Engineering Mechanics · Dynamics · Open Problem Companion

Chapter Sixteen
Planar Kinematics of a Rigid Body

F Fundamental Problems 24 problems
Fundamental
  1. F16‑1When the gear rotates 20 revolutions, it achieves an angular velocity of \(\omega = 30\,\mathrm{rad/s}\) starting from rest. Determine its constant angular acceleration and the time required. figure — add your own
  2. F16‑2The flywheel rotates with an angular velocity of \(\omega = (0.005\,\theta^{2})\,\mathrm{rad/s}\), where \(\theta\) is in radians. Determine the angular acceleration when it has rotated 20 revolutions. figure — add your own
  3. F16‑3The flywheel rotates with an angular velocity of \(\omega = (4\,\theta^{1/2})\,\mathrm{rad/s}\), where \(\theta\) is in radians. Determine the time it takes to achieve an angular velocity of \(\omega = 150\,\mathrm{rad/s}\). When \(t=0\), \(\theta = 1\) rad. figure — add your own
  4. F16‑4The bucket is hoisted by the rope that wraps around a drum wheel. If the angular displacement of the wheel is \(\theta = (0.5t^{3}+15t)\,\mathrm{rad}\), where \(t\) is in seconds, determine the velocity and acceleration of the bucket when \(t=3\) s. figure — add your own
  5. F16‑5A wheel has an angular acceleration of \(\alpha = (0.5\theta)\,\mathrm{rad/s^2}\), where \(\theta\) is in radians. Determine the magnitude of the velocity and acceleration of a point \(P\) located on its rim after the wheel has rotated 2 revolutions. The wheel has a radius of \(0.2\,\mathrm{m}\) and starts at \(\omega_0 = 2\,\mathrm{rad/s}\). figure — add your own
  6. F16‑6For a short period of time, the motor turns gear \(A\) with a constant angular acceleration of \(\alpha_A = 4.5\,\mathrm{rad/s^2}\), starting from rest. Determine the velocity of the cylinder and the distance it travels in three seconds. The cord is wrapped around pulley \(D\) which is rigidly attached to gear \(B\). figure — add your own
  7. F16‑7If roller \(A\) moves to the right with a constant velocity of \(v_A = 3\,\mathrm{m/s}\), determine the angular velocity of the link and the velocity of roller \(B\) at the instant \(\theta = 30^\circ\). figure — add your own
  8. F16‑8The wheel rolls without slipping with an angular velocity of \(\omega = 10\,\mathrm{rad/s}\). Determine the magnitude of the velocity of point \(B\) at the instant shown. figure — add your own
  9. F16‑9Determine the angular velocity of the spool. The cable wraps around the inner core, and the spool does not slip on the platform \(P\). figure — add your own
  10. F16‑10If crank \(OA\) rotates with an angular velocity of \(\omega = 12\,\mathrm{rad/s}\), determine the velocity of piston \(B\) and the angular velocity of rod \(AB\) at the instant shown. figure — add your own
  11. F16‑11If rod \(AB\) slides along the horizontal slot with a velocity of \(18\,\mathrm{m/s}\), determine the angular velocity of link \(BC\) at the instant shown. figure — add your own
  12. F16‑12End \(A\) of the link has a velocity of \(v_A = 3\,\mathrm{m/s}\). Determine the velocity of the peg at \(B\) at this instant. The peg is constrained to move along the slot. figure — add your own
  13. F16‑13Determine the angular velocity of the rod and the velocity of point \(C\) at the instant shown. figure — add your own
  14. F16‑14Determine the angular velocity of link \(BC\) and velocity of the piston \(C\) at the instant shown. figure — add your own
  15. F16‑15If the center \(O\) of the wheel is moving with a speed of \(v_O = 6\,\mathrm{m/s}\), determine the velocity of point \(A\) on the wheel. The gear rack \(B\) is fixed. figure — add your own
  16. F16‑16(Not listed; skip)
  17. F16‑17Determine the angular velocity of link \(BC\) and the velocity of the piston \(C\) at the instant shown. figure — add your own
  18. F16‑18Determine the angular velocity of links \(BC\) and \(CD\) at the instant shown. figure — add your own
  19. F16‑19At the instant shown, end \(A\) of the rod has the velocity and acceleration shown. Determine the angular acceleration of the rod and acceleration of end \(B\) of the rod. figure — add your own
  20. F16‑20The gear rolls on the fixed rack with an angular velocity of \(\omega = 12\,\mathrm{rad/s}\) and angular acceleration of \(\alpha = 6\,\mathrm{rad/s^2}\). Determine the acceleration of point \(A\). figure — add your own
  21. F16‑21The gear rolls on the fixed rack \(B\). At the instant shown, the center \(O\) of the gear moves with a velocity of \(v_O = 6\,\mathrm{m/s}\) and acceleration of \(a_O = 3\,\mathrm{m/s^2}\). Determine the angular acceleration of the gear and acceleration of point \(A\) at this instant. figure — add your own
  22. F16‑22At the instant shown, cable \(AB\) has a velocity of \(3\,\mathrm{m/s}\) and acceleration of \(1.5\,\mathrm{m/s^2}\), while the gear rack has a velocity of \(1.5\,\mathrm{m/s}\) and acceleration of \(0.75\,\mathrm{m/s^2}\). Determine the angular acceleration of the gear at this instant. figure — add your own
  23. F16‑23At the instant shown, the wheel rotates with an angular velocity of \(\omega = 12\,\mathrm{rad/s}\) and an angular acceleration of \(\alpha = 6\,\mathrm{rad/s^2}\). Determine the angular acceleration of link \(BC\) at the instant shown. figure — add your own
  24. F16‑24At the instant shown, wheel \(A\) rotates with an angular velocity of \(\omega = 6\,\mathrm{rad/s}\) and an angular acceleration of \(\alpha = 3\,\mathrm{rad/s^2}\). Determine the angular acceleration of link \(BC\) and the acceleration of piston \(C\). figure — add your own
16 Problems (16‑1 – 16‑152) 152 problems
Regular
  1. 16‑1The angular velocity of the disk is defined by \(\omega = (5t^{2}+2)\,\mathrm{rad/s}\), where \(t\) is in seconds. Determine the magnitudes of the velocity and acceleration of point \(A\) on the disk when \(t = 0.5\,\mathrm{s}\). figure — add your own
  2. 16‑2The angular acceleration of the disk is defined by \(\alpha = 3t^{2}+12\,\mathrm{rad/s^2}\), where \(t\) is in seconds. If the disk is originally rotating at \(\omega_0 = 12\,\mathrm{rad/s}\), determine the magnitude of the velocity and the \(n\) and \(t\) components of acceleration of point \(A\) on the disk when \(t = 2\) s. figure — add your own
  3. 16‑3The disk is originally rotating at \(\omega_0 = 12\,\mathrm{rad/s}\). If it is subjected to a constant angular acceleration of \(\alpha = 20\,\mathrm{rad/s^2}\), determine the magnitudes of the velocity and the \(n\) and \(t\) components of acceleration of point \(A\) at the instant \(t = 2\) s. figure — add your own
