Equilíbrio de Corpos Rígidos 3D - Diagrama de Corpo Livre e Reações

196 T he equilibrium of a threedimensional body was considered in the sections you just completed It is again most important that you draw a complete freebody diagram as the first step of your solution 1 As you draw the freebody diagram pay particular attention to the reac tions at the supports The number of unknowns at a support can range from one to six Fig 410 To decide whether an unknown reaction or reaction component exists at a support ask yourself whether the support prevents motion of the body in a certain direction or about a certain axis a If motion is prevented in a certain direction include in your freebody diagram an unknown reaction or reaction component that acts in the same direction b If a support prevents rotation about a certain axis include in your free body diagram a couple of unknown magnitude that acts about the same axis 2 The external forces acting on a threedimensional body form a system equivalent to zero Writing oF 5 0 and oMA 5 0 about an appropriate point A and setting the coefficients of i j k in both equations equal to zero will provide you with six scalar equations In general these equations will contain six unknowns and may be solved for these unknowns 3 After completing your freebody diagram you may want to seek equations involving as few unknowns as possible The following strategies may help you a By summing moments about a ballandsocket support or a hinge you will obtain equations from which three unknown reaction components have been elimi nated Sample Probs 48 and 49 b If you can draw an axis through the points of application of all but one of the unknown reactions summing moments about that axis will yield an equation in a single unknown Sample Prob 410 4 After drawing your freebody diagram you may find that one of the following situations exists a The reactions involve fewer than six unknowns the body is said to be partially constrained and motion of the body is possible However you may be able to determine the reactions for a given loading condition Sample Prob 47 b The reactions involve more than six unknowns the reactions are said to be statically indeterminate Although you may be able to calculate one or two reactions you cannot determine all of the reactions Sample Prob 410 c The reactions are parallel or intersect the same line the body is said to be improperly constrained and motion can occur under a general loading condition SOLVING PROBLEMS ON YOUR OWN bee29400ch04156217indd Page 196 112908 33358 PM users172 bee29400ch04156217indd Page 196 112908 33358 PM users172 Volumes204MHDQ076work0indd0 Volumes204MHDQ076work0indd0 PROBLEMS 197 491 A 4 3 8ft sheet of plywood weighing 34 lb has been temporarily placed among three pipe supports The lower edge of the sheet rests on small collars at A and B and its upper edge leans against pipe C Neglecting friction at all surfaces determine the reactions at A B and C 4 ft y z B A x 1 ft 375 ft 3 ft 5 ft 3 ft 4 ft C Fig P491 TB x D C B A z y 90 mm 90 mm 120 mm TC Fig P492 492 Two tape spools are attached to an axle supported by bearings at A and D The radius of spool B is 30 mm and the radius of spool C is 40 mm Knowing that TB 5 80 N and that the system rotates at a constant rate determine the reactions at A and D Assume that the bearing at A does not exert any axial thrust and neglect the weights of the spools and axle 493 Solve Prob 492 assuming that the spool C is replaced by a spool of radius 50 mm bee29400ch04156217indd Page 197 112908 33400 PM users172 bee29400ch04156217indd Page 197 112908 33400 PM users172 Volumes204MHDQ076work0indd0 Volumes204MHDQ076work0indd0 198 Equilibrium of Rigid Bodies 494 Two transmission belts pass over sheaves welded to an axle supported by bearings at B and D The sheave at A has a radius of 25 in and the sheave at C has a radius of 2 in Knowing that the system rotates at a constant rate determine a the tension T b the reactions at B and D Assume that the bearing at D does not exert any axial thrust and neglect the weights of the sheaves and axle T 720 N y 80 mm 120 mm 120 mm 200 mm A E B C D x z 40 mm Fig P495 495 A 200mm lever and a 240mmdiameter pulley are welded to the axle BE that is supported by bearings at C and D If a 720N verti cal load is applied at A when the