20-1. At a given instant, the satellite dish has an

At a given instant, the satellite dish has an angular motion \(\omega_1=6\mathrm{\ rad}/\mathrm{s}\text{ and }\dot{\omega}_1=3\ \mathrm{rad}/\mathrm{s}^2\) about the z axis. At this same instant \(\theta=25^{\circ}\), the angular motion about the x axis is \(\omega_2=2\ \mathrm{rad}/\mathrm{s},\text{ and }\dot{\omega}_2=1.5\mathrm{\ rad}/\mathrm{s}^2\). Determine the velocity and acceleration of the signal horn A at this instant.

20-2. Gears A and B are fixed, while gears C and D are free to rotate about the shaft S. If the shaft turns about the z axis at a constant rate of = 4 rad/s, determine the angular velocity and angular acceleration of gear C

20-3. The ladder of the fire truck rotates around the z, axis with an angular velocity co\ = 0.15 rad/s. which is increasing at 0.8 rad/s2. At the same instant it is rotating upward at a constant rate u)2 = 0.6 rad/s. Determine the velocity and acceleration of point A located at the top of the ladder at this instant.

The ladder of the fire truck rotates around the z axis with an angular velocity of \(\omega_1=0.15 \mathrm{rad} / \mathrm{s}\), which is increasing at \(0.2 \mathrm{rad} / \mathrm{s}^2\). At the same instant it is rotating upwards at \(\omega_2=0.6 \mathrm{rad} / \mathrm{s}\) while increasing at \(0.4 \mathrm{rad} / \mathrm{s}^2\). Determine the velocity and acceleration of point A located at the top of the ladder at this instant.

20-5. Gear B is connected to the rotating shaft, while the plate gear A is fixed. If the shaft is turning at a constant rate of u). = 10 rad/s about the z axis, determine the magnitudes of the angular velocity and the angular acceleration of gear B. Also, determine the magnitudes of the velocity and acceleration of point P.

20-6. Gear A is fixed while gear B is free to rotate on the shaft S. If the shaft is turning about the z axis at co. = 5 rad/s, while increasing at 2 rad/s2. determine the velocity and acceleration of point P at the instant showm. The face of gear B lies in a vertical plane.

At a given instant, the antenna has an angular motion \(\omega_1=3\ \mathrm{rad}/\mathrm{s}\text{ and }\dot{\omega}_1=2\mathrm{\ rad}/\mathrm{s}^2\) about the z axis. At this same instant \(\theta=30^{\circ}\), the angular motion about the x axis is \(\omega_{2}=1.5 \mathrm{\ rad} / \mathrm{s}, \text { and } \omega_{2}=4 \mathrm{\ rad} / \mathrm{s}^{2}\). Determine the velocity and acceleration of the signal horn A at this instant. The distance from O to A is d = 3 ft.

*20-8. The cone rolls without slipping such that at the instant shown w: = 4 rad/s and 6). = 3 rad/s2. Determine the velocity and acceleration of point A at this instant.

20-9. The cone rolls without slipping such that at the instant showrn cd. = 4 rad/s and co. = 3 rad/s2. Determine the velocity and acceleration of point B at this instant.

20-10. At the instant w'hen 0 = 90. the satellite's body is rotating with an angular velocity of a>, = 15 rad/s and angular acceleration of &> | = 3 rad/s2. Simultaneously, the solar panels rotate with an angular velocity of w2 = 6 rad/s and angular acceleration of io2 = 1.5 rad/s2. Determine the velocity and acceleration of point B on the solar panel at this instant.

20-11. At the instant wrhen 0 = 90, the satellite's body travels in the a direction with a velocity of v(; = { 500i} m/s and acceleration of a0 = {50i} m/s2. Simultaneously, the body also rotates with an angular velocity of (o{ = 15 rad/s and angular acceleration of w| = 3 rad/s2. At the same time, the solar panels rotate with an angular velocity of co2 = 6 rad/s and angular acceleration of io2 = 1.5 rad/s2 Determine the velocity and acceleration of point B on the solar panel.

*20-12. The disk is free to rotate on the shaft S. If the shaft is turning about the z axis at ojz = 2 rad/s. while increasing at 8 rad/s2, determine the velocity and acceleration of point A at the instant shown.

20-13. The disk spins about the arm with an angular velocity of ios = 8 rad/s. which is increasing at a constant rate of ojs = 3 rad/s2 at the instant shown. If the shaft rotates with a constant angular velocity of cup = 6 rad/s, determine the velocity and acceleration of point A located on the rim of the disk at this instant.

