F18-9. The 60-kg rod OA is released from rest when H = 0.

F18-1. The 80-kg wheel has a radius of gyration about its mass center 0 of k0 = 400 mm. Determine its angular velocity after it has rotated 20 revolutions starting from rest. ,0.6 m P = 50 N

F18-2. The uniform 50-lb slender rod is subjected to a couple moment of M = 100 lb ft. If the rod is at rest when 0 = 0. determine its angular velocity when 0 = 90.

FI8-3. The uniform 50-kg slender rod is at rest in the position shown when P = 600 N is applied. Determine the angular velocity of the rod when the rod reaches the vertical position.

F18-4. The 50-kg wheel is subjected to a force of 50 N. If the wheel starts from rest and rolls without slipping, determine its angular velocity after it has rotated I0 revolutions. The radius of gyration of the wheel about its mass center O is k0 = 0.3 m.

F18-5. If the uniform 30-kg slender rod starts from rest at the position shown, determine its angular velocity after it has rotated 4 revolutions. The forces remain perpendicular to the rod.

The 20-kg wheel has a radius of gyration about its center O of \(k_{O}=300 \mathrm{\ mm}\). When it is subjected to a couple moment of \(M=50 \mathrm{\ N} \cdot \mathrm{m}\), it rolls without slipping. Determine the angular velocity of the wheel after its center O has traveled through a distance of \(s_{O}=20 \mathrm{\ m}\), starting from rest.

F18-7. If the 30-kg disk is released from rest when 0 = 0, determine its angular velocity when 0 = 90.

FI8-8. The 50-kg reel has a radius of gyration about its center O of kQ = 300 mm. If it is released from rest, determine its angular velocity when its center O has traveled 6 m down the smooth inclined plane.

F18-9. The 60-kg rod OA is released from rest when H = 0. Determine its angular velocity when 0 = 45. The spring remains vertical during the motion and is unstretched when 0 = 0.

F18-10. The 30-kg rod is released from rest when 0 = 0. Determine the angular velocity of the rod when 0 = 90. The spring is unstretched when 0 = 0.

F18-11. The 30-kg rod is released from rest when 0 = 45. Determine the angular velocity of the rod when 0 = 0.The spring is unstretched when 0 = 45.

F18-I2. The 20-kg rod is released from rest when 0 = 0. Determine its angular velocity when 0 = 90. The spring has an unstretched length of 0.5 in.

18-1. At a given instant the body of mass m has an angular velocity to and its mass center has a velocity vG. Show that its kinetic energy can be represented as T = where ltc is the moment of inertia of the body determined about the instantaneous axis of zero velocity, located a distance rGjjC from the mass center as shown.

18-2. The wheel is made from a 5-kg thin ring and two 2-kg slender rods. If the torsional spring attached to the wheel's center has a stiffness k = 2 N m/rad, and the wheel is rotated until the torque Af = 25N-m is developed, determine the maximum angular velocity of the wheel if it is released from rest.

18-3. The wheel is made from a 5-kg thin ring and two 2-kg slender rods. If the torsional spring attached to the wheels center has a stiffness k = 2 N-m/rad. so that the torque on the center of the wheel is M (20) N m. where 0 is in radians, determine the maximum angular velocity of the wheel if it is rotated two revolutions and then released from rest.

*18-4. The 50-kg flywheel has a radius of gyration of ku = 200 mm about its center of mass. If it is subjected to a torque of M = (9tf1,/2) N m. where 0 is in radians, determine its angular velocity when it has rotated 5 revolutions, starting from rest.

The spool has a mass of 60 kg and a radius of gyration \(k_{G}=0.3 \mathrm{\ m}\). If it is released from rest, determine how far its center descends down the smooth plane before it attains an angular velocity of \(\omega=6 \mathrm{\ rad} / \mathrm{s}\) Neglect friction and the mass of the cord which is wound around the central core.

18-6. Solve Prob. 18-5 if the coefficient of kinetic friction between the spool and plane at A is /xk = 0.2.

18-7. The double pulley consists of two parts that are attached to one another. It has a weight of 50 lb and a centroidal radius of gyration of k0 = 0.6 ft and is turning with an angular velocity of 20 rad/s clockwise. Determine the kinetic energy of the system. Assume that neither cable slips on the pulley.