lever is horizontal determine a the tension in the cord b the reactions at C and D Assume that the bearing at D does not exert any axial thrust 30 lb T 24 lb 18 lb y 8 in 6 in A B C D x z 6 in Fig P494 496 Solve Prob 495 assuming that the axle has been rotated clockwise in its bearings by 30 and that the 720N load remains vertical bee29400ch04156217indd Page 198 112908 33402 PM users172 bee29400ch04156217indd Page 198 112908 33402 PM users172 Volumes204MHDQ076work0indd0 Volumes204MHDQ076work0indd0 199 Problems 497 An opening in a floor is covered by a 1 3 12m sheet of plywood of mass 18 kg The sheet is hinged at A and B and is maintained in a position slightly above the floor by a small block C Determine the vertical component of the reaction a at A b at B c at C y z x A B E C 015 m 02 m 02 m 06 m 12 m D Fig P497 x C A 60 in 60 in 60 in 30 in 15 in 15 in B z y Fig P499 and P4100 498 Solve Prob 497 assuming that the small block C is moved and placed under edge DE at a point 015 m from corner E 499 The rectangular plate shown weighs 80 lb and is supported by three vertical wires Determine the tension in each wire 4100 The rectangular plate shown weighs 80 lb and is supported by three vertical wires Determine the weight and location of the lightest block that should be placed on the plate if the tensions in the three wires are to be equal bee29400ch04156217indd Page 199 112908 33402 PM users172 bee29400ch04156217indd Page 199 112908 33402 PM users172 Volumes204MHDQ076work0indd0 Volumes204MHDQ076work0indd0 200 Equilibrium of Rigid Bodies 4101 Two steel pipes AB and BC each having a mass per unit length of 8 kgm are welded together at B and supported by three wires Knowing that a 5 04 m determine the tension in each wire 4102 For the pipe assembly of Prob 4101 determine a the largest permissible value of a if the assembly is not to tip b the corre sponding tension in each wire 4103 The 24lb square plate shown is supported by three vertical wires Determine a the tension in each wire when a 5 10 in b the value of a for which the tension in each wire is 8 lb 4104 The table shown weighs 30 lb and has a diameter of 4 ft It is sup ported by three legs equally spaced around the edge A vertical load P of magnitude 100 lb is applied to the top of the table at D Determine the maximum value of a if the table is not to tip over Show on a sketch the area of the table over which P can act without tipping the table B A C D y x z a 12 m 06 m Fig P4101 y x B C A a 30 in a 30 in z Fig P4103 A B C D a P Fig P4104 bee29400ch04156217indd Page 200 112908 33403 PM users172 bee29400ch04156217indd Page 200 112908 33403 PM users172 Volumes204MHDQ076work0indd0 Volumes204MHDQ076work0indd0 201 Problems 4105 A 10ft boom is acted upon by the 840lb force shown Determine the tension in each cable and the reaction at the ballandsocket joint at A x y E G A B C D 225 ft 165 ft 5 ft 66 ft 5 ft 13 ft 6 ft 28 ft 32 ft F H z 66 ft Fig P4108 4106 A 24m boom is held by a ballandsocket joint at C and by two cables AD and AE Determine the tension in each cable and the reaction at C 4107 Solve Prob 4106 assuming that the 36kN load is applied at point A 4108 A 600lb crate hangs from a cable that passes over a pulley B and is attached to a support at H The 200lb boom AB is supported by a ballandsocket joint at A and by two cables DE and DF The center of gravity of the boom is located at G Determine a the tension in cables DE and DF b the reaction at A 840 lb x y z E A B C D 4 ft 6 ft 7 ft 6 ft 6 ft Fig P4105 A B C x y z D E 36 kN 12 m 12 m 12 m 06 m 08 m 08 m Fig P4106 bee29400ch04156217indd Page 201 112908 33403 PM users172 bee29400ch04156217indd Page 201 112908 33403 PM users172 Volumes204MHDQ076work0indd0 Volumes204MHDQ076work0indd0 202 Equilibrium of Rigid Bodies 4109 A 3m pole is supported by a ballandsocket joint at A and by the cables CD and CE Knowing that the 5kN force acts vertically downward f 5 0 determine a the tension in cables CD and CE b the reaction at A x y z E D C A 15 m 2 m 1 m 12 m 12 m B 5 kN f Fig P4109 and P4110 4110 A 3m pole is supported by a ballandsocket joint at A and by the cables CD and CE Knowing that the line of action of the 5kN force forms an angle f 5 30 with the vertical xy plane determine a the tension in cables CD and CE b the reaction at A 4111 A 48in boom is held by a