DepartamentodeEngenhariaMecânicaÁreaCientíficadeMecânicadosMeiosSólidos

RMM–2015/16

MecânicaAplicada

Cap.151-Cinemáticaemmovimentoplano

Problema15.57

Nosistemabiela-manivelademotormostradonafigura,𝑙 = 20 𝑐𝑚e𝑏 = 7,5 𝑐𝑚.Sabendo que a manivela𝐴𝐵 gira com uma velocidade angular constante de1000 𝑟𝑝𝑚nosentidohorário,determineavelocidadedopistão𝑃eavelocidadeangulardabielaquando:

(a) 𝜃 = 0;(b) 𝜃 = 90°.

Problema15.65

Na posição mostrada nas figuras, a barra𝐴𝐵tem uma velocidadeangular de4 𝑟𝑎𝑑/𝑠no sentido horário. Determine as velocidadesangularesdasbarras𝐵𝐷e𝐷𝐸.

Problema15.75

Um tambor de60 𝑚𝑚 de raio está rigidamente preso a um tambor de100 𝑚𝑚de raio como ilustra a figura. Um dos tambores rola sem deslizarsobre a superfície mostrada e uma corda é enrolada ao redor de outrotambor. Sabendo que a extremidade𝐸da corda é puxada para a esquerdacomumavelocidadede120 𝑚𝑚/𝑠,determine:

(a) avelocidadeangulardostambores;(b) avelocidadedocentrodostambores;(c) ocomprimentodacordaenroladaoudesenroladaporsegundo.

Problema15.83

Uma porta basculante é guiada por roletes em𝐴e𝐵 que rolamnumapistahorizontaleumavertical.Sabendoque,quando𝜃 = 40°,avelocidadedorolete𝐵éde0,6 𝑚/𝑠paracima,determine:

(a) avelocidadeangulardaporta;(b) avelocidadedaextremidade𝐷daporta.

1Osproblemasapresentadosreferem-seaolivro“MecânicaVetorialparaEngenheiros–Dinâmica,FerdinandP.Beer,E.RussellJohnstonJr.,WilliamE.Clausen,7ªEdMcGraw-Hill”

943Problems 15.57 In the engine system shown, l 5 160 mm and b 5 60 mm. Knowing that the crank AB rotates with a constant angular velocity of 1000 rpm clockwise, determine the velocity of the piston P and the angular velocity of the connecting rod when (a) u 5 0, (b) u 5 90°.

15.58 In the engine system shown in Fig. P15.57 and P15.58, l 5 160 mm and b 5 60 mm. Knowing that crank AB rotates with a constant angular velocity of 1000 rpm clockwise, determine the velocity of the piston F and the angular velocity of the connecting rod when u 5 60°.

15.59 A straight rack rests on a gear of radius r and is attached to a block B as shown. Denoting by vD the clockwise angular velocity of gear D and by u the angle formed by the rack and the horizontal, derive expressions for the velocity of block B and the angular velocity of the rack in terms of r, u, and vD.

P

D

AB

l

q

b

Fig. P15.57 and P15.58A

DB

q

r

Fig. P15.59, P15.60, and P15.61

2 in.

OA

Bq

8 in.in.12

Fig. P15.62

15.60 A straight rack rests on a gear of radius r 5 75 mm and is attached to a block B as shown. Knowing that at the instant shown the angular velocity of gear D is 15 rpm counterclockwise and u 5 20°, determine (a) the velocity of block B, (b) the angular velocity of the rack.

15.61 A straight rack rests on a gear of radius r 5 60 mm and is attached to a block B as shown. Knowing that at the instant shown the velocity of block B is 200 mm/s to the right and u 5 25°, determine (a) the angular velocity of gear D, (b) the angular velocity of the rack.

15.62 In the eccentric shown, a disk of 2-in.-radius revolves about shaft O that is located 0.5 in. from the center A of the disk. The distance between the center A of the disk and the pin at B is 8 in. Knowing that the angular velocity of the disk is 900 rpm clockwise, deter-mine the velocity of the block when u 5 30°.

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944 Kinematics of Rigid Bodies 15.63 through 15.65 In the position shown, bar AB has an angular velocity of 4 rad/s clockwise. Determine the angular velocity of bars BD and DE.