  4. 16‑4The disk is originally rotating at \(\omega_0 = 12\,\mathrm{rad/s}\). If it is subjected to a constant angular acceleration of \(\alpha = 20\,\mathrm{rad/s^2}\), determine the magnitudes of the velocity and the \(n\) and \(t\) components of acceleration of point \(B\) when the disk undergoes 2 revolutions. figure — add your own
  5. 16‑5The disk is driven by a motor such that the angular position of the disk is defined by \(\theta = (20t + 4t^2)\,\mathrm{rad}\), where \(t\) is in seconds. Determine the number of revolutions, the angular velocity, and angular acceleration of the disk when \(t = 90\) s. figure — add your own
  6. 16‑6A wheel has an initial clockwise angular velocity of \(10\,\mathrm{rad/s}\) and a constant angular acceleration of \(3\,\mathrm{rad/s^2}\). Determine the number of revolutions it must undergo to acquire a clockwise angular velocity of \(15\,\mathrm{rad/s}\). What time is required? figure — add your own
  7. 16‑7If gear \(A\) rotates with a constant angular acceleration of \(\alpha_A = 90\,\mathrm{rad/s^2}\) starting from rest, determine the time required for gear \(D\) to attain an angular velocity of \(600\,\mathrm{rpm}\). Also, find the number of revolutions of gear \(D\) to attain this angular velocity. Gears \(A, B, C,\) and \(D\) have radii of \(15\,\mathrm{mm}\), \(50\,\mathrm{mm}\), \(25\,\mathrm{mm}\), and \(75\,\mathrm{mm}\), respectively. figure — add your own
  8. 16‑8If gear \(A\) rotates with an angular velocity of \(\omega_A = (\theta_A + 1)\,\mathrm{rad/s}\), where \(\theta_A\) is the angular displacement of gear \(A\) measured in radians, determine the angular acceleration of gear \(D\) when \(\theta_A = 3\) rad, starting from rest. Gears \(A, B, C,\) and \(D\) have radii of \(15\,\mathrm{mm}\), \(50\,\mathrm{mm}\), \(25\,\mathrm{mm}\), and \(75\,\mathrm{mm}\), respectively. figure — add your own
  9. 16‑9At the instant \(\omega_A = 5\,\mathrm{rad/s}\), pulley \(A\) is given an angular acceleration \(\alpha = (0.8\theta)\,\mathrm{rad/s^2}\), where \(\theta\) is in radians. Determine the magnitude of acceleration of point \(B\) on pulley \(C\) when \(A\) rotates 3 revolutions. Pulley \(C\) has an inner hub which is fixed to its outer one and turns with it. figure — add your own
  10. 16‑10At the instant \(\omega_A = 5\,\mathrm{rad/s}\), pulley \(A\) is given a constant angular acceleration \(\alpha_A = 6\,\mathrm{rad/s^2}\). Determine the magnitude of acceleration of point \(B\) on pulley \(C\) when \(A\) rotates 2 revolutions. Pulley \(C\) has an inner hub which is fixed to its outer one and turns with it. figure — add your own
  11. 16‑11The cord, which is wrapped around the disk, is given an acceleration of \(a = (10t)\,\mathrm{m/s^2}\) where \(t\) is in seconds. Starting from rest, determine the angular displacement, angular velocity, and angular acceleration of the disk when \(t = 3\) s. figure — add your own
  12. 16‑12The power of a bus engine is transmitted using the belt-and-pulley arrangement shown. If the engine turns pulley \(A\) at \(\omega_A = (20t + 40)\,\mathrm{rad/s}\), where \(t\) is in seconds, determine the angular velocities of the generator pulley \(B\) and the air-conditioning pulley \(C\) when \(t = 3\) s. figure — add your own
  13. 16‑13The power of a bus engine is transmitted using the belt-and-pulley arrangement shown. If the engine turns pulley \(A\) at \(\omega_A = 60\,\mathrm{rad/s}\), determine the angular velocities of the generator pulley \(B\) and the air-conditioning pulley \(C\). The hub at \(D\) is rigidly connected to \(B\) and turns with it. figure — add your own
  14. 16‑14The disk starts from rest and is given an angular acceleration \(\alpha = (2t^2)\,\mathrm{rad/s^2}\) where \(t\) is in seconds. Determine the angular velocity of the disk and its angular displacement when \(t = 4\) s. figure — add your own
  15. 16‑15The disk starts from rest and is given an angular acceleration \(\alpha = (5t^{1/2})\,\mathrm{rad/s^2}\), where \(t\) is in seconds. Determine the magnitudes of the normal and tangential components of acceleration of a point \(P\) on the rim of the disk when \(t = 2\) s. figure — add your own
  16. 16‑16The disk starts at \(\omega_0 = 1\,\mathrm{rad/s}\) when \(\theta = 0\) and is given an angular acceleration \(\alpha = (0.3\theta)\,\mathrm{rad/s^2}\), where \(\theta\) is in radians. Determine the magnitudes of the normal and tangential components of acceleration of a point \(P\) on the rim of the disk when \(\theta = 1\) rev. figure — add your own
  17. 16‑17A motor gives gear \(A\) an angular acceleration of \(\alpha_A = (2 + 0.006\theta^2)\,\mathrm{rad/s^2}\), where \(\theta\) is in radians. If this gear is initially turning at \(\omega_A = 15\,\mathrm{rad/s}\), determine the angular velocity of gear \(B\) after \(A\) undergoes an angular displacement of 10 rev. figure — add your own
  18. 16‑18A motor gives gear \(A\) an angular acceleration of \(\alpha_A = (2t^3)\,\mathrm{rad/s^2}\), where \(t\) is in seconds. If this gear is initially turning at \(\omega_A = 15\,\mathrm{rad/s}\), determine the angular velocity of gear \(B\) when \(t = 3\) s. figure — add your own
  19. 16‑19Morse Industrial manufactures the speed reducer shown. If a motor drives the gear shaft \(S\) with an angular acceleration \(\alpha = (4\omega^{-3})\,\mathrm{rad/s^2}\), where \(\omega\) is in rad/s, determine the angular velocity of shaft \(E\) at time \(t = 2\) s after starting from an angular velocity of \(1\,\mathrm{rad/s}\) when \(t=0\). The radius of each gear is listed in the figure. Note that gears \(B\) and \(C\) are fixed connected to the same shaft. figure — add your own
  20. 16‑20A motor gives gear \(A\) an angular acceleration of \(\alpha_A = (4t^3)\,\mathrm{rad/s^2}\), where \(t\) is in seconds. If this gear is initially turning at \((\omega_A)_0 = 20\,\mathrm{rad/s}\), determine the angular velocity of gear \(B\) when \(t = 2\) s. figure — add your own
  21. 16‑21The motor turns the disk with an angular velocity of \(\omega = (5t^2 + 3t)\,\mathrm{rad/s}\), where \(t\) is in seconds. Determine the magnitudes of the velocity and the \(n\) and \(t\) components of acceleration of the point \(A\) on the disk when \(t = 3\) s. figure — add your own
  22. 16‑22If the motor turns gear \(A\) with an angular acceleration of \(\alpha_A = 2\,\mathrm{rad/s^2}\) when the angular velocity is \(\omega_A = 20\,\mathrm{rad/s}\), determine the angular acceleration and angular velocity of gear \(D\). figure — add your own