ballandsocket joint at C and by two cables BF and DAE cable DAE passes around a frictionless pulley at A For the loading shown determine the tension in each cable and the reaction at C A B C F x y z D E 20 in 16 in 320 lb 30 in 20 in 48 in Fig P4111 bee29400ch04156217indd Page 202 112908 33404 PM users172 bee29400ch04156217indd Page 202 112908 33404 PM users172 Volumes204MHDQ076work0indd0 Volumes204MHDQ076work0indd0 203 Problems 4112 Solve Prob 4111 assuming that the 320lb load is applied at A 4113 A 20kg cover for a roof opening is hinged at corners A and B The roof forms an angle of 30 with the horizontal and the cover is maintained in a horizontal position by the brace CE Determine a the magnitude of the force exerted by the brace b the reac tions at the hinges Assume that the hinge at A does not exert any axial thrust 4114 The bent rod ABEF is supported by bearings at C and D and by wire AH Knowing that portion AB of the rod is 250 mm long determine a the tension in wire AH b the reactions at C and D Assume that the bearing at D does not exert any axial thrust 250 mm 50 mm 300 mm 400 N C D E F x z 50 mm 250 mm A B H y 30 Fig P4114 E C D z A y x B 09 m 09 m 06 m 30 Fig P4113 bee29400ch04156217indd Page 203 112908 33404 PM users172 bee29400ch04156217indd Page 203 112908 33404 PM users172 Volumes204MHDQ076work0indd0 Volumes204MHDQ076work0indd0 204 Equilibrium of Rigid Bodies 4116 Solve Prob 4115 assuming that cable DCE is replaced by a cable attached to point E and hook C 4117 The rectangular plate shown weighs 75 lb and is held in the posi tion shown by hinges at A and B and by cable EF Assuming that the hinge at B does not exert any axial thrust determine a the tension in the cable b the reactions at A and B x y z D H F E A B C 25 in 20 in 4 in 12 in 8 in 4 in 30 in Fig P4117 4115 A 100kg uniform rectangular plate is supported in the position shown by hinges A and B and by cable DCE that passes over a frictionless hook at C Assuming that the tension is the same in both parts of the cable determine a the tension in the cable b the reactions at A and B Assume that the hinge at B does not exert any axial thrust 690 mm 960 mm x y z E D A B C 675 mm 90 mm 450 mm 270 mm 90 mm Fig P4115 4118 Solve Prob 4117 assuming that cable EF is replaced by a cable attached at points E and H bee29400ch04156217indd Page 204 112908 33405 PM users172 bee29400ch04156217indd Page 204 112908 33405 PM users172 Volumes204MHDQ076work0indd0 Volumes204MHDQ076work0indd0 205 Problems 4119 Solve Prob 4114 assuming that the bearing at D is removed and that the bearing at C can exert couples about axes parallel to the y and z axes 4120 Solve Prob 4117 assuming that the hinge at B is removed and that the hinge at A can exert couples about axes parallel to the y and z axes 4121 The assembly shown is used to control the tension T in a tape that passes around a frictionless spool at E Collar C is welded to rods ABC and CDE It can rotate about shaft FG but its motion along the shaft is prevented by a washer S For the loading shown determine a the tension T in the tape b the reaction at C 4122 The assembly shown is welded to collar A that fits on the vertical pin shown The pin can exert couples about the x and z axes but does not prevent motion about or along the y axis For the load ing shown determine the tension in each cable and the reaction at A T A B C F D E S G x y z 6 lb 2 in 16 in 42 in 24 in T Fig P4121 480 N A C D E F x y z 60 mm 45 mm 90 mm 120 mm 80 mm Fig P4122 bee29400ch04156217indd Page 205 112908 33405 PM users172 bee29400ch04156217indd Page 205 112908 33405 PM users172 Volumes204MHDQ076work0indd0 Volumes204MHDQ076work0indd0 206 Equilibrium of Rigid Bodies 4123 The rigid Lshaped member ABC is supported by a ballandsocket joint at A and by three cables If a 450lb load is applied at F determine the tension in each cable x z y A B C D E F 450 lb 24 in 32 in 21 in 21 in 42 in 42 in Fig P4123 4124 Solve Prob 4123 assuming that the 450lb load is applied at C 4125 Frame ABCD is supported by a ballandsocket joint at A and by three cables For a 5 150 mm determine the tension in each cable and the reaction at A 4126 Frame ABCD is supported by a ballandsocket joint at A and by three cables Knowing that the 350N load is applied at D a 5 300 mm determine the tension in each cable and the