Resumo sobre Centro de Gravidade e Centroide em Corpos Rígidos 2D

274 REVIEW AND SUMMARY This chapter was devoted chiefly to the determination of the center of gravity of a rigid body ie to the determination of the point G where a single force W called the weight of the body can be applied to represent the effect of the earths attraction on the body In the first part of the chapter we considered twodimensional bodies such as flat plates and wires contained in the xy plane By adding force components in the vertical z direction and moments about the horizontal y and x axes Sec 52 we derived the relations W 5 dW xW 5 x dW yW 5 y dW 52 which define the weight of the body and the coordinates x and y of its center of gravity In the case of a homogeneous flat plate of uniform thickness Sec 53 the center of gravity G of the plate coincides with the centroid C of the area A of the plate the coordinates of which are defined by the relations xA 5 x dA yA 5 y dA 53 Similarly the determination of the center of gravity of a homoge neous wire of uniform cross section contained in a plane reduces to the determination of the centroid C of the line L representing the wire we have xL 5x dL yL 5y dL 54 The integrals in Eqs 53 are referred to as the first moments of the area A with respect to the y and x axes and are denoted by Qy and Qx respectively Sec 54 We have Qy 5 xA Qx 5 yA 56 The first moments of a line can be defined in a similar way The determination of the centroid C of an area or line is simplified when the area or line possesses certain properties of symmetry If the area or line is symmetric with respect to an axis its centroid C Center of gravity of a twodimensional body Centroid of an area or line First moments Properties of symmetry bee29400ch05218283indd Page 274 12108 33016 PM users172 bee29400ch05218283indd Page 274 12108 33016 PM users172 Volumes204MHDQ076work0indd0 Volumes204MHDQ076work0indd0 275 Review and Summary lies on that axis if it is symmetric with respect to two axes C is located at the intersection of the two axes if it is symmetric with respect to a center O C coincides with O The areas and the centroids of various common shapes are tabulated in Fig 58 When a flat plate can be divided into several of these shapes the coordinates X and Y of its center of gravity G can be determined from the coordinates x1 x2 and y1 y2 of the centers of gravity G1 G2 of the various parts Sec 55 Equating moments about the y and x axes respectively Fig 524 we have XwoW 5 oxwW YwoW 5 oywW 57 Center of gravity of a composite body x y z x y z O O G X Y ΣW G1 G2 G3 W1 W2 W3 Fig 524 If the plate is homogeneous and of uniform thickness its center of gravity coincides with the centroid C of the area of the plate and Eqs 57 reduce to Qy 5 XwoA 5 oxwA Qx 5 YwoA 5 oywA 58 These equations yield the first moments of the composite area or they can be solved for the coordinates X and Y of its centroid Sam ple Prob 51 The determination of the center of gravity of a com posite wire is carried out in a similar fashion Sample Prob 52 When an area is bounded by analytical curves the coordinates of its centroid can be determined by integration Sec 56 This can be done by evaluating either the double integrals in Eqs 53 or a sin gle integral which uses one of the thin rectangular or pieshaped elements of area shown in Fig 512 Denoting by xel and yel the coordinates of the centroid of the element dA we have Qy 5 xA 5 xel dA Qx 5 yA 5 yel dA 59 It is advantageous to use the same element of area to compute both of the first moments Qy and Qx the same element can also be used to determine the area A Sample Prob 54 Determination of centroid by integration bee29400ch05218283indd Page 275 112908 45518 PM users172 bee29400ch05218283indd Page 275 112908 45518 PM users172 Volumes204MHDQ076work0indd0 Volumes204MHDQ076work0indd0 Distributed Forces Centroids and Centers of Gravity Theorems of PappusGuldinus Distributed loads Center of gravity of a threedimensional body Centroid of a volume The theorems of PappusGuldinus relate the determination of the area of a surface of revolution or the volume of a body of revolution to the determination of the centroid of the generating curve or area Sec 57 The area A of the surface generated by rotating a curve of length L about a fixed axis Fig 525a is A 2πȳL 510 where ȳ represents the distance from the centroid C of the curve to the fixed axis Similarly the volume V of the body generated by rotating an area A about a fixed axis Fig 525b is V 2πȳA 511 where ȳ represents the distance from the centroid C of the area to the fixed axis The concept of centroid of an area can also be used to solve problems other than those dealing with the weight of flat plates For example to determine the reactions at the supports of a beam Sec 58 we can