  23. 16‑23If the motor turns gear \(A\) with an angular acceleration of \(\alpha_A = 3\,\mathrm{rad/s^2}\) when the angular velocity is \(\omega_A = 60\,\mathrm{rad/s}\), determine the angular acceleration and angular velocity of gear \(D\). figure — add your own
  24. 16‑24The \(50\,\mathrm{mm}\)-radius pulley \(A\) of the clothes dryer rotates with an angular acceleration \(\alpha_A = (27\theta_A^{1/2})\,\mathrm{rad/s^2}\) where \(\theta_A\) is in radians. Determine its angular acceleration when \(t = 1\) s, starting from rest. figure — add your own
  25. 16‑25If the \(50\,\mathrm{mm}\)-radius motor pulley \(A\) of the clothes dryer rotates with an angular acceleration of \(\alpha_A = (10 + 50t)\,\mathrm{rad/s^2}\), where \(t\) is in seconds, determine its angular velocity when \(t = 3\) s, starting from rest. figure — add your own
  26. 16‑26The pinion gear \(A\) on the motor shaft is given a constant angular acceleration \(\alpha = 3\,\mathrm{rad/s^2}\). If the gears \(A\) and \(B\) have the dimensions shown, determine the angular velocity and angular displacement of the output shaft \(C\), when \(t = 2\) s starting from rest. The shaft is fixed to \(B\) and turns with it. figure — add your own
  27. 16‑27A stamp \(S\), located on the revolving drum, is used to label canisters. If the canisters are centered \(200\,\mathrm{mm}\) apart on the conveyor, determine the radius \(r_A\) of the driving wheel \(A\) and the radius \(r_B\) of the conveyor belt drum so that for each revolution of the stamp it marks the top of a canister. How many canisters are marked per minute if the drum at \(B\) is rotating at \(\omega_B = 0.2\,\mathrm{rad/s}\)? Note that the driving belt is twisted as it passes between the wheels. figure — add your own
  28. 16‑28At the instant shown, gear \(A\) is rotating with a constant angular velocity of \(\omega_A = 6\,\mathrm{rad/s}\). Determine the largest angular velocity of gear \(B\) and the maximum speed of point \(C\). figure — add your own
  29. 16‑29For a short time a motor of the random-orbit sander drives the gear \(A\) with an angular velocity of \(\omega_A = 40(t^3 + 6t)\,\mathrm{rad/s}\), where \(t\) is in seconds. This gear is connected to gear \(B\), which is fixed connected to the shaft \(CD\). The end of this shaft is connected to the eccentric spindle \(EF\) and pad \(P\), which causes the pad to orbit around shaft \(CD\) at a radius of \(15\,\mathrm{mm}\). Determine the magnitudes of the velocity and the tangential and normal components of acceleration of the spindle \(EF\) when \(t = 2\) s after starting from rest. figure — add your own
  30. 16‑30Determine the distance the load \(W\) is lifted in \(t = 5\) s using the hoist. The shaft of the motor \(M\) turns with an angular velocity \(\omega = 100(4 + t)\,\mathrm{rad/s}\), where \(t\) is in seconds. figure — add your own
  31. 16‑31The driving belt is twisted so that pulley \(B\) rotates in the opposite direction to that of drive wheel \(A\). If the angular displacement of \(A\) is \(\theta_A = (5t^3 + 10t^2)\,\mathrm{rad}\), where \(t\) is in seconds, determine the angular velocity and angular acceleration of \(B\) when \(t = 3\) s. figure — add your own
  32. 16‑32The driving belt is twisted so that pulley \(B\) rotates in the opposite direction to that of drive wheel \(A\). If \(A\) has a constant angular acceleration of \(\alpha_A = 30\,\mathrm{rad/s^2}\), determine the tangential and normal components of acceleration of a point located at the rim of \(B\) when \(t = 3\) s, starting from rest. figure — add your own
  33. 16‑33The rope of diameter \(d\) is wrapped around the tapered drum which has the dimensions shown. If the drum is rotating at a constant rate of \(\omega\), determine the upward acceleration of the block. Neglect the small horizontal displacement of the block. figure — add your own
  34. 16‑34A tape having a thickness \(s\) wraps around the wheel which is turning at a constant rate \(\omega\). Assuming the unwrapped portion of tape remains horizontal, determine the acceleration of point \(P\) of the unwrapped tape when the radius of the wrapped tape is \(r\). Hint: Since \(v_P = \omega r\), take the time derivative and note that \(dr/dt = \omega (s/2\pi)\). figure — add your own
  35. 16‑35If the shaft and plate rotates with a constant angular velocity of \(\omega = 14\,\mathrm{rad/s}\), determine the velocity and acceleration of point \(C\) located on the corner of the plate at the instant shown. Express the result in Cartesian vector form. figure — add your own
  36. 16‑36At the instant shown, the shaft and plate rotates with an angular velocity of \(\omega = 14\,\mathrm{rad/s}\) and angular acceleration of \(\alpha = 7\,\mathrm{rad/s^2}\). Determine the velocity and acceleration of point \(D\) located on the corner of the plate at this instant. Express the result in Cartesian vector form. figure — add your own
  37. 16‑37The rod assembly is supported by ball-and-socket joints at \(A\) and \(B\). At the instant shown it is rotating about the \(y\) axis with an angular velocity \(\omega = 5\,\mathrm{rad/s}\) and has an angular acceleration \(\alpha = 8\,\mathrm{rad/s^2}\). Determine the magnitudes of the velocity and acceleration of point \(C\) at this instant. Solve the problem using Cartesian vectors and Eqs. 16–9 and 16–13. figure — add your own
  38. 16‑38The mechanism for a car window winder is shown in the figure. Here the handle turns the small cog \(C\), which rotates the spur gear \(S\), thereby rotating the fixed-connected lever \(AB\) which raises track \(D\) in which the window rests. The window is free to slide on the track. If the handle is wound at \(0.5\,\mathrm{rad/s}\), determine the speed of points \(A\) and \(E\) and the speed \(v_w\) of the window at the instant \(\theta = 30^\circ\). figure — add your own
  39. 16‑39The end \(A\) of the bar is moving downward along the slotted guide with a constant velocity \(\mathbf{v}_A\). Determine the angular velocity \(\omega\) and angular acceleration \(\alpha\) of the bar as a function of its position \(y\). figure — add your own
  40. 16‑40At the instant \(\theta = 60^\circ\), the slotted guide rod is moving to the left with an acceleration of \(2\,\mathrm{m/s^2}\) and a velocity of \(5\,\mathrm{m/s}\). Determine the angular acceleration and angular velocity of link \(AB\) at this instant. figure — add your own
  41. 16‑41At the instant \(\theta = 50^\circ\), the slotted guide is moving upward with an acceleration of \(3\,\mathrm{m/s^2}\) and a velocity of \(2\,\mathrm{m/s}\). Determine the angular acceleration and angular velocity of link \(AB\) at this instant. Note: The upward motion of the guide is in the negative \(y\) direction. figure — add your own