20-14. The wheel is spinning about shaft AB with an angular velocity of <os = 10 rad/s, which is increasing at a constant rate of io2 = 6 rad/s2, while the frame precesses about the z axis with an angular velocity of wp 12 rad/s, which is increasing at a constant rate of u)p = 3 rad/s2. Determine the velocity and acceleration of point C located on the rim of the wheel at this instant.

20-15. At the instant shown, the tower crane rotates about the z axis with an angular velocity = 0.25 rad/s. which is increasing at 0.6 rad/s2. The boom OA rotates downward with an angular velocity o>2 = 0.4 rad/s, which is increasing at 0.8 rad/s2. Determine the velocity and acceleration of point A located at the end of the boom at this instant.

*20-16. If the top gear B rotates at a constant rate of to, determine the angular velocity of gear A. which is free to rotate about the shaft and rolls on the bottom fixed gear C.

20-17. When 0 = 0, the radar disk rotates about the y axis with an angular velocity of 0=2 rad/s.increasing at a constant rate of () = 1.5 rad/s2. Simultaneously, the disk also prccesses about the z axis with an angular velocity of (op = 5 rad/s, increasing at a constant rate of iop = 3 rad/s2. Determine the velocity and acceleration of the receiver A at this instant.

Gear A is fixed to the crankshaft S, while gear C is fixed. Gear B and the propeller are free to rotate. The crankshaft is turning at 80 rad/s about its axis. Determine the magnitudes of the angular velocity of the propeller and the angular acceleration of gear B.

20-19. Shaft BD is connected to a ball-and-socket joint at B. and a beveled gear A is attached to its other end.The gear is in mesh with a fixed gear C. If the shaft and gear A are spinning with a constant angular velocity (o{ = 8 rad/s, determine the angular velocity and angular acceleration of gear A.

*20-20. Gear B is driven by a motor mounted on turntable C. Tf gear A is held fixed, and the motor shaft rotates with a constant angular velocity of toY = 30 rad/s. determine the angular velocity and angular acceleration of gear B.

20-21. Gear B is driven by a motor mounted on turntable C. If gear A and the motor shaft rotate with constant angular speeds of coA = 110k) rad/s and o)Y= (30j) rad/s. respectively, determine the angular velocity and angular acceleration of gear B.

20-22. The crane boom OA rotates about the z axis with a constant angular velocity of a>, = 0.15rad/s. while it is rotating downward with a constant angular velocity of co2 = 0.2 rad/s. Determine the velocity and acceleration of point A located at the end of the boom at the instant shown.

20-23. The differential of an automobile allows the two rear wheels to rotate at different speeds when the automobile travels along a curve. For operation, the rear axles are attached to the wheels at one end and have beveled gears A and B on their other ends. The differential case D is placed over the left axle but can rotate about C independent of the axle. The case supports a pinion gear E on a shaft, which meshes with gears A and B. Finally, a ring gear G is fixed to the differential case so that the case rotates with the ring gear when the latter is driven by the drive pinion //.This gear, like the differential case, is free to rotate about the left wheel axle. If the drive pinion is turning at wH = 100 rad/s and the pinion gear E is spinning about its shaft at oje = 30 rad/s. determine the angular velocity. <aA and a>/yf of each axle.

*20-24. The truncated cone rotates about the z axis at a constant rate io: = 0.4 rad/s without slipping on the horizontal plane. Determine the velocity and acceleration of point A on the cone.

20-25. Disk A rotates at a constant angular velocity of 10 rad/s. If rod BC is joined to the disk and a collar by ball-and-socket joints, determine the velocity of collar B at the instant shown. Also, what is the rod's angular velocity <oBC if it is directed perpendicular to the axis of the rod?

20-26. If the rod is attached with ball-and-socket joints to smooth collars A and B at its end points, determine the speed of B at the instant shown if A is moving downward at a constant speed of vA = 8 ft/s. Also, determine the angular velocity of the rod if it is directed perpendicular to the axis of the rod.

20-27. If the collar at A is moving downward with an acceleration a4 = {-5k} ft/s2, at the instant its speed is vA = 8 ft/s, determine the acceleration of the collar at B at this instant.

*20-28. If wheel C rotates with a constant angular velocity of o)C = 10 rad/s. determine the velocity of the collar at B when rod AB is in the position shown.

