NEW 375 Lecture14(2) - FIRST MOMENT OF AREA LET US CONSIDER...

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Unformatted text preview: FIRST MOMENT OF AREA LET US CONSIDER A GENERIC AREA A LOCATED IN THE PLANE x-y . IF THE ELEMENTARY AREA dA HAS COORDINATES x AND y , WE DEFINE THE FIRST MOMENT OF AREA WITH RESPECT TO THE x AXIS AS: ∫ = A x dA y Q NOTE THAT THE UNITS OF MEASUREMENT FOR MOMENTS OF AREA ARE [LENGTH] 3 . FIRST MOMENT OF AREA SIMILARLY, THE FIRST MOMENT OF AREA WITH RESPECT TO THE y AXIS IS: ∫ = A y dA x Q NOTE THAT THESE INTEGRALS CAN BE POSITIVE, NEGATIVE OR ZERO DEPENDING ON THE POSITION OF THE AXES. DEFINITION OF CENTROID WE DEFINE THE CENTROID OF AN AREA A THE POINT C OF CO- ORDINATES x C AND y C WHICH SATISFY THE FOLLOWING RELATIONS: ∫ ∫ = = ⋅ = = ⋅ A x y A Q dA y y A Q dA x x A NOTE THAT THE FIRST MOMENTS OF AREA CAN BE EXPRESSED AS THE PRODUCT OF THE AREA AND THE RELEVANT COORDINATES OF THE CENTROID SOME PROPERTIES IF THE AREA POSSESSES AN AXIS OF SYMMETRY , THE FIRST MOMENT OF AREA WITH RESPECT TO THAT AXIS WILL BE ZERO, I.E. THE CENTROID WILL BE FOUND ALONG THAT AXIS IF THE AREA POSSESSES TWO OR MORE AXES OF SYMMETRY, THE CENTROID WILL BE FOUND AT THE INTERSECTION OF THE AXES. SOME PROPERTIES IF THE AREA POSSESSES A CENTER OF SYMMETRY , THE FIRST MOMENT OF AREA WITH RESPECT TO ANY AXIS THROUGH THAT POINT WILL BE ZERO, I.E. THE CENTROID IS THE CENTER OF SYMMETRY SOME PROPERTIES THEREFORE, IF THE CENTROID OF AN AREA CAN BE LOCATED BY ANY OF THESE PROPERTIES, THE FIRST MOMENT OF AREA ABOUT ANY AXIS CAN BE EASILY OBTAINED USING THESE EQUATIONS: x A Q y A Q y x ⋅ = ⋅ = A QUICK EXAMPLE TO CALCULATE THE FIRST MOMENTS OF AREA ABOUT AXES x AND y IN FIGURE, KNOWING THAT THE CENTROID IS AT MID-HEIGHT, MID-WIDTH OF THE RECTANGLE (INTERSE- CTION OF AXES OF SYMMETRY): ( 29 ( 29 2 2 2 2 2 2 h b b bh x A Q bh h bh y A Q y x = = ⋅ = = = ⋅ = COMPOSITE AREAS...
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This note was uploaded on 03/27/2008 for the course ENGIN 375 taught by Professor Miller during the Spring '08 term at University of Cincinnati.

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NEW 375 Lecture14(2) - FIRST MOMENT OF AREA LET US CONSIDER...

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