by gravity
and ends up on the line connecting the pin with the center of the earth
and that this is the same line segment created by a weight hanging by string
from the pin (see Figure 2). So we know that the centroid of the
region lies on the line created by the hanging string.
Figure 2.
If we repeat hanging the region from another
point near its edge, we get another line that must contain the centroid
of the region. Thus the centroid of our region must lie on both lines,
so it lies at their intersection, and we are done. (See Figure 3.)
Figure 3.
As a check, I always do this a third time
to get a third line, and, if everything was working correctly (and swinging
freely), the three lines intersect at a point or very close to a point
(see Figure 4). Then I show that we actually found the centroid by
showing that I can balance the region on my finger at the centroid.
Figure 4.
After this demonstration, I usually give them
an assignment (to be turned in a couple days) to find the centroid of a
city, state or country that interests them (with the restriction that no
one can be interested in a rectangular state such as Nevada). Since
I have a lot of foreign students, I get to see the centroids of many countries
-- did you know the centroid of Brazil is very close to the capital city
Brasilia, and that was part of the reason the location was chosen to be
the capital?
Center of Population:
After the “uniform thickness, uniform density"
case, I ask how we could find the location of the “center of population"
of our state, and someone usually suggests attaching weights proportional
to the populations of the major cities at the locations of the cities.
Sometimes we simply discuss that idea and sometimes I have some fishing
weights that I attach to our crazy shape (see Figure 5). Then I quickly
take the “weighted region" and determine a couple lines using the string
to find the 'balance point' for a populated region. (The shiny squares
in Figure 5 are clear mounting squares which make it easy to adjoin weights
to the figure.)
Figure 5.
One other situation:
If there are only a few minutes left in
class, I demonstrate that the centroid of a region does not have to actually
be in the region (Figures 6 and 7).
Figure 6.
Figure 7.
Materials:
a piece of rigid cardboard or foamboard
a piece of string
a pin
something to act as a weight on the string
Comments:
This is a simple and quick demonstration
that seems to appeal to the “hands on" students in the class. And
they seem to appreciate that I am willing to do something “real" for them
after all of the derivations and calculations that we had been doing to
find centroids of regions determined by formulas in our calculus class.
Even some of the more “theory" oriented students seem to have a bit of
fun with the assignment of finding the centroid of a region of interest
to them.
I require the students to turn in a paragraph
telling me what region they chose, why they chose that region, and describing
the method they used to determine the centroid. I bring a box to
class on the day the assignments are due because they also have to turn
in their materials. (I look for pin holes on their regions to see
that they actually did the experiment.)
Sometimes as part of the same assignment
I have them use the “official math department teeter-totter" to find their
own center of mass.
This hands on activity can also be used
in a precalculus course or as part of a geometry course for education majors
at all levels.
Other resources:
Centroid of a
triangle.
The following
link lets you drag to change the vertices of a triangle and shows that
the centroid remains at the intersection of the medians.
http://www.keypress.com/sketchpad/javasketchpad/gallery/pages/centroid.php
'This is an interesting link where students
can see an illustration of how a centroid of a triangle is calculated.
Instructor or student can click on any of t he three vertices and change
the shape of the triangle as desired. A good visual aid for students to
understand theory and math of centroids, pictorially.' (by Un Jung Sin,
student at Temple University, Spring 2001)
The next link briefly discuss
the centroid of a tetrahedron in three space.
http://www.pballew.net/centroid.html
A table of centroids.
A table of centroid locations
for a few common line segments, plane regions, and solids. Includes formulas
and pictures.
http://www.engineering.com/content/ContentDisplay?contentId=41005016
Credits:
This demo was submitted by
Dale
Hoffman
Department of Mathematics
Bellevue Community College
and is included in Demos
with Positive Impact with his permission.
We also appreciate the work of Un Jung
Sin in reviewing the web sites mentioned above and recommending that several
be included as other resources.
