Centróides de Áreas e Linhas - Problemas Resolvidos e Teoremas de Pappus-Guldinus

243 SOLVING PROBLEMS ON YOUR OWN In the problems for this lesson you will use the equations xA 5 x dA yA 5 y dA 53 xL 5 x dL yL 5 y dL 54 to locate the centroids of plane areas and lines respectively You will also apply the theorems of PappusGuldinus Sec 57 to determine the areas of surfaces of revolution and the volumes of bodies of revolution 1 Determining by direct integration the centroids of areas and lines When solving problems of this type you should follow the method of solution shown in Sample Probs 54 and 55 compute A or L determine the first moments of the area or the line and solve Eqs 53 or 54 for the coordinates of the centroid In addition you should pay particular attention to the following points a Begin your solution by carefully defining or determining each term in the applicable integral formulas We strongly encourage you to show on your sketch of the given area or line your choice for dA or dL and the distances to its centroid b As explained in Sec 56 the x and the y in the above equations represent the coordinates of the centroid of the differential elements dA and dL It is important to recognize that the coordinates of the centroid of dA are not equal to the coordi nates of a point located on the curve bounding the area under consideration You should carefully study Fig 512 until you fully understand this important point c To possibly simplify or minimize your computations always examine the shape of the given area or line before defining the differential element that you will use For example sometimes it may be preferable to use horizontal rectangular elements instead of vertical ones Also it will usually be advantageous to use polar coordinates when a line or an area has circular symmetry d Although most of the integrations in this lesson are straightforward at times it may be necessary to use more advanced techniques such as trigonometric sub stitution or integration by parts Of course using a table of integrals is the fastest method to evaluate difficult integrals 2 Applying the theorems of PappusGuldinus As shown in Sample Probs 56 through 58 these simple yet very useful theorems allow you to apply your knowl edge of centroids to the computation of areas and volumes Although the theorems refer to the distance traveled by the centroid and to the length of the generating curve or to the generating area the resulting equations Eqs 510 and 511 contain the products of these quantities which are simply the first moments of a line y L and an area y A respectively Thus for those problems for which the generating line or area consists of more than one common shape you need only determine y L or y A you do not have to calculate the length of the generating curve or the generating area bee29400ch05218283indd Page 243 112908 45437 PM users172 bee29400ch05218283indd Page 243 112908 45437 PM users172 Volumes204MHDQ076work0indd0 Volumes204MHDQ076work0indd0 244 PROBLEMS 534 through 536 Determine by direct integration the centroid of the area shown Express your answer in terms of a and h x y h a Fig P534 x y y mx y kx2 h a Fig P535 x y y kx3 h a Fig P536 537 through 539 Determine by direct integration the centroid of the area shown x y b a x2 a2 y2 b2 1 Fig P537 x y r1 r2 Fig P538 x y a a 2 a 2 a a Fig P539 540 and 541 Determine by direct integration the centroid of the area shown Express your answer in terms of a and b x y b a y kx a2 Fig P540 x y b a y1 k1x2 y2 k2 x4 Fig P541 bee29400ch05218283indd Page 244 112908 45439 PM users172 bee29400ch05218283indd Page 244 112908 45439 PM users172 Volumes204MHDQ076work0indd0 Volumes204MHDQ076work0indd0 245 Problems 542 Determine by direct integration the centroid of the area shown 543 and 544 Determine by direct integration the centroid of the area shown Express your answer in terms of a and b y x L L a y a 1 x L x2 L2 Fig P542 x y b 2 b 2 a 2 a 2 x ky2 Fig P543 x y y kx2 a a b b Fig P544 545 and 546 A homogeneous wire is bent into the shape shown Determine by direct integration the x coordinate of its centroid x y a a π 0 2 θ x a cos3 y a sin3 θ θ Fig P545 y x r 45 45 Fig P546 547 A homogeneous wire is bent into the shape shown Determine by direct integration the x coordinate of its centroid Express your answer in terms of a 548 and 549 Determine by direct integration the centroid of the area shown x y a a y kx 3 2 Fig P547 x y a y a cos px 2L L 2 L 2 Fig P548 x y q r a eq Fig P549 bee29400ch05218283indd Page 245 112908 45442 PM users172 bee29400ch05218283indd Page 245 112908 45442 PM users172 Volumes204MHDQ076work0indd0 Volumes204MHDQ076work0indd0 246 Distributed Forces Centroids and Centers of Gravity 550 Determine the centroid of the area shown when a 5 2 in 551 Determine the value of a for which the ratio xyy is 9 552 Determine the volume and the surface area of the solid obtained by rotating the area of Prob 51 about a the line x 5 240 mm b the y axis 553 Determine the volume and the surface area of the solid obtained by rotating the area of Prob 52 about a the line y 5 60 mm b the y axis 554 Determine the volume and the surface area of the solid obtained by rotating the area of Prob 58 about a the x axis b the y axis 555 Determine the volume of the solid generated by rotating the para bolic area shown about a the x axis b the axis AA9 556 Determine the volume and the surface area of the chain link shown which is made from a 6mmdiameter bar if R 5 10 mm and L 5 30 mm y x a y 1 1 in 1 x Fig P550 and P551 L R R Fig P556 90 3 4 in 1 4 in 1 1 in Fig P558 R R Fig P559 557 Verify that the expressions for the volumes of the first four shapes in Fig 521 on page 260 are correct 558 A 3 4indiameter hole is drilled in a piece of 1inthick steel the hole is then countersunk as shown Determine the volume of steel removed during the countersinking process 559 Determine the capacity in liters of the punch bowl shown if R 5 250 mm x y h a a a A A Fig P555 bee29400ch05218283indd Page 246 112908 45443 PM users172 bee29400ch05218283indd Page 246 112908 45443 PM users172 Volumes204MHDQ076work0indd0 Volumes204MHDQ076work0indd0 247 Problems 560 Three different drive belt profiles are to be studied If at any given time each belt makes contact with onehalf of the circumference of its pulley determine the contact area between the belt and the pulley for each design 0625 in a b c 008 in r 025 in 40 40 0375 in 0125 in 3 in 3 in 3 in Fig P560 561 The aluminum shade for the small highintensity lamp shown has a uniform thickness of 1 mm Knowing that the density of alumi num is 2800 kgm3 determine the mass of the shade 32 mm 26 mm 32 mm 56 mm 28 mm 66 mm 8 mm Fig P561 562 The escutcheon a decorative plate placed on a pipe where the pipe exits from a wall shown is cast from brass Knowing that the density of brass is 8470 kgm3 determine the mass of the escutcheon 563 A manufacturer is planning to produce 20000 wooden pegs having the shape shown Determine how many gallons of paint should be ordered knowing that each peg will be given two coats of paint and that one gallon of paint covers 100 ft2 564 The wooden peg shown is turned from a dowel 1 in in diameter and 4 in long Determine the percentage of the initial volume of the dowel that becomes waste 565 The shade for a wallmounted light is formed from a thin sheet of translucent plastic Determine the surface area of the outside of the shade knowing that it has the parabolic cross section shown 75 mm 25 mm 75 mm 26 26 Fig P562 r 01875 in r 0875 in 300 in 100 in 050 in 050 in 0625 in Fig P563 and P564 100 mm y x y kx2 250 mm Fig P565 bee29400ch05218283indd Page 247 12108 33001 PM users172 bee29400ch05218283indd Page 247 12108 33001 PM users172 Volumes204MHDQ076work0indd0 Volumes204MHDQ076work0indd0