  42. 16‑42At the instant shown, \(\theta = 60^\circ\), and rod \(AB\) is subjected to a deceleration of \(16\,\mathrm{m/s^2}\) when the velocity is \(10\,\mathrm{m/s}\). Determine the angular velocity and angular acceleration of link \(CD\) at this instant. figure — add your own
  43. 16‑43The crank \(AB\) is rotating with a constant angular velocity of \(4\,\mathrm{rad/s}\). Determine the angular velocity of the connecting rod \(CD\) at the instant \(\theta = 30^\circ\). figure — add your own
  44. 16‑44Determine the velocity and acceleration of the follower rod \(CD\) as a function of \(\theta\) when the contact between the cam and follower is along the straight region \(AB\) on the face of the cam. The cam rotates with a constant counterclockwise angular velocity \(\omega\). figure — add your own
  45. 16‑45Determine the velocity of rod \(R\) for any angle \(\theta\) of the cam \(C\) if the cam rotates with a constant angular velocity \(\omega\). The pin connection at \(O\) does not cause an interference with the motion of \(A\) on \(C\). figure — add your own
  46. 16‑46The circular cam rotates about the fixed point \(O\) with a constant angular velocity \(\omega\). Determine the velocity \(v\) of the follower rod \(AB\) as a function of \(\theta\). figure — add your own
  47. 16‑47Determine the velocity of the rod \(R\) for any angle \(\theta\) of cam \(C\) as the cam rotates with a constant angular velocity \(\omega\). The pin connection at \(O\) does not cause an interference with the motion of plate \(A\) on \(C\). figure — add your own
  48. 16‑48Determine the velocity and acceleration of the peg \(A\) which is confined between the vertical guide and the rotating slotted rod. figure — add your own
  49. 16‑49Bar \(AB\) rotates uniformly about the fixed pin \(A\) with a constant angular velocity \(\omega\). Determine the velocity and acceleration of block \(C\), at the instant \(\theta = 60^\circ\). figure — add your own
  50. 16‑50The center of the cylinder is moving to the left with a constant velocity \(\mathbf{v}_0\). Determine the angular velocity \(\omega\) and angular acceleration \(\alpha\) of the bar. Neglect the thickness of the bar. figure — add your own
  51. 16‑51The pins at \(A\) and \(B\) are confined to move in the vertical and horizontal tracks. If the slotted arm is causing \(A\) to move downward at \(\mathbf{v}_A\), determine the velocity of \(B\) at the instant shown. figure — add your own
  52. 16‑52The crank \(AB\) has a constant angular velocity \(\omega\). Determine the velocity and acceleration of the slider at \(C\) as a function of \(\theta\). Suggestion: Use the \(x\) coordinate to express the motion of \(C\) and the \(\phi\) coordinate for \(CB\), \(x=0\) when \(\phi=0^\circ\). figure — add your own
  53. 16‑53If the wedge moves to the left with a constant velocity \(\mathbf{v}\), determine the angular velocity of the rod as a function of \(\theta\). figure — add your own
  54. 16‑54The crate is transported on a platform which rests on rollers, each having a radius \(r\). If the rollers do not slip, determine their angular velocity if the platform moves forward with a velocity \(\mathbf{v}\). figure — add your own
  55. 16‑55Arm \(AB\) has an angular velocity of \(\omega\) and an angular acceleration of \(\alpha\). If no slipping occurs between the disk \(D\) and the fixed curved surface, determine the angular velocity and angular acceleration of the disk. figure — add your own
  56. 16‑56The bridge girder \(G\) of a bascule bridge is raised and lowered using the drive mechanism shown. If the hydraulic cylinder \(AB\) shortens at a constant rate of \(0.15\,\mathrm{m/s}\), determine the angular velocity of the bridge girder at the instant \(\theta = 60^\circ\). figure — add your own
  57. 16‑57The wheel is rotating with an angular velocity \(\omega = 8\,\mathrm{rad/s}\). Determine the velocity of the collar \(A\) at the instant \(\theta = 30^\circ\) and \(\phi = 60^\circ\). Also, sketch the location of bar \(AB\) when \(\theta = 0^\circ, 30^\circ\), and \(60^\circ\) to show its general plane motion. figure — add your own
  58. 16‑58The slider block \(C\) moves at \(8\,\mathrm{m/s}\) down the inclined groove. Determine the angular velocities of links \(AB\) and \(BC\), at the instant shown. figure — add your own
  59. 16‑59At the instant shown, the truck is traveling to the right at \(3\,\mathrm{m/s}\), while the pipe is rolling counterclockwise at angular \(\omega = 8\,\mathrm{rad/s}\) without slipping at \(B\). Determine the velocity of the pipe's center \(G\). figure — add your own
  60. 16‑60At the instant shown, the truck is traveling to the right at \(8\,\mathrm{m/s}\). If the spool does not slip at \(B\), determine its angular velocity so that its mass center \(G\) appears to an observer on the ground to remain stationary. figure — add your own
  61. 16‑61The link \(AB\) has an angular velocity of \(3\,\mathrm{rad/s}\). Determine the velocity of block \(C\) and the angular velocity of link \(BC\) at the instant \(\theta = 45^\circ\). Also, sketch the position of link \(BC\) when \(\theta = 60^\circ, 45^\circ\), and \(30^\circ\) to show its general plane motion. figure — add your own
  62. 16‑62If the gear rotates with an angular velocity of \(\omega = 10\,\mathrm{rad/s}\) and the gear rack moves at \(v_C = 5\,\mathrm{m/s}\), determine the velocity of the slider block \(A\) at the instant shown. figure — add your own
  63. 16‑63Knowing that angular velocity of link \(AB\) is \(\omega_{AB} = 4\,\mathrm{rad/s}\), determine the velocity of the collar at \(C\) and the angular velocity of link \(CB\) at the instant shown. Link \(CB\) is horizontal at this instant. figure — add your own
  64. 16‑64The cylinder \(B\) rolls on the fixed cylinder \(A\) without slipping. If the connected bar \(CD\) is rotating with an angular velocity of \(\omega_{CD} = 5\,\mathrm{rad/s}\), determine the angular velocity of cylinder \(B\). figure — add your own
  65. 16‑65The angular velocity of link \(AB\) is \(\omega_{AB} = 5\,\mathrm{rad/s}\). Determine the velocity of block \(C\) and the angular velocity of link \(BC\) at the instant \(\theta = 45^\circ\) and \(\phi = 30^\circ\). Also, sketch the position of link \(CB\) when \(\theta = 45^\circ, 60^\circ\), and \(75^\circ\) to show its general plane motion. figure — add your own
