Centroids and Center of Gravity - Centroid of a Plane Region
? Maplesoft, a division of Waterloo Maple Inc., 2008
Introduction
This application is one of a collection of educational engineering examples using Maple. These applications use Clickable Engineering? methods to solve problems interactively. Steps are given at every stage of the solution, and many are illustrated using short video clips.
Click on the 
buttons to watch the videos.
The steps in the document can be repeated to solve similar problems.
Problem Statement
|
Sheet metal of uniform thickness is cut to form a perfect quarter circle with radius R.
Find the centroid of the piece in terms of R.
|
Figure 1
|
Solution
|
Step
|
Result
|
|
The centroid  for a plane region can be found by subdividing the area into differential elements dA and computing the 'moment' of this element about each of the coordinate axes. The centroid is the location where concentrating the total area generates the same moments as the distributed area. Hence, the centroid is given by
 
where the terms  ,  represent the "moment arms" for the centroid of the differential element that is used.
|
|
The graph in Figure 2 shows the differential element in polar coordinates. The gray sector is approximated as a triangle, with the angle  measured in radians. The area of this approximating triangle is
Since it is known that the centroid of a triangle lies along its median,  of its length from the base, the centroid of the differential area will be:
|
Figure 2
|
|
From the formula for  the x-component of the centroid can be computed as the ratio shown to the right.
To enter  , press [Ctrl][Shift]["] and then press the underscore (_) key.
Use the right arrow (→) to move back to the baseline. Use the assignment operator  (a colon followed by an equal sign) to define the variable.
To calculate a definite integral, click on the definite integral template from the Expression palette.
Overwrite  and  with 0 and  respectively, and overwrite  with the appropriate expression.
Obtain π and θ from the Greek palette.
Press [Enter] to evaluate.
|
 |
(1) |
|
|
From the formula for  , the y-component of the centroid can be computed as the ratio shown to the right.
To calculate a definite integral, click on the definite integral template from the Expression palette.
Press [Enter] evaluate.
Right-click the output and select Approximate > 5.
The two coordinates are numerically the same because the area is symmetric.
|
 |
(2) |
|
Legal Notice: The copyright for this application is owned by Maplesoft. The application is intended to demonstrate the use of Maple to solve a particular problem. It has been made available for product evaluation purposes only and may not be used in any other context without the express permission of Maplesoft.
|
