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chapter17 - 394 CHAPTER 17 PLANAR KlNETlCS OF A RIGtD BODY...

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Unformatted text preview: 394 CHAPTER 17 PLANAR KlNETlCS OF A RIGtD BODY: FORCE AND ACCELERATION PROBLEM ..,1Waywamwwflwwwwunwaawfi‘.Tuvalu“.,y ..~y,.w.l.......... “was,“ .... ), ,...,,,, .1”, 17-1. The right circular cone is formed by revolving the 17-3. The solid is formed by revolving the shaded area shaded area around the x axis. Determine the moment of around the y axis. Determine the radius of gyration ky. The inertia I X and express the result in terms of the total mass m specific weight of the material is y a 3801b/ft3. of the cone. The cone has a constant density p. E? B i ' K, y‘ = 9x x L3 in. , Prob. 17-3 17—2. Determine the moment of inertia of the thin ring *17-4. Determine the moment of inertia Ix of the sphere about the z axis. The ring has a mass m. and express the result in terms of the total mass m of the sphere. The sphere has a constant density p. Prob. 17—2 Prob. 17—4 PROBLEMS 3 9 5 17—5. Determine the radius of gyration kx of the 17-7. Determine the radius of gyration kx of the body. The paraboloid. The density of the material is p = 5 Mg/m3. specific weight of the material is y = 3801b/ft3. y T l 100 mm 200 mm ——---% Prob. 17—5 Prob. 17—7 17—6. Determine the moment of inertia of the semiellipsoid *17—8. Determine the moment of inertia of the ellipsoid with respect to the x axis and express the result in terms of with respect to the x axis and express the result in terms of the massm of the semiellipsoid.The material has a constant the mass m of the ellipsoid. The material has a constant density p. density p. Prob. 17-6 Prob. 17—8 396 CHAPTER 17 PLANAR KINETICS OF A RIGID BODY: FORCE AND ACCELERATION 17—9. Determine the moment of inertia of the homogeneous 17—11. Determine the moment of inertia of the thin plate pyramid of mass m with respect to the z axis.The density of the about an axis perpendicular to the page and passing material is p. Suggestion: Use a rectangular plate element through the pin at 0. The plate has a hole in its center. Its havingavolume of dV = (2x)(2y) dz. thickness is 50 mm, and the material has a density of p = 50 kg/m3. Prob. 17—11 Prob. 17-9 17—10. The concrete shape is formed by rotating the *17—12. Determine the moment of inertia I: of the shaded area about the y axis. Determine the moment of frustum of the cone which has a conical depression. The inertia 1),. The specific weight of concrete is y = 150 lb/ft3. material has a density of 200 kg/m3. Prob. 17—10 Prob. 17—12 17—13. Determine the moment of inertia of the assembly about an axis which is perpendicular to the page and passes through the center of mass G. The material has a specific weight ofy = 901b/ft3. 17—14. Determine the moment of inertia of the assembly about an axis which is perpendicular to the page and passes through point 0. The material has a specific weight of y = 90 lb/ft3. Probs. 17—13I14 17—15. The wheel consists of a thin ring having a mass of 10 kg and four spokes made from slender rods, each having a mass of 2 kg. Determine the wheel’s moment of inertia about an axis perpendicular to the page and passing through point A. PROBLEMS 3 97 *17—16. The slender rods have a weight of 3 lb/ ft. Determine the moment of inertia of the assembly about an axis perpendicular to the page and passing through point A. I l 1.5 ft—+— 1.5 ft~4 Prob. 17—16 17—17. Each of the three rods has a mass m. Determine the moment of inertia of the assembly about an axis which is perpendicular to the page and passes through the center point 0. Prob. 17—15 Prob. 17—17 398 CHAPTER 17 17—18. The slender rods have a weight of 3 lb/ft. Determine the moment of inertia of the assembly about an axis perpendicular to the page and passing through point A. 1.5 ft .1 Prob. 17—18 17—19. The pendulum consists of a plate having a weight of 12 lb and a slender rod having a weight of 4 lb. Determine the radius of gyration of the pendulum about an axis perpendicular to the page and passing through point 0. Prob. 17—19 *17—20. Determine the moment of inertia of the overhung crank about the x axis. The material is steel having a density ofp = 7.85 Mg/m3. 17—21. Determine the moment of inertia of the overhung crank about the x’ axis. The material is steel having a density of p = 7.85 Mg/m3. 180 mm Probs. 17—20/21 PLANAR KINETICS OF A RIGID BODY: FORCE AND ACCELERATION 17-22. Determine the moment of inertia of the solid steel assembly about the x axis. Steel has a specific weight of y” = 4901b/ft3. < 2ft 5“ 3ft 4 Prob. 17—22 17—23. The pendulum consists of two slender rods AB and 0C which have a mass of 3 kg/m. The thin plate has a mass of 12 kg/m2. Determine the location y of the center of mass G of the pendulum, then calculate the moment of inertia of the pendulum about an axis perpendicular to the page and passing through G. ‘ B Prob. 17-23 PROBLEMS *17-24. Determine the greatest possible acceleration of the 975~kg race car so that its front tires do not leave the ground or the tires slip on the track. The coefficients of static and kinetic friction are M = 0.8 and lurk = 0.6, respectively. Neglect the mass of the tires. The car has rear- wheel drive and the front tires are free to roll. 17—25. Determine the greatest possible acceleration of the 975-kg race car so that its front wheels do not leave the ground or the tires slip on the track. The coefficients of static and kinetic friction are #5 = 0.8 and .ka = 0.6, respectively. Neglect the mass of the tires. The car has four- wheel drive. Probs. 