  66. 16‑66The shaper mechanism is designed to give a slow cutting stroke and a quick return to a blade attached to the slider at \(C\). Determine the velocity of the slider block \(C\) at the instant \(\theta = 60^\circ\), if link \(AB\) is rotating at \(4\,\mathrm{rad/s}\). figure — add your own
  67. 16‑67Determine the velocity of the slider block at \(C\) at the instant \(\theta = 45^\circ\), if link \(AB\) is rotating at \(4\,\mathrm{rad/s}\). figure — add your own
  68. 16‑68Rod \(AB\) is rotating with an angular velocity of \(\omega_{AB} = 60\,\mathrm{rad/s}\). Determine the velocity of the slider \(C\) at the instant \(\theta = 60^\circ\) and \(\phi = 45^\circ\). Also, sketch the position of bar \(BC\) when \(\theta = 30^\circ, 60^\circ\) and \(90^\circ\) to show its general plane motion. figure — add your own
  69. 16‑69If the slider block \(C\) is moving at \(v_C = 3\,\mathrm{m/s}\), determine the angular velocity of \(BC\) and the crank \(AB\) at the instant shown. figure — add your own
  70. 16‑70Determine the velocity of the center \(O\) of the spool when the cable is pulled to the right with a velocity of \(\mathbf{v}\). The spool rolls without slipping. figure — add your own
  71. 16‑71Determine the velocity of point \(A\) on the outer rim of the spool at the instant shown when the cable is pulled to the right with a velocity of \(\mathbf{v}\). The spool rolls without slipping. figure — add your own
  72. 16‑72If the flywheel is rotating with an angular velocity of \(\omega_A = 6\,\mathrm{rad/s}\), determine the angular velocity of rod \(BC\) at the instant shown. figure — add your own
  73. 16‑73The epicyclic gear train consists of the sun gear \(A\) which is in mesh with the planet gear \(B\). This gear has an inner hub \(c\) which is fixed to \(B\) and in mesh with the fixed ring gear \(R\). If the connecting link \(DE\) pinned to \(B\) and \(c\) is rotating at \(\omega_{DE} = 18\,\mathrm{rad/s}\) about the pin at \(E\), determine the angular velocities of the planet and sun gears. figure — add your own
  74. 16‑74If the slider block \(A\) is moving downward at \(v_A = 4\,\mathrm{m/s}\), determine the velocities of blocks \(B\) and \(C\) at the instant shown. figure — add your own
  75. 16‑75If the slider block \(A\) is moving downward at \(v_A = 4\,\mathrm{m/s}\), determine the velocity of point \(E\) at the instant shown. figure — add your own
  76. 16‑76The planetary gear system is used in an automatic transmission for an automobile. By locking or releasing certain gears, it has the advantage of operating the car at different speeds. Consider the case where the ring gear \(R\) is held fixed, \(\omega_R = 0\) and the sun gear \(S\) is rotating at \(\omega_S = 5\,\mathrm{rad/s}\). Determine the angular velocity of each of the planet gears \(P\) and shaft \(A\). figure — add your own
  77. 16‑77The mechanism is used on a machine for the manufacturing of a wire product. Because of the rotational motion of link \(AB\) and the sliding of block \(F\), the segmental gear lever \(DE\) undergoes general plane motion. If \(AB\) is rotating at \(\omega_{AB} = 5\,\mathrm{rad/s}\), determine the velocity of point \(E\) at the instant shown. figure — add your own
  78. 16‑78The similar links \(AB\) and \(CD\) rotate about the fixed pins at \(A\) and \(C\). If \(AB\) has an angular velocity \(\omega_{AB} = 8\,\mathrm{rad/s}\), determine the angular velocity of \(BDP\) and the velocity of point \(P\). figure — add your own
  79. 16‑79If the ring gear \(A\) rotates clockwise with an angular velocity of \(\omega_A = 30\,\mathrm{rad/s}\), while link \(BC\) rotates clockwise with an angular velocity of \(\omega_{BC} = 15\,\mathrm{rad/s}\), determine the angular velocity of gear \(D\). figure — add your own
  80. 16‑80The mechanism shown is used in a riveting machine. It consists of a driving piston \(A\), three links, and a riveter which is attached to the slider block \(D\). Determine the velocity of \(D\) at the instant shown, when the piston at \(A\) is traveling at \(v_A = 20\,\mathrm{m/s}\). figure — add your own
  81. 16‑81In each case show graphically how to locate the instantaneous center of zero velocity of link \(AB\). Assume the geometry is known. figure — add your own
  82. 16‑82If crank \(AB\) is rotating with an angular velocity of \(\omega_{AB} = 6\,\mathrm{rad/s}\), determine the velocity of the center \(O\) of the gear at the instant shown. figure — add your own
  83. 16‑83The square plate is confined within the slots at \(A\) and \(B\). When \(\theta = 30^\circ\), point \(A\) is moving at \(v_A = 8\,\mathrm{m/s}\). Determine the velocity of point \(C\) at this instant. figure — add your own
  84. 16‑84The square plate is confined within the slots at \(A\) and \(B\). When \(\theta = 30^\circ\), point \(A\) is moving at \(v_A = 8\,\mathrm{m/s}\). Determine the velocity of point \(D\) at this instant. figure — add your own
  85. 16‑85The shaper mechanism is designed to give a slow cutting stroke and a quick return to a blade attached to the slider at \(C\). Determine the angular velocity of the link \(CB\) at the instant shown, if the link \(AB\) is rotating at \(4\,\mathrm{rad/s}\). figure — add your own
  86. 16‑86At the instant shown, the disk is rotating at \(\omega = 4\,\mathrm{rad/s}\). Determine the velocities of points \(A, B\) and \(C\). figure — add your own
  87. 16‑87Member \(AB\) is rotating at \(\omega_{AB} = 6\,\mathrm{rad/s}\). Determine the velocity of point \(D\) and the angular velocity of members \(BPD\) and \(CD\). figure — add your own
  88. 16‑88Member \(AB\) is rotating at \(\omega_{AB} = 6\,\mathrm{rad/s}\). Determine the velocity of point \(P\) and the angular velocity of member \(BPD\). figure — add your own
  89. 16‑89If rod \(CD\) is rotating with an angular velocity \(\omega_{CD} = 4\,\mathrm{rad/s}\), determine the angular velocities of rods \(AB\) and \(CB\) at the instant shown. figure — add your own
  90. 16‑90If bar \(AB\) has an angular velocity \(\omega_{AB} = 6\,\mathrm{rad/s}\), determine the velocity of the slider block \(C\) at the instant shown. figure — add your own
  91. 16‑91The mechanism used in a marine engine consists of a single crank \(AB\) and two connecting rods \(BC\) and \(BD\). Determine the velocity of the piston at \(C\) the instant the crank is in the position shown and has an angular velocity of \(5\,\mathrm{rad/s}\). figure — add your own
  92. 16‑92The mechanism used in a marine engine consists of a single crank \(AB\) and two connecting rods \(BC\) and \(BD\). Determine the velocity of the piston at \(D\) the instant the crank is in the position shown and has an angular velocity of \(5\,\mathrm{rad/s}\). figure — add your own
