When forces are continuously distributed over a region of a structure ...

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c05.qxd 10/29/07 1:17 PM Page 255Article 5/3 Problems 255 5/41 Determine the y-coordinate of the centroid of theplane area shown. Set h 0 in your result and comparewith the result y for a full semicircular4a3area (see Sample Problem 5/3 and Table D/3). Alsoaevaluate your result for the conditions h and4ah 2 .Ans. y 23 [a2 h 2 ] 3/2a 2 2 sin1 h a ha2 h 2 5/43 Determine the x-coordinate of the mass center of thesolid homogeneous body shown.Ans. x 1.542R2RRh a 4 : y 0.562a, h a 2 : y 0.705ayx4RhaProblem 5/41 5/42 Determine the x-coordinate of the mass center of thehomogeneous hemisphere with the smaller hemisphericalportion removed.Ans. x 45112 RyxProblem 5/43 5/44 Determine the x-coordinate of the mass center of thecylindrical shell of small uniform thickness.Ans. x 1.583R2RRRxxR–– 2z4RProblem 5/42Problem 5/44
c05.qxd 10/29/07 1:17 PM Page 256256 Chapter 5 Distributed Forces5/4 COMPOSITE B ODIES AND F IGURES;A PPROXIMATIONSWhen a body or figure can be conveniently divided into several partswhose mass centers are easily determined, we use the principle of momentsand treat each part as a finite element of the whole. Such a body isillustrated schematically in Fig. 5/13. Its parts have masses m 1 , m 2 , m 3with the respective mass-center coordinates x 1 , x 2 , x 3 in the x-direction.The moment principle gives(m 1 m 2 m 3 )X m 1 x 1 m 2 x 2 m 3 x 3where X is the x-coordinate of the center of mass of the whole. Similarrelations hold for the other two coordinate directions.We generalize, then, for a body of any number of parts and expressthe sums in condensed form to obtain the mass-center coordinatesX ΣmxΣmY ΣmyΣmZ ΣmzΣm(5/7)Analogous relations hold for composite lines, areas, and volumes, wherethe m’s are replaced by L’s, A’s, and V’s, respectively. Note that if a holeor cavity is considered one of the component parts of a composite bodyor figure, the corresponding mass represented by the cavity or hole istreated as a negative quantity.An Approximation MethodIn practice the boundaries of an area or volume might not be expressiblein terms of simple geometrical shapes or as shapes which canbe represented mathematically. For such cases we must resort to amethod of approximation. As an example, consider the problem of locat-–x 3––x 2– XG 1GGx 1m 1G 2m 23m 3Figure 5/13
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When forces are continuously distributed over a region of a structure ...

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