*18-8. The double pulley consists of two parts that are attached to one another. It has a weight of 50 lb and a centroidal radius of gyration of kG = 0.6 ft and is turning with an angular velocity of 20 rad/s clockwise. Determine the angular velocity of the pulley at the instant the 20-lb weight moves 2 ft downward.

18-9. If the cable is subjected to force of P 300 N. and the spool starts from rest, determine its angular velocity after its center of mass O has moved 1.5 m.The mass of the spool is 100 kg and its radius of gyration about its center of mass is ka = 275 mm. Assume that the spool rolls without slipping.

18-10. The two tugboats each exert a constant force F on the ship. These forces are always directed perpendicular to the ships centerline. If the ship has a mass m and a radius of gyration about its center of mass G of k0. determine the angular velocity of the ship after it turns 90. The ship is originally at rest.

18-11. At the instant shown, link AB has an angular velocity io AB = 2 rad/s. If each link is considered as a uniform slender bar with a weight of 0.5 lb/in.. determine the total kinetic energy of the system.

*18-12. Determine the velocity of the 50-kg cylinder after it has descended a distance of 2 m. Initially, the system is at rest. The reel has a mass of 25 kg and a radius of gyration about its center of mass A of kA = 125 mm.

18-13. The wheel and the attached reel have a combined weight of 50 lb and a radius of gyration about their center of kA = 6 in. If pulley B attached to the motor is subjected to a torque of M = 40(2 - e~0Aa) lb ft, where H is in radians, determine the velocity of the 200-lb crate after it has moved upwards a distance of 5 ft. starting from rest. Neglect the mass of pulley B.

18-14. The wheel and the attached reel have a combined weight of 50 lb and a radius of gyration about their center of kA = 6 in. If pulley B that is attached to the motor is subjected to a torque of M 50 lb*ft. determine the velocity of the 200-lb crate after the pulley has turned 5 revolutions. Neglect the mass of the pulley.

18-15. The 50-kg gear has a radius of gyration of 125 mm about its center of mass O. If gear rack B is stationary, while the 25-kg gear rack C is subjected to a horizontal force of P = 150 N, determine the speed of C after the gears center O has moved to the right a distance of 0.3 m, starting from rest.

*18-16. Gear B is rigidly attached to drum A and is supported by two small rollers at E and D. Gear B is in mesh with gear C and is subjected to a torque of M = 50 N-m. Determine the angular velocity of the drum after C. has rotated 10 revolutions, starting from rest. Gear B and the drum have 100 kg and a radius of gyration about their rotating axis of 250 mm. Gear C has a mass of 30 kg and a radius of gyration about its rotating axis of 125 mm.

18-17. The center O of the thin ring of mass m is given an angular velocity of co{). If the ring rolls without slipping, determine its angular velocity after it has traveled a distance of s down the plane. Neglect its thickness.

18-18. If the end of the cord is subjected to a force of P = 75 lb. determine the speed of the 100-lb block C after P has moved a distance of 4 ft, starting from rest. Pulleys A and B are identical, each of which has a weight of 10 lb and a radius of gyration of k = 3 in. about its center of mass.

1819. When 0 = 0, the assembly is held at rest, and the torsional spring is untwisted. If the assembly is released and falls downward, determine its angular velocity at the instant 0 = 90. Rod AB has a mass of 6 kg, and disk C has a mass of 9 kg.

If P = 200 N and the 15-kg uniform slender rod starts from rest at \(\theta=0^{\circ}\), determine the rod's angular velocity at the instant just before \(\theta=45^{\circ}\).

A yo-yo has a weight of 0.3 lb and a radius of gyration \(k_O=0.06\mathrm{\ ft}\). If it is released from rest, determine how far it must descend in order to attain an angular velocity \(\omega=70\ \mathrm{rad}/\mathrm{s}\). Neglect the mass of the string and assume that the string is wound around the central peg such that the mean radius at which it unravels is r = 0.02 ft.

18-22. If the 50-lb bucket is released from rest, determine its velocity after it has fallen a distance of 10 ft. The windlass A can be considered as a 30-lb cylinder, while the spokes are slender rods, each having a weight of 2 lb. Neglect the pulley's weight.

18-23. The combined weight of the load and the platform is 200 lb, with the center of gravity located at G. If a couple moment of M = 900 lb ft is applied to link AB. determine the angular velocity of links AB and CD at the instant 0 = 60. The system is at rest when 0 = 0. Neglect the weight of the links.