20-29. At the instant rod AB is in the position shown wheel C rotates with an angular velocity of ojc = 10 rad/s and has an angular acceleration of ac = 1.5 rad/s2. Determine the acceleration of collar B at this instant.

20-30. If wheel D rotates with an angular velocity of (i>D = 6 rad/s, determine the angular velocity of the follower link BC at the instant shown.The link rotates about the z axis at z = 2 ft.

20-31. Rod AB is attached to the rotating arm using ball-and-socket joints. If AC is rotating with a constant angular velocity of 8 rad/'s about the pin at C. determine the angular velocity of link BD at the instant shown.

*20-32. Rod AB is attached to the rotating arm using ball-and-socket joints. If AC is rotating about point C with an angular velocity of 8 rad/'s and has an angular acceleration of aAc = {6k} rad/'s2 at the instant shown, determine the angular velocity and angular acceleration of link BD at this instant.

20-33. Rod AB is attached to collars at its ends by ball-and-socket joints. If collar A moves upward with a velocity of \A = {8k} ft/s, determine the angular velocity of the rod and the speed of collar B at the instant shown. Assume that the rods angular velocity is directed perpendicular to the rod.

20-34. Rod AB is attached to collars at its ends by ball-and-socket joints. If collar A moves upward with an acceleration of a4 = {4k }ft/s2. determine the angular acceleration of rod A B and the magnitude of acceleration of collar B. Assume that the rods angular acceleration is directed perpendicular to the rod, and use the result of Prob. 20-33 for oja[{.

20-35. Solve Prob. 20-25 if the connection at B consists of a pin as shown in the figure below, rather than a ball-and-socket joint. ////: The constraint allows rotation of the rod both about bar DE (j direction) and about the axis of the pin (n direction). Since there is no rotational component in the u direction, i.e.. perpendicular to n and j where u = j X n. an additional equation for solution can be obtained from oj u = 0. The vector n is in the same direction as vR!c X rD/c.

*20-36. The rod assembly is supported at B by a ball-and-socket joint and at A by a clevis. If the collar at B moves in the x-z plane with a speed vR = 5 ft/s, determine the velocity of points A and C on the rod assembly at the instant shown. Hint: See Prob. 20-35.

20-37. Solve Example 20.5 such that the .v. y. z axes move with curvilinear translation, ft = 0 in which case the collar appears to have both an angular velocity ft<v: = io\ 4- cj2 and radial motion.

20-38. Solve Example 20.5 by fixing x, y, z axes to rod BD so that ft = + to2. In this case the collar appears only to move radially outward along BD: hence ft 1V. = 0.

At the instant \(\theta=60^{\circ}\), the telescopic boom AB of the construction lift is rotating with a constant angular velocity about the z axis of \(\omega_1=0.5 \mathrm{rad} / \mathrm{s}\) and about the pin at A with a constant angular speed of \(\omega_2=0.25 \mathrm{rad} / \mathrm{s}\). Simultaneously, the boom is extending with a velocity of \(1.5 \mathrm{ft} / \mathrm{s}\), and it has an acceleration of \(0.5 \mathrm{ft} / \mathrm{s}^2\), both measured relative to the construction lift. Determine the velocity and acceleration of point B located at the end of the boom at this instant.

*20-40. At the instants = 60, the construction lift is rotating about the z axis with an angular velocity of cot = 0.5 rad/s and an angular acceleration of tb\ = 0.25 rad/s2 while the telescopic boom AB rotates about the pin at A with an angular velocity of to2 = 0.25 rad/s and angular acceleration of d)2 = 0.1 rad/s2. Simultaneously, the boom is extending with a velocity of 1.5 ft/s. and it has an acceleration of 0.5 ft/s2, both measured relative to the frame. Determine the velocity and acceleration of point B located at the end of the boom at this instant.

20-41. At a given instant, rod BD is rotating about the y axis with an angular velocity toBD = 2 rad/s and an angular acceleration doBD = 5 rad/s2. Also, when 6 = 60 link AC is rotating downward such that 0 = 2 rad/s and () ~ 8 rad/s2. Determine the velocity and acceleration of point A on the link at this instant.

20-42. At the instant 0 = 30. the frame of the crane and the boom AB rotate with a constant angular velocity of to| = 1.5 rad/s and to2 = 0.5 rad/s, respectively. Determine the velocity and acceleration of point B at this instant.