  93. 16‑93Show that if the rim of the wheel and its hub maintain contact with the three tracks as the wheel rolls, it is necessary that slipping occurs at the hub \(A\) if no slipping occurs at \(B\). Under these conditions, what is the speed at \(A\) if the wheel has angular velocity \(\omega\)? figure — add your own
  94. 16‑94The pinion gear \(A\) rolls on the fixed gear rack \(B\) with an angular velocity \(\omega = 8\,\mathrm{rad/s}\). Determine the velocity of the gear rack \(C\). figure — add your own
  95. 16‑95The cylinder \(B\) rolls on the fixed cylinder \(A\) without slipping. If connected bar \(CD\) is rotating with an angular velocity \(\omega_{CD} = 5\,\mathrm{rad/s}\), determine the angular velocity of cylinder \(B\). Point \(C\) is a fixed point. figure — add your own
  96. 16‑96Determine the angular velocity of the double-tooth gear and the velocity of point \(C\) on the gear. figure — add your own
  97. 16‑97If the hub gear \(H\) and ring gear \(R\) have angular velocities \(\omega_H = 5\,\mathrm{rad/s}\) and \(\omega_R = 20\,\mathrm{rad/s}\), respectively, determine the angular velocity \(\omega_S\) of the spur gear \(S\) and the angular velocity of its attached arm \(OA\). figure — add your own
  98. 16‑98If the hub gear \(H\) has an angular velocity \(\omega_H = 5\,\mathrm{rad/s}\), determine the angular velocity of the ring gear \(R\) so that the arm \(OA\) attached to the spur gear \(S\) remains stationary \((\omega_{OA} = 0)\). What is the angular velocity of the spur gear? figure — add your own
  99. 16‑99The crankshaft \(AB\) rotates at \(\omega_{AB} = 50\,\mathrm{rad/s}\) about the fixed axis through point \(A\) and the disk at \(C\) is held fixed in its support at \(E\). Determine the angular velocity of rod \(CD\) at the instant shown. figure — add your own
  100. 16‑100Cylinder \(A\) rolls on the fixed cylinder \(B\) without slipping. If bar \(CD\) is rotating with an angular velocity of \(\omega_{CD} = 3\,\mathrm{rad/s}\), determine the angular velocity of \(A\). figure — add your own
  101. 16‑101The planet gear \(A\) is pin connected to the end of the link \(BC\). If the link rotates about the fixed point \(B\) at \(4\,\mathrm{rad/s}\), determine the angular velocity of the ring gear \(R\). The sun gear \(D\) is fixed from rotating. figure — add your own
  102. 16‑102Solve Prob. 16‑101 if the sun gear \(D\) is rotating clockwise at \(\omega_D = 5\,\mathrm{rad/s}\) while link \(BC\) rotates counterclockwise at \(\omega_{BC} = 4\,\mathrm{rad/s}\). figure — add your own
  103. 16‑103Pulley \(A\) rotates with the angular velocity and angular acceleration shown. Determine the angular acceleration of pulley \(B\) at the instant shown. figure — add your own
  104. 16‑104Pulley \(A\) rotates with the angular velocity and angular acceleration shown. Determine the acceleration of block \(E\) at the instant shown. figure — add your own
  105. 16‑105Member \(AB\) has the angular motions shown. Determine the velocity and acceleration of the slider block \(C\) at this instant. figure — add your own
  106. 16‑106At a given instant the roller \(A\) on the bar has the velocity and acceleration shown. Determine the velocity and acceleration of the roller \(B\) and the bar's angular velocity and angular acceleration at this instant. figure — add your own
  107. 16‑107Bar \(AB\) has the angular motions shown. Determine the velocity and acceleration of the slider block \(C\) at this instant. figure — add your own
  108. 16‑108The flywheel rotates with angular velocity \(\omega = 2\,\mathrm{rad/s}\) and angular acceleration \(\alpha = 6\,\mathrm{rad/s^2}\). Determine the angular acceleration of links \(AB\) and \(BC\) at this instant. figure — add your own
  109. 16‑109A cord is wrapped around the inner spool of the gear. If it is pulled with a constant velocity \(\mathbf{v}\), determine the velocities and accelerations of points \(A\) and \(B\). The gear rolls on the fixed gear rack. figure — add your own
  110. 16‑110The reel of rope has the angular motion shown. Determine the velocity and acceleration of point \(A\) at the instant shown. figure — add your own
  111. 16‑111The reel of rope has the angular motion shown. Determine the velocity and acceleration of point \(B\) at the instant shown. figure — add your own
  112. 16‑112The disk has an angular acceleration \(\alpha = 8\,\mathrm{rad/s^2}\) and angular velocity \(\omega = 3\,\mathrm{rad/s}\) at the instant shown. If it does not slip at \(A\), determine the acceleration of point \(B\). figure — add your own
  113. 16‑113The disk has an angular acceleration \(\alpha = 8\,\mathrm{rad/s^2}\) and angular velocity \(\omega = 3\,\mathrm{rad/s}\) at the instant shown. If it does not slip at \(A\), determine the acceleration of point \(C\). figure — add your own
  114. 16‑114Member \(AB\) has the angular motions shown. Determine the angular velocity and angular acceleration of members \(CB\) and \(DC\). figure — add your own
  115. 16‑115Determine the angular acceleration of link \(CD\) if link \(AB\) has the angular velocity and angular acceleration shown. figure — add your own
  116. 16‑116At a given instant the slider block \(A\) is moving to the right with the motion shown. Determine the angular acceleration of link \(AB\) and the acceleration of point \(B\) at this instant. figure — add your own
  117. 16‑117The slider block has the motion shown. Determine the angular velocity and angular acceleration of the wheel at this instant. figure — add your own
  118. 16‑118(Not listed in detail; assume problem exists) figure — add your own
  119. 16‑119If member \(AB\) has the angular motion shown, determine the angular velocity and angular acceleration of member \(CD\) at the instant shown. figure — add your own
  120. 16‑120If member \(AB\) has the angular motion shown, determine the velocity and acceleration of point \(C\) at the instant shown. figure — add your own
  121. 16‑121The wheel rolls without slipping such that at the instant shown it has an angular velocity \(\omega\) and angular acceleration \(\alpha\). Determine the velocity and acceleration of point \(B\) on the rod at this instant. figure — add your own
  122. 16‑122A single pulley having both an inner and outer rim is pin connected to the block at \(A\). As cord \(CF\) unwinds from the inner rim of the pulley with the motion shown, cord \(DE\) unwinds from the outer rim. Determine the angular acceleration of the pulley and the acceleration of the block at the instant shown. figure — add your own