*18-24. The tub of the mixer has a weight of 70 lb and a radius of gyration k$ = 1.3 ft about its center of gravity. If a constant torque M = 60 lb ft is applied to the dumping wheel, determine the angular velocity of the tub when it has rotated 0 = 90. Originally the tub is at rest when 0 = 0.

18-25. The tub of the mixer has a weight of 70 lb and a radius of gyration k(} = 1.3 ft about its center of gravity. If a constant torque M = 60 lb ft is applied to the tub. determine its angular velocity when it has rotated 0 45. Originally the tub is at rest when 0 = 0.

18-26. Two wheels of negligible weight are mounted at corners A and B of the rectangular 75-lb plate. If the plate is released from rest at 0 = 90, determine its angular velocity at the instant just before 0 = 0.

18-27. The 100-lb block is transported a short distance by using two cylindrical rollers, each having a weight of 35 lb. If a horizontal force P = 25 lb is applied to the block, determine the blocks speed after it has been displaced 2 ft to the left. Originally the block is at rest. No slipping occurs.

*18-28. The hand winch is used to lift the 50-kg load. Determine the work required to rotate the handle five revolutions, starting and ending at rest.The gear at A has a radius of 20 mm.

18-29. A motor supplies a constant torque or twist of M = 120 lb ft to the drum. If the drum has a weight of 30 lb and a radius of gyration of k0 = 0.8 ft. determine the speed of the 15-lb crate A after it rises s = 4 ft starting from rest. Neglect the mass of the cord.

18-30. Motor M exerts a constant force of P = 750 N on the rope. If the 100-kg post is at rest when 0 = 0, determine the angular velocity of the post at the instant 0 = 60. Neglect the mass of the pulley and its size, and consider the post as a slender rod.

18-31. The uniform bar has a mass m and length /. If it is released from rest when 0 = 0, determine its angular velocity as a function of the angle 0 before it slips.

*18-32. The uniform bar has a mass m and length /. If it is released from rest when 0 = 0. determine the angle 0 at which it first begins to slip. The coefficient of static friction at O is fxs = 0.3.

18-33. The two 2-kg gears A and B are attached to the ends of a 3-kg slender bar. The gears roll within the fixed ring gear C, which lies in the horizontal plane. If a 10-N m torque is applied to the center of the bar as shown, determine the number of revolutions the bar must rotate starting from rest in order for it to have an angular velocity of u)AH = 20rad/s. For the calculation, assume the gears can be approximated by thin disks. What is the result if the gears lie in the vertical plane?

18-34. A ball of mass m and radius r is cast onto the horizontal surface such that it rolls without slipping. Determine its angular velocity at the instant 0 = 90, if it has an initial speed of vc, as shown.

18-35. A ball of mass m and radius r is cast onto the horizontal surface such that it rolls without slipping. Determine the minimum speed vG of its mass center G so that it rolls completely around the loop of radius R + r without leaving the track.

18-36. At the instant shown, the 50-lb bar rotates clockwise at 2 rad/s. The spring attached to its end always remains vertical due to the roller guide at C. If the spring has an unstretched length of 2 ft and a stiffness of k = 6 lb/ft. determine the angular velocity of the bar the instant it has rotated 30 clockwise.

18-37. At the instant shown, the 50-lb bar rotates clockwise at 2 rad/s. The spring attached to its end always remains vertical due to the roller guide at C. If the spring has an unstretched length of 2 ft and a stiffness of k = 12 lb/ft. determine the angle 0. measured from the horizontal, to which the bar rotates before it momentarily stops.

18-38. The spool has a mass of 50 kg and a radius of gyration k0 = 0.280 m. If the 20-kg block A is released from rest, determine the distance the block must fall in order for the spool to have an angular velocity = 5 rad/s. Also, what is the tension in the cord while the block is in motion? Neglect the mass of the cord.

18-39. The spool has a mass of 50 kg and a radius of gyration k0 = 0.280 m. If the 20-kg block A is released from rest, determine the velocity of the block when it descends 0.5 m.

*18-40. An automobile tire has a mass of 7 kg and radius of gyration k0 = 0.3 m. If it is released from rest at A on the incline, determine its angular velocity when it reaches the horizontal plane.The tire rolls without slipping.

18-41. The system consists of a 20-lb disk A. 4-lb slender rod BC, and a 1-lb smooth collar C. If the disk rolls without slipping, determine the velocity of the collar at the instant the rod becomes horizontal, i.e.. 0 = 0. The system is released from rest when f) = 45.

