Concept explainers
Show that as r1 approaches r2, the location of the centroid approaches that for an arc of circle of radius (r1 + r2)/2.
To show that as
Answer to Problem 5.17P
The location of the centroid for an arc of circle of radius
Explanation of Solution
Write the expression for the y-coordinate of the centroid of the sector with radius
Here
Write the expression for the area of the sector with radius
Write the expression for the y-coordinate of the centroid of the sector with radius
Here
Write the expression for the area of the sector with radius
Write the expression for
Substitute (I), (II), (III) and (IV) in the above equation to calculate
Write the expression for
Substitute (II) and (IV) in the above equation to calculate
Write the expression to calculate the y-coordinate of the centroid of the given area.
Substitute (V) and (VI) in the above equation to calculate
Write the expression for the y-coordinate of the centroid with an arc of radius
Rewrite the expression
Let
Substitute the above two expressions in (VIII) to rewrite.
Apply the limits
Rewrite r using
Conclusion:
Substitute (XII) in (VII) to calculate
Therefore, the above expression is same as that of the expression given in (VIII).
Thus, the location of the centroid for an arc of circle of radius
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Chapter 5 Solutions
Vector Mechanics for Engineers: Statics
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- International Edition---engineering Mechanics: St...Mechanical EngineeringISBN:9781305501607Author:Andrew Pytel And Jaan KiusalaasPublisher:CENGAGE L