20-43. At the instant 0 30. the frame of the crane is rotating with an angular velocity of = 1.5 rad/s and angular acceleration of = 0.5 rad/s2. while the boom AB rotates with an angular velocity of co2 = 0.5 rad/s and angular acceleration of io2 = 0.25 rad/s2. Determine the velocity and acceleration of point B at this instant.

*20-44. At the instant shown, the boom is rotating about the z axis with an angular velocity o>, = 2 rad/s and angular acceleration w, = 0.8 rad/s2. At this same instant the swivel is rotating at a>2 = 3 rad/s when cb2 = 2 rad/s2, both measured relative to the boom. Determine the velocity and acceleration of point P on the pipe at this instant.

20-45. During the instant shown the frame of the X-ray camera is rotating about the vertical axis at a>. = 5 rad/s and (i). = 2 rad/s2. Relative to the frame the arm is rotating at a)^, = 2 rad/s and = 1 rad/s2. Determine the velocity and acceleration of the center of the camera C at this instant.

20-46. The boom A B of the crane is rotating about the z axis with an angular velocity cjz = 0.75rad/s, which is increasing at ib: = 2 rad/s2. At the same instant, 0 = 60 and the boom is rotating upward at a constant rate f) = 0.5 rad/s2. Determine the velocity and acceleration of the tip B of the boom at this instant.

*20-47. The boom AB of the crane is rotating about the z axis with an angular velocity of co: 0.75 rad/s, which is increasing at w. = 2 rad/s2. At the same instant, 0 = 60 and the boom is rotating upward at 0 = 0.5 rad/s2, which is increasing at 0 = 0.75 rad/s2. Determine the velocity and acceleration of the tip B of the boom at this instant.

20-48. At the instant shown, the motor rotates about the z axis with an angular velocity of to, = 3 rad/s and angular acceleration of to, = 1.5 rad/s2. Simultaneously, shaft OA rotates with an angular velocity of to2 = 6 rad/s. and angular acceleration of to2 = 3 rad/s2, and collar C slides along rod A B with a velocity and acceleration of 6 m/s and 3 m/s2. Determine the velocity and acceleration of collar Cat this instant.

20-49. The motor rotates about the z axis with a constant angular velocity of w, =3 rad/s. Simultaneously, shaft OA rotates with a constant angular velocity of oj2 = 6 rad/s. Also, collar C slides along rod AB with a velocity and acceleration of 6 m/s and 3 m/s2. Determine the velocity and acceleration of collar C at the instant shown.

20-50. At the instant shown, the arm OA of the conveyor belt is rotating about the z axis with a constant angular velocity a>, = 6 rad/s, while at the same instant the arm is rotating upward at a constant rate co2 = 4 rad/s. If the conveyor is running at a constant rate r = 5 ft/s, determine the velocity and acceleration of the package P at the instant shown. Neglect the size of the package.

20-51. At the instant shown, the arm OA of the conveyor belt is rotating about the z axis with a constant angular velocity oj\ = 6 rad/s, while at the same instant the arm is rotating upward at a constant rate oj2 = 4 rad/s. If the conveyor is running at a rate r = 5 ft/s, which is increasing at V = 8 ft/s2, determine the velocity and acceleration of the package P at the instant shown. Neglect the size of the package.

*20-52. 'Hie boom AB of the locomotive crane is rotating about the z axis with an angular velocity o)[ = 0.5 rad/s, which is increasing at d)] = 3 rad/s2. At this same instant. 0 = 30 and the boom is rotating upward at a constant rate of 0 =3 rad/s. Determine the velocity and acceleration of the tip B of the boom at this instant.

The locomotive crane is traveling to the right at 2 m/s and has an acceleration of \(1.5 \mathrm{\ m} / \mathrm{s}^{2}\), while the boom is rotating about the z axis with an angular velocity \(\omega_{1}=0.5 \mathrm{\ rad} / \mathrm{s}\), which is increasing at \(\dot{\omega}_{1}=3 \mathrm{\ rad} / \mathrm{s}^{2}\). At this same instant, \(\theta=30^{\circ}\) and the boom is rotating upward at a constant rate \(\dot {\theta}=3 \mathrm{\ rad} / \mathrm{s}\). Determine the velocity and acceleration of the tip B of the boom at this instant.

20-54. The robot shown has four degrees of rotational freedom, namely, arm OA rotates about the x and z axes, arm AB rotates about the x axis, and CB rotates about the y axis. At the instant shown. io2 = 1.5 rad/s, co2 = 1 rad/s2.