  123. 16‑123The collar is moving downward with the motion shown. Determine the angular velocity and angular acceleration of the gear at the instant shown as it rolls along the fixed gear rack. figure — add your own
  124. 16‑124The tied crank and gear mechanism gives rocking motion to crank \(AC\), necessary for the operation of a printing press. If link \(DE\) has the angular motion shown, determine the respective angular velocities of gear \(F\) and crank \(AC\) at this instant, and the angular acceleration of crank \(AC\). figure — add your own
  125. 16‑125The center \(O\) of the gear and the gear rack \(P\) move with the velocities and accelerations shown. Determine the angular acceleration of the gear and the acceleration of point \(B\) located at the rim of the gear at the instant shown. figure — add your own
  126. 16‑126The hoop is cast on the rough surface such that it has angular velocity \(\omega = 4\,\mathrm{rad/s}\) and an angular acceleration \(\alpha = 5\,\mathrm{rad/s^2}\). Also, its center has a velocity \(v_O = 5\,\mathrm{m/s}\) and a deceleration \(a_O = 2\,\mathrm{m/s^2}\). Determine the acceleration of point \(A\) at this instant. figure — add your own
  127. 16‑127Determine the angular acceleration of link \(AB\) if link \(CD\) has the angular velocity and angular deceleration shown. figure — add your own
  128. 16‑128The mechanism produces intermittent motion of link \(AB\). If the sprocket \(S\) is turning with an angular acceleration \(\alpha_S = 2\,\mathrm{rad/s^2}\) and has an angular velocity \(\omega_S = 6\,\mathrm{rad/s}\) at the instant shown, determine the angular velocity and angular acceleration of link \(AB\) at this instant. The sprocket \(S\) is mounted on a shaft which is separate from a collinear shaft attached to \(AB\) at \(A\). The pin at \(C\) is attached to one of the chain links such that it moves vertically downward. figure — add your own
  129. 16‑129At the instant shown, ball \(B\) is rolling along the slot in the disk with a velocity of \(600\,\mathrm{mm/s}\) and an acceleration of \(150\,\mathrm{mm/s^2}\), both measured relative to the disk and directed away from \(O\). If at the same instant the disk has the angular velocity and angular acceleration shown, determine the velocity and acceleration of the ball at this instant. figure — add your own
  130. 16‑130Block \(A\), which is attached to a cord, moves along the slot of a horizontal forked rod. At the instant shown, the cord is pulled down through the hole at \(O\) with an acceleration of \(4\,\mathrm{m/s^2}\) and its velocity is \(2\,\mathrm{m/s}\). Determine the acceleration of the block at this instant. The rod rotates about \(O\) with a constant angular velocity \(\omega = 4\,\mathrm{rad/s}\). figure — add your own
  131. 16‑131Ball \(C\) moves with a speed of \(3\,\mathrm{m/s}\) which is increasing at a constant rate of \(1.5\,\mathrm{m/s^2}\), both measured relative to the circular plate and directed as shown. At the same instant the plate rotates with the angular velocity and angular acceleration shown. Determine the velocity and acceleration of the ball at this instant. figure — add your own
  132. 16‑132Particles \(B\) and \(A\) move along the parabolic and circular paths, respectively. If \(B\) has a velocity of \(7\,\mathrm{m/s}\) in the direction shown and its speed is increasing at \(4\,\mathrm{m/s^2}\) while \(A\) has a velocity of \(8\,\mathrm{m/s}\) in the direction shown and its speed is decreasing at \(6\,\mathrm{m/s^2}\), determine the relative velocity and relative acceleration of \(B\) with respect to \(A\). figure — add your own
  133. 16‑133Rod \(AB\) rotates counterclockwise with a constant angular velocity \(\omega = 3\,\mathrm{rad/s}\). Determine the velocity of point \(C\) located on the double collar when \(\theta = 30^\circ\). The collar consists of two pin-connected slider blocks which are constrained to move along the circular path and the rod \(AB\). figure — add your own
  134. 16‑134Rod \(AB\) rotates counterclockwise with a constant angular velocity \(\omega = 3\,\mathrm{rad/s}\). Determine the velocity and acceleration of point \(C\) located on the double collar when \(\theta = 45^\circ\). The collar consists of two pin-connected slider blocks which are constrained to move along the circular path and the rod \(AB\). figure — add your own
  135. 16‑135A girl stands at \(A\) on a platform which is rotating with an angular acceleration \(\alpha = 0.2\,\mathrm{rad/s^2}\) and at the instant shown has an angular velocity \(\omega = 0.5\,\mathrm{rad/s}\). If she walks at a constant speed \(v = 0.75\,\mathrm{m/s}\) measured relative to the platform, determine her acceleration (a) when she reaches point \(D\) in going along the path \(ADC\), \(d=1\,\mathrm{m}\), and (b) when she reaches point \(B\) if she follows the path \(ABC\), \(r=3\,\mathrm{m}\). figure — add your own
  136. 16‑136If the piston is moving with a velocity \(v_A = 3\,\mathrm{m/s}\) and acceleration of \(a_A = 1.5\,\mathrm{m/s^2}\), determine the angular velocity and angular acceleration of the slotted link at the instant shown. Link \(AB\) slides freely along its slot on the fixed peg \(C\). figure — add your own
  137. 16‑137Water leaves the impeller of the centrifugal pump with a velocity of \(25\,\mathrm{m/s}\) and acceleration of \(30\,\mathrm{m/s^2}\), both measured relative to the impeller along the blade line \(AB\). Determine the velocity and acceleration of a water particle at \(A\) as it leaves the impeller at the instant shown. The impeller rotates with a constant angular velocity of \(\omega = 15\,\mathrm{rad/s}\). figure — add your own
  138. 16‑138Peg \(B\) on the gear slides freely along the slot in link \(AB\). If the gear's center \(O\) moves with the velocity and acceleration shown, determine the angular velocity and angular acceleration of the link at this instant. figure — add your own
  139. 16‑139The collar \(C\) is pinned to rod \(CD\) while it slides on rod \(AB\). If rod \(AB\) has an angular velocity of \(2\,\mathrm{rad/s}\) and an angular acceleration of \(8\,\mathrm{rad/s^2}\), both acting counterclockwise, determine the angular velocity and the angular acceleration of rod \(CD\) at the instant shown. figure — add your own
  140. 16‑140At the instant shown, the robotic arm \(AB\) is rotating counterclockwise at \(\omega = 5\,\mathrm{rad/s}\) and has an angular acceleration \(\alpha = 2\,\mathrm{rad/s^2}\). Simultaneously, the grip \(BC\) is rotating counterclockwise at \(\omega' = 6\,\mathrm{rad/s}\) and \(\alpha' = 2\,\mathrm{rad/s^2}\), both measured relative to a fixed reference. Determine the velocity and acceleration of the object held at the grip \(C\). figure — add your own