15.66 In the position shown, bar DE has a constant angular velocity of 10 rad/s clockwise. Knowing that h 5 500 mm, determine (a) the angular velocity of bar FBD, (b) the velocity of point F.

8 in.

7 in.4 in.

3 in.

A

B

D

E

Fig. P15.63

A B

D

E

250 mm 150 mm

100 mm

60 mm

Fig. P15.64

400 mm400 mm

A

BD

E

300 mm

500 mm

Fig. P15.65

A

B

D

E

100 mm

F

300 mm100 mm

200 mm

120 mm

h

Fig. P15.66 and P15.67

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953Problems 15.77 Solve Sample Prob. 15.2, assuming that the lower rack is not stationary but moves to the left with a velocity of 0.6 m/s.

15.78 A double pulley is attached to a slider block by a pin at A. The 30-mm-radius inner pulley is rigidly attached to the 60-mm-radius outer pulley. Knowing that each of the two cords is pulled at a constant speed as shown, determine (a) the instantaneous center of rotation of the double pulley, (b) the velocity of the slider block, (c) the number of millimeters of cord wrapped or unwrapped on each pulley per second.

15.79 Solve Prob. 15.78, assuming that cord E is pulled upward at a speed of 160 mm/s and cord F is pulled downward at a speed of 200 mm/s.

15.80 and 15.81 A 3-in.-radius drum is rigidly attached to a 5-in.- radius drum as shown. One of the drums rolls without sliding on the surface shown, and a cord is wound around the other drum. Knowing that end E of the cord is pulled to the left with a velocity of 6 in./s, determine (a) the angular velocity of the drums, (b) the velocity of the center of the drums, (c) the length of cord wound or unwound per second. 160 mm/s

AB D

F

E

200 mm/s

Fig. P15.78

A

BE

D

3 in.5 in.

Fig. P15.80

A

B

E D

3 in.5 in.

Fig. P15.81

15.82 Knowing that at the instant shown the angular velocity of rod AB is 15 rad/s clockwise, determine (a) the angular velocity of rod BD, (b) the velocity of the midpoint of rod BD.

A

B

D E

0.2 m

0.2 m

0.25 m

0.6 m

Fig. P15.82 and P15.83

15.83 Knowing that at the instant shown the velocity of point D is 2.4 m/s upward, determine (a) the angular velocity of rod AB, (b) the velocity of the midpoint of rod BD.

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954 Kinematics of Rigid Bodies 15.84 Rod ABD is guided by wheels at A and B that roll in horizontal and vertical tracks. Knowing that at the instant b 5 60° and the velocity of wheel B is 40 in./s downward, determine (a) the angular velocity of the rod, (b) the velocity of point D.

15.85 An overhead door is guided by wheels at A and B that roll in hori-zontal and vertical tracks. Knowing that when u 5 40° the velocity of wheel B is 1.5 ft/s upward, determine (a) the angular velocity of the door, (b) the velocity of end D of the door.

A

B

D

15 in.

15 in.

b

Fig. P15.84 A

B

D

q

5 ft

5 ft

Fig. P15.85

15.86 Knowing that at the instant shown the angular velocity of rod BE is 4 rad/s counterclockwise, determine (a) the angular velocity of rod AD, (b) the velocity of collar D, (c) the velocity of point A.

15.87 Knowing that at the instant shown the velocity of collar D is 1.6 m/s upward, determine (a) the angular velocity of rod AD, (b) the velocity of point B, (c) the velocity of point A.

15.88 Rod AB can slide freely along the floor and the inclined plane. Denoting by vA the velocity of point A, derive an expression for (a) the angular velocity of the rod, (b) the velocity of end B.

A

D

B

E

192 mm

360 mm

240 mm30°

Fig. P15.86 and P15.87

vA qbA

Bl

Fig. P15.88 and P15.89

15.89 Rod AB can slide freely along the floor and the inclined plane. Knowing that u 5 20°, b 5 50°, l 5 0.6 m, and vA 5 3 m/s, determine (a) the angular velocity of the rod, (b) the velocity of end B.

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60 𝑚𝑚 100

2 𝑚

2 𝑚