Centroids and Center of Gravity - Centroid of a Curve
? 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.
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The steps in the document can be repeated to solve similar problems.
Problem Statement
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A thin rod is bent into the smooth curve  for  .
Find the centroid (i.e., the geometric center) of the curved rod.
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   Figure 1
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Solution
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Step
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Result
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If the geometry of an object is in the form of a curve, the balance of moments of the differential elements dL about each of the coordinate axes yields  , the coordinates of the centroid of the curve:
  
where the terms  ,  ,  represent the "moment arms" or the centroid for the differential element that is used.
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Figure 2 shows the differential element, dL. The length of this element can be expressed in terms of dx and dy using Pythagoras' theorem.
Since,  , then  so
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Figure 2
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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 -1 and 2 respectively, and overwrite  with the appropriate expression. The square root function can be found in the Expression palette.
Press [Enter] evaluate.
Right-click the output and select Approximate > 5.
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(3.1) |
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(3.2) |
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From the formula for  , the y-component of the centroid can be computed as the ratio shown to the right. The y must be replaced by  since the integration is taking place with respect to x.
To calculate a definite integral, click on the definite integral template from the Expression palette. Follow the instructions from the previous step.
Press [Enter] evaluate.
Right-click the output and select Approximate > 5.
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(3.4) |
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