18-42. The system consists of a 20-lb disk A, 4-lb slender rod BC. and a 1-lb smooth collar C. If the disk rolls without slipping, determine the velocity of the collar at the instant 0 = 30. The system is released from rest when 0 = 45.

The door is made from one piece, whose sides move along the horizontal and vertical tracks. If the door is in the open position, \(\boldsymbol{\theta}=0^{\circ}\), and then released, determine the speed at which its end A strikes the stop at C. Assume the door is a 180-lb thin plate having a width of 10 ft.

*18-44. Determine the speed of the 50-kg cylinder after it has descended a distance of 2 m. starting from rest. Gear A has a mass of 10 kg and a radius of gyration of 125 mm about its center of mass. Gear B and drum C have a combined mass of 30 kg and a radius of gyration about their center of mass of 150 mm.

18-45. The disk A is pinned at O and weighs 15 lb. A 1 -ft rod weighing 2 lb and a 1-ft-diameter sphere weighing 10 lb arc welded to the disk, as shown. If the spring is orginally stretched 1 ft and the sphere is released from the position shown, determine the angular velocity of the disk when it has rotated 90.

18-46. The disk A is pinned at O and weighs 15 lb. A 1 -ft rod weighing 2 lb and a 1-ft-diameter sphere weighing 10 lb are welded to the disk, as shown. If the spring is originally stretched 1 ft and the sphere is released from the position shown, determine the angular velocity of the disk when it has rotated 45u.

18-47. At the instant the spring becomes undeformed, the center of the 40-kg disk has a speed of 4 m/s. From this point determine the distance d the disk moves down the plane before momentarily stopping. The disk rolls without slipping.

*18-48. A chain that has a negligible mass is draped over the sprocket which has a mass of 2 kg and a radius of gyration of k0 = 50 mm. If the 4-kg block A is released from rest in the position s = 1 m. determine the angular velocity of the sprocket at the instant s = 2 m.

18-49. Solve Prob. 18-48 if the chain has a mass of 0.8 kg/m. For the calculation neglect the portion of the chain that wraps over the sprocket.

The compound disk pulley consists of a hub and attached outer rim. If it has a mass of 3 kg and a radius of gyration \(k_{G}=45 \mathrm{\ mm}\). determine the speed of block A after A descends 0.2 m from rest. Blocks A and B each have a mass of 2 kg. Neglect the mass of the cords.

A spring having a stiffness of k = 300 N/m is attached to the end of the 15-kg rod, and it is unstretched when \(\theta=0^{\circ}\). If the rod is released from rest when \(\theta=0^{\circ}\). determine its angular velocity at the instant \(\theta=30^{\circ}\). The motion is in the vertical plane.

*18-52. The two bars are released from rest at the position 0. Determine their angular velocities at the instant they become horizontal. Neglect the mass of the roller at C. Each bar has a mass m and length L.

18-53. The two bars are released from rest at the position 0 = 90. Determine their angular velocities at the instant they become horizontal. Neglect the mass of the roller at C. Each bar has a mass m and length L.

If the 250-lb block is released from rest when the spring is unstretched, determine the velocity of the block after it has descended 5 ft. The drum has a weight of 50 lb and a radius of gyration of \(k_{O}=0.5 \mathrm{\ ft}\) about its center of mass O.

18-55. The 6-kg rod A BCis connected to the 3-kg rod CD. If the system is released from rest when 0 = 0. determine the angular velocity of rod ABC at the instant it becomes horizontal.

*18-56. If the chain is released from rest from the position shown, determine the angular velocity of the pulley after the end B has risen 2 ft.The pulley has a weight of 50 lb and a radius of gyration of 0.375 ft about its axis. The chain weighs 6 Ib/ft.

18-57. If the gear is released from rest, determine its angular velocity after its center of gravity O has descended a distance of 4 ft. The gear has a weight of 100 lb and a radius of gyration about its center of gravity of k = 0.75 ft.

18-58. When the slender 10-kg bar A B is horizontal it is at rest and the spring is unstretched. Determine the stiffness k of the spring so that the motion of the bar is momentarily stopped when it has rotated clockwise 90.

18-59. When the slender 10-kg bar AB is horizontal it is at rest and the spring is unstretched. Determine the stiffness k of the spring so that the motion of the bar is momentarily stopped when it has rotated clockwise 45.