replace a distributed load w by a concentrated load W equal in magnitude to the area A under the load curve and passing through the centroid C of that area Fig 526 The same approach can be used to determine the resultant of the hydrostatic forces exerted on a rectangular plate submerged in a liquid Sec 59 The last part of the chapter was devoted to the determination of the center of gravity G of a threedimensional body The coordinates x ȳ z of G were defined by the relations xW x dW ȳW y dW zW z dW 517 In the case of a homogeneous body the center of gravity G coincides with the centroid C of the volume V of the body the coordinates of C are defined by the relations xV x dV ȳV y dV zV z dV 519 If the volume possesses a plane of symmetry its centroid C will lie in that plane if it possesses two planes of symmetry C will be located on the line of intersection of the two planes if it possesses three planes of symmetry which intersect at only one point C will coincide with that point Sec 510 The volumes and centroids of various common threedimensional shapes are tabulated in Fig 521 When a body can be divided into several of these shapes the coordinates X Y Z of its center of gravity G can be determined from the corresponding coordinates of the centers of gravity of its various parts Sec 511 We have XΣW Σx W YΣW Σȳ W ZΣW Σz W 520 If the body is made of a homogeneous material its center of gravity coincides with the centroid C of its volume and we write Sample Probs 511 and 512 XΣV Σx V YΣV Σȳ V ZΣV Σz V 521 When a volume is bounded by analytical surfaces the coordinates of its centroid can be determined by integration Sec 512 To avoid the computation of the triple integrals in Eqs 519 we can use elements of volume in the shape of thin filaments as shown in Fig 527 Denoting by xel ȳel and zel the coordinates of the centroid of the element dV we rewrite Eqs 519 as xV xel dV ȳV ȳel dV zV zel dV 523 which involve only double integrals If the volume possesses two planes of symmetry its centroid C is located on their line of intersection Choosing the x axis to lie along that line and dividing the volume into thin slabs parallel to the yz plane we can determine C from the relation xV xel dV 524 with a single integration Sample Prob 513 For a body of revolution these slabs are circular and their volume is given in Fig 528 278 REVIEW PROBLEMS 5137 and 5138 Locate the centroid of the plane area shown x y 54 mm 72 mm 30 mm 54 mm 48 mm Fig P5137 x y a 8 in x ky2 b 4 in Fig P5138 5139 The frame for a sign is fabricated from thin flat steel bar stock of mass per unit length 473 kgm The frame is supported by a pin at C and by a cable AB Determine a the tension in the cable b the reaction at C A C B R 075 m 08 m 02 m 135 m 06 m Fig P5139 x y y mx b y k1 cx2 h a Fig P5140 5140 Determine by direct integration the centroid of the area shown Express your answer in terms of a and h bee29400ch05218283indd Page 278 112908 45520 PM users172 bee29400ch05218283indd Page 278 112908 45520 PM users172 Volumes204MHDQ076work0indd0 Volumes204MHDQ076work0indd0 279 Review Problems 5141 Determine by direct integration the centroid of the area shown Express your answer in terms of a and b A B C 3 m 1 m 900 Nm Fig P5143 A wA wBC wDE B C D E F 6 m 31 m 06 m 10 m 08 m 1200 Nm Fig P5144 30 A B 18 ft d Fig P5145 5145 The square gate AB is held in the position shown by hinges along its top edge A and by a shear pin at B For a depth of water d 5 35 ft determine the force exerted on the gate by the shear pin 4 in 4 in 10 in Fig P5142 x y b a 2b y 2b cx2 y kx2 Fig P5141 5142 Knowing that two equal caps have been removed from a 10in diameter wooden sphere determine the total surface area of the remaining portion 5143 Determine the reactions at the beam supports for the given loading 5144 A beam is subjected to a linearly distributed downward load and rests on two wide supports BC and DE which exert uniformly distributed upward loads as shown Determine the values of wBC and wDE corresponding to equilibrium when wA 5 600 Nm bee29400ch05218283indd Page 279 112908 45523 PM users172 bee29400ch05218283indd Page 279 112908 45523 PM users172 Volumes204MHDQ076work0indd0 Volumes204MHDQ076work0indd0 280 Distributed Forces Centroids and Centers of Gravity 5148 Locate the centroid of the volume obtained by rotating the shaded area about the x axis y a z x b 2 L h b Fig P5146 5146 Consider the composite body shown Determine a the value of x when h 5 L2 b the ratio hL for which x 5 L x y z 016 m 02 m 012 m 01 m 005 m r 018 m Fig P5147 y x 1 m 3 m y 1 1 x Fig P5148 5147 Locate the center of gravity of the sheetmetal form shown bee29400ch05218283indd Page 280 112908 45527 PM users172 bee29400ch05218283indd Page 280 112908 45527 PM users172 Volumes204MHDQ076work0indd0 Volumes204MHDQ076work0indd0