  141. 16‑141At the instant shown rod \(AB\) has an angular velocity \(\omega_{AB} = 4\,\mathrm{rad/s}\) and an angular acceleration \(\alpha_{AB} = 2\,\mathrm{rad/s^2}\). Determine the angular velocity and angular acceleration of rod \(CD\) at this instant. The collar at \(C\) is pin connected to \(CD\) and slides freely along \(AB\). figure — add your own
  142. 16‑142Collar \(B\) moves to the left with a speed of \(5\,\mathrm{m/s}\) which is increasing at a constant rate of \(1.5\,\mathrm{m/s^2}\), relative to the hoop, while the hoop rotates with the angular velocity and angular acceleration shown. Determine the magnitudes of the velocity and acceleration of the collar at this instant. figure — add your own
  143. 16‑143Block \(D\) of the mechanism is confined to move within the slot of member \(CB\). If link \(AD\) is rotating at a constant rate of \(\omega_{AD} = 4\,\mathrm{rad/s}\), determine the angular velocity and angular acceleration of member \(CB\) at the instant shown. figure — add your own
  144. 16‑144A ride in an amusement park consists of a rotating arm \(AB\) having a constant angular velocity \(\omega_{AB} = 2\,\mathrm{rad/s}\) about point \(A\) and a car mounted at the end of the arm which has a constant angular velocity \(\omega' = \{-0.5\,\mathbf{k}\}\,\mathrm{rad/s}\), measured relative to the arm. At the instant shown, determine the velocity and acceleration of the passenger at \(C\). figure — add your own
  145. 16‑145A ride in an amusement park consists of a rotating arm \(AB\) that has an angular acceleration of \(\alpha_{AB} = 1\,\mathrm{rad/s^2}\) when \(\omega_{AB} = 2\,\mathrm{rad/s}\) at the instant shown. Also at this instant the car mounted at the end of the arm has a relative angular acceleration of \(\alpha' = \{-0.6\,\mathbf{k}\}\,\mathrm{rad/s^2}\) when \(\omega' = \{-0.5\,\mathbf{k}\}\,\mathrm{rad/s}\). Determine the velocity and acceleration of the passenger \(C\) at this instant. figure — add your own
  146. 16‑146If the slider block \(C\) is fixed to the disk that has a constant counterclockwise angular velocity of \(4\,\mathrm{rad/s}\), determine the angular velocity and angular acceleration of the slotted arm \(AB\) at the instant shown. figure — add your own
  147. 16‑147At the instant shown, car \(A\) travels with a speed of \(25\,\mathrm{m/s}\), which is decreasing at a constant rate of \(2\,\mathrm{m/s^2}\), while car \(C\) travels with a speed of \(15\,\mathrm{m/s}\), which is increasing at a constant rate of \(3\,\mathrm{m/s^2}\). Determine the velocity and acceleration of car \(A\) with respect to car \(C\). figure — add your own
  148. 16‑148At the instant shown, car \(B\) travels with a speed of \(15\,\mathrm{m/s}\), which is increasing at a constant rate of \(2\,\mathrm{m/s^2}\), while car \(C\) travels with a speed of \(15\,\mathrm{m/s}\), which is increasing at a constant rate of \(3\,\mathrm{m/s^2}\). Determine the velocity and acceleration of car \(B\) with respect to car \(C\). figure — add your own
  149. 16‑149A ride in an amusement park consists of a rotating platform \(P\) having constant angular velocity \(\omega_P = 1.5\,\mathrm{rad/s}\), and four cars, \(C\) mounted on the platform, which have constant angular velocities \(\omega_{C/P} = 2\,\mathrm{rad/s}\) measured relative to the platform. Determine the velocity and acceleration of the passenger at \(B\) at the instant shown. figure — add your own
  150. 16‑150(Not listed in detail; assume problem exists) figure — add your own
  151. 16‑151The disk rotates with the angular motion shown. Determine the angular velocity and angular acceleration of the slotted link \(AC\) at this instant. The peg at \(B\) is fixed to the disk. figure — add your own
  152. 16‑152The "quick-return" mechanism consists of a crank \(AB\), slider block \(B\), and slotted link \(CD\). If the crank has the angular motion shown, determine the angular motion of the slotted link at this instant. figure — add your own
R Review Problems 8 problems
Review
  1. R16‑1The hoisting gear \(A\) has an initial angular velocity of \(60\,\mathrm{rad/s}\) and a constant deceleration of \(1\,\mathrm{rad/s^2}\). Determine the velocity and deceleration of the block which is being hoisted by the hub on gear \(B\) when \(t = 3\,\mathrm{s}\). figure — add your own
  2. R16‑2Starting at \((\omega_A)_0 = 3\,\mathrm{rad/s}\) when \(\theta = 0\), \(s = 0\), pulley \(A\) is given an angular acceleration \(\alpha = (0.6\theta)\,\mathrm{rad/s^2}\) where \(\theta\) is in radians. Determine the speed of block \(B\) when it has risen \(s = 0.5\,\mathrm{m}\). The pulley has an inner hub \(D\) which is fixed to \(C\) and turns with it. figure — add your own
  3. R16‑3The board rests on the surface of two drums. At the instant shown, it has an acceleration of \(0.5\,\mathrm{m/s^2}\) to the right, while at the same instant, points on the outer rim of each drum have an acceleration with a magnitude of \(3\,\mathrm{m/s^2}\). If the board does not slip on the drums, determine its speed due to the motion. figure — add your own
  4. R16‑4If bar \(AB\) has an angular velocity \(\omega_{AB} = 6\,\mathrm{rad/s}\), determine the velocity of the slider block \(C\) at the instant shown. figure — add your own
  5. R16‑5The center of the pulley is being lifted vertically with an acceleration of \(4\,\mathrm{m/s^2}\) at the instant it has a velocity of \(2\,\mathrm{m/s}\). If the cable does not slip on the pulley's surface, determine the accelerations of the cylinder \(B\) and point \(C\) on the pulley. figure — add your own
  6. R16‑6At the instant shown, link \(AB\) has an angular velocity \(\omega_{AB} = 2\,\mathrm{rad/s}\) and an angular acceleration \(\alpha_{AB} = 6\,\mathrm{rad/s^2}\). Determine the acceleration of the pin at \(C\) and the angular acceleration of link \(CB\) at this instant, when \(\theta = 60^\circ\). figure — add your own
  7. R16‑7The disk is moving to the left such that it has an angular acceleration \(\alpha = 8\,\mathrm{rad/s^2}\) and angular velocity \(\omega = 3\,\mathrm{rad/s}\) at the instant shown. If it does not slip at \(A\), determine the acceleration of point \(B\). figure — add your own
  8. R16‑8At the given instant member \(AB\) has the angular motions shown. Determine the velocity and acceleration of the slider block \(C\) at this instant. figure — add your own