If the 40-kg gear B is released from rest at \(\theta=0^{\circ}\), determine the angular velocity of the 20-kg gear A at the instant \(\theta=90^{\circ}\). The radii of gyration of gears A and B about their respective centers of mass are \(k_{A}=125 \mathrm{\ mm}\) and \(k_{B}=175 \mathrm{\ mm}\). The outer gear ring P is fixed.

18-61. A uniform ladder having a weight of 30 lb is released from rest when it is in the vertical position. If it is allowed to fall freely, determine the angle 0 at which the bottom end A starts to slide to the right of A. For the calculation, assume the ladder to be a slender rod and neglect friction at A.

18-62. The 50-lb wheel has a radius of gyration about its center of gravity G of kc = 0.7 ft. If it rolls without slipping, determine its angular velocity when it has rotated clockwise 90c from the position shown. The spring AB has a stiffness k = 1.20 Ib/ft and an unstretched length of 0.5 ft.The wheel is released from rest.

18-63. The uniform window shade A B has a total weight of 0.4 lb. When it is released, it winds up around the spring-loaded core O. Motion is caused by a spring within the core, which is coiled so that it exerts a torque M = 0.3( I O_3)0 lb ft. where 0 is in radians, on the core. If the shade is released from rest, determine the angular velocity of the core at the instant the shade is completely rolled up. i.e.. after 12 revolutions. When this occurs, the spring becomes uncoiled and the radius of gyration of the shade about the axle at O is kQ = 0.9 in. Note: The elastic potential energy of the torsional spring is Ve = \k02, where M = kOandk = 0.3(10~3) lb ft/rad.

*18-64. The motion of the uniform 80-lb garage door is guided at its ends by the track. Determine the required initial stretch in the spring when the door is open. 0 = 0,so that when it falls freely it comes to rest when it just reaches the fully closed position. 0 = 90. Assume the door can be treated as a thin plate, and there is a spring and pulley system on each of the two sides of the door.

18-65. The motion of the uniform 80-lb garage door is guided at its ends by the track. If it is released from rest at 0 = 0. determine the doors angular velocity at the instant 0 = 30. The spring is originally stretched 1 ft when the door is held open. 0 = 0. Assume the door can be treated as a thin plate, and there is a spring and pulley system on each of the two sides of the door.

18-66. The end A of the garage door AB travels along the horizontal track, and the end of member BC is attached to a spring at C. If the spring is originally unstretched, determine the stiffness k so that when the door falls downward from rest in the position shown, it will have zero angular velocity the moment it closes, i.e.. when it and BC become vertical. Neglect the mass of member BC and assume the door is a thin plate having a weight of 200 lb and a width and height of 12 ft. There is a similar connection and spring on the other side of the door.

18-67. Determine the stiffness k of the torsional spring at A, so that if the bars are released from rest when 0 = 0. bar A# has an angular velocity of 0.5 rad/s at the closed position, 0 = 90. The spring is uncoiled when 0 = 0.The bars have a mass per unit length of 10 kg/m.

*18-68. The torsional spring at A has a stiffness of k 900 N m/rad and is uncoiled when 0 = 0. Determine the angular velocity of the bars. AB and BC. when 0 = 0. if they are released from rest at the closed position. 0 = 90. The bars have a mass per unit length of 10 kg/m.

P18-1. The bicycle and rider start from rest at the top of the hill. Show how to determine the speed of the rider when he freely coasts down the hill. Use appropriate dimensions of the wheels, and the mass of the rider, frame and wheels of the bicycle to explain your results.

P18-2. Two torsional springs. M = kO. are used to assist in opening and closing the hood of this truck. Assuming the springs are uncoiled (0 = 0) when the hood is opened, determine the stiffness k (N -m/rad) of each spring so that the hood can easily be lifted, i.e.. practically no force applied to it. when it is closed in the unlocked position. Use appropriate numerical values to explain your result.

P18-3. The operation of this garage door is assisted using two springs AB and side members BCD. which are pinned at C. Assuming the springs are unstretched when the door is in the horizontal (open) position and ABCD is vertical, determine each spring stiffness k so that when the door falls to the vertical (closed) position, it will slowly come to a stop. Use appropriate numerical values to explain your result.

PI8-4. Determine the counterweight of A needed to balance the weight of the bridge deck when 0 = 0. Show that this weight will maintain equilibrium of the deck by considering the potential energy of the system when the deck is in the arbitrary position 0. Both the deck and AB are horizontal when f) = 0. Neglect the weights of the other members. Use appropriate numerical values to explain this result.