reaction at A 4127 Three rods are welded together to form a corner that is supported by three eyebolts Neglecting friction determine the reactions at A B and C when P 5 240 lb a 5 12 in b 5 8 in and c 5 10 in 4128 Solve Prob 4127 assuming that the force P is removed and is replaced by a couple M 5 1600 lb inj acting at B A B C H D E F G x y z 140 mm 350 N 300 mm 140 mm 200 mm a 480 mm Fig P4125 and P4126 x y z b c A B C P a Fig P4127 bee29400ch04156217indd Page 206 112908 33405 PM users172 bee29400ch04156217indd Page 206 112908 33405 PM users172 Volumes204MHDQ076work0indd0 Volumes204MHDQ076work0indd0 207 Problems 4129 In order to clean the clogged drainpipe AE a plumber has discon nected both ends of the pipe and inserted a power snake through the opening at A The cutting head of the snake is connected by a heavy cable to an electric motor that rotates at a constant speed as the plumber forces the cable into the pipe The forces exerted by the plumber and the motor on the end of the cable can be rep resented by the wrench F 5 248 Nk M 5 290 N mk Deter mine the additional reactions at B C and D caused by the cleaning operation Assume that the reaction at each support consists of two force components perpendicular to the pipe 05 m x y z A B D E O 3 m 1 m 2 m 1 m C Fig P4129 4130 Solve Prob 4129 assuming that the plumber exerts a force F 5 248 Nk and that the motor is turned off M 5 0 4131 The assembly shown consists of an 80mm rod AF that is welded to a cross consisting of four 200mm arms The assembly is supported by a ballandsocket joint at F and by three short links each of which forms an angle of 45 with the vertical For the loading shown deter mine a the tension in each link b the reaction at F x y z E F A B P C D 45º 45º 45º 200 mm 200 mm 200 mm 200 mm 80 mm Fig P4131 bee29400ch04156217indd Page 207 112908 33406 PM users172 bee29400ch04156217indd Page 207 112908 33406 PM users172 Volumes204MHDQ076work0indd0 Volumes204MHDQ076work0indd0 208 Equilibrium of Rigid Bodies 4132 The uniform 10kg rod AB is supported by a ballandsocket joint at A and by the cord CG that is attached to the midpoint G of the rod Knowing that the rod leans against a frictionless vertical wall at B determine a the tension in the cord b the reactions at A and B x y z G O A B C 150 mm 150 mm 400 mm 600 mm Fig P4132 x y z A B C D E F 8 in 7 in 9 in 60 lb 11 in 16 in 10 in 14 in Fig P4133 x y z A B C D E 240 mm 400 mm 400 mm 200 mm 200 mm 480 mm 240 mm Fig P4135 4133 The bent rod ABDE is supported by ballandsocket joints at A and E and by the cable DF If a 60lb load is applied at C as shown determine the tension in the cable 4134 Solve Prob 4133 assuming that cable DF is replaced by a cable connecting B and F 4135 The 50kg plate ABCD is supported by hinges along edge AB and by wire CE Knowing that the plate is uniform determine the ten sion in the wire 4136 Solve Prob 4135 assuming that wire CE is replaced by a wire connecting E and D bee29400ch04156217indd Page 208 112908 33407 PM users172 bee29400ch04156217indd Page 208 112908 33407 PM users172 Volumes204MHDQ076work0indd0 Volumes204MHDQ076work0indd0 209 Problems 4137 Two rectangular plates are welded together to form the assembly shown The assembly is supported by ballandsocket joints at B and D and by a ball on a horizontal surface at C For the loading shown determine the reaction at C B A y z D C x 6 in 12 in 8 in 9 in 80 lb Fig P4137 z 2 ft 2 ft 2 ft 2 ft 2 ft x x y y A B C O H D E F 12 lb 12 lb Fig P4138 640 N x y z A B C D E F 240 mm 160 mm 480 mm 200 mm 490 mm Fig P4140 4138 Two 2 3 4ft plywood panels each of weight 12 lb are nailed together as shown The panels are supported by ballandsocket joints at A and F and by the wire BH Determine a the location of H in the xy plane if the tension in the wire is to be minimum b the corresponding minimum tension 4139 Solve Prob 4138 subject to the restriction that H must lie on the y axis 4140 The pipe ACDE is supported by ballandsocket joints at A and E and by the wire DF Determine the tension in the wire when a 640N load is applied at B as shown 4141 Solve Prob 4140 assuming that wire DF is replaced by a wire connecting C and F bee29400ch04156217indd Page 209 112908 33407 PM users172 bee29400ch04156217indd Page 209 112908 33407 PM users172 Volumes204MHDQ076work0indd0 Volumes204MHDQ076work0indd0