17-24125 17—26. The Z-lb bottle rests on the check-out conveyor at a grocery store. If the coefficient of static friction is as = 0.2, determine the largest acceleration the conveyor can have without causing the bottle to slip or tip. The center of gravity is at G. Prob. 17—26 PROBLEMS 409 17—27. The assembly has a mass of 8 Mg and is hoisted using the boom and pulley system. If the winch at B draws in the cable with an acceleration of 2 m/sz, determine the compressive force in the hydraulic cylinder needed to support the boom.The boom has a mass of 2 Mg and mass center at G. Prob. 17—27 *17—28. The jet aircraft has a total mass of 22 Mg and a center of mass at G. Initially at take-off the engines provide a thrust 2T = 4 kN and T' = 1.5 kN. Determine the acceleration of the plane and the normal reactions on the nose wheel and each of the two wing wheels located at B. Neglect the mass of the wheels and, due to low velocity, neglect any lift caused by the wings. T’ .5T,\ /G Prob. 17—28 410 CHAPTER 17 17—29. The lift truck has a mass of 70 kg and mass center at G. If it lifts the 120—kg spool with an acceleration of 3 m/sz, determine the reactions of each of the four wheels on the ground. The loading is symmetric. Neglect the mass of the movable arm CD. 17—30. The lift truck has a mass of 70 kg and mass center at G. Determine the largest upward acceleration of the 120—kg spool so that no reaction of the wheels on the ground exceeds 600 N. Probs. 17—29/30 17—31. The door has a weight of 200 lb and a center of gravity at G. Determine how far the door moves in 2 s, starting from rest, if a man pushes on it at C with a horizontal force F = 30 lb. Also. find the vertical reactions at the rollers A and B. *17—32. The door has a weight of 200 lb and a center of gravity at G. Determine the constant force F that must be applied to the door to push it open 12 ft to the right in 5 5, starting from rest. Also, find the vertical reactions at the rollers A and B. l f i l i t i i l E i I l i «WW—E wwwrmmt Probs. 17—31/32 PLANAR KINETICS OF A RIGID BODY: FORCE AND ACCELERATION 17—33. The fork lift has a boom with a mass of 800 kg and a mass center at G. If the vertical acceleration of the boom is 4 m/sz, determine the horizontal and vertical reactions at the pin A and on the short link BC when the 1.25—Mg load is lifted. Prob. 17—33 17—34. The pipe has a mass of 800 kg and is being towed behind the truck. If the acceleration of the truck is a, = 0.5 m/s2, determine the angle 0 and the tension in the cable. The coefficient of kinetic friction between the pipe and the ground is uk 2 0.1. 17—35. The pipe has a mass of 800 kg and is being towed behind a truck. If the angle 0 = 300, determine the acceleration of the truck and the tension in the cable. The coefficient of kinetic friction between the pipe and the ground is ,uk = 0.1. Probs. 17—34/35 *1’7—36. The pipe has a mass of 460 kg and is held in place on the truck bed using the two boards A and 8. Determine the greatest acceleration of the truck so that the pipe begins to lose contact at A and the bed of the truck and starts to pivot about B. Assume board B will not slip on the bed of the truck, and the pipe is smooth. Also, what force does board B exert on the pipe during the acceleration? PROBLEMS 41 1 17—38. The sports car has a mass of 1.5 Mg and a center of mass at G. Determine the shortest time it takes for it to reach a speed of 80 km/h, starting from rest, if the engine only drives the rear wheels, whereas the front wheels are free rolling. The coefficient of static friction between the wheels and the road is M = 0.2. Neglect the mass of the wheels for the calculation. If driving power could be supplied to all four wheels, what would be the shortest time for the car to reach a speed of 80 km/ h? Prob. 17—36 17—37. The drop gate at the end of the trailer has a mass of 1.25 Mg and mass center at G. If it is supported by the cable AB and hinge at C, determine the tension in the cable when the truck begins to accelerate at 5 m/sz. Also, what are the horizontal and vertical components of reaction at the hinge C? Prob. 17—37 Prob. 17—38 17—39. The crate of mass m is supported on a cart of negligible mass. Determine the maximum force P that can be applied a distance d from the cart bottom without causing the crate to tip on the cart. Prob. 17-39 412 CHAPTER 17 *17—40. The car accelerates uniformly from rest to 88 ft/s in 15 seconds. If it has a weight of 3800 lb and a center of gravity at G, determine the normal reaction of each wheel on the pavement during the motion. Power is developed at the rear wheels, whereas the front wheels are free to roll. Neglect the mass of the wheels and take the coefficients of static and kinetic friction to be m = 0.4 and ,th = 0.2, respectively. Prob. 17—40 17—41. Block A weighs 50 lb and the platform weighs 10 lb. If P = 100 lb. determine the normal force exerted by block A on B. Neglect the weight of the pulleys and bars of the triangular frame. Prob. 17—41 PLANAR KINETICS OF A RIGID BODY: FORCE AND ACCELERATION 17-42. The 1.6-Mg car shown has been “raked" by increasing the height of its center of mass to h = 0.2 m. This was done by raising the springs on the rear axle. If the coefficient of kinetic friction between the rear wheels and the ground is Mk = 0.3, show that the car can accelerate ( slightly faster than its counterpart for which h = 0. Neglect the mass of the wheels and driver and assume the front wheels at B are free to roll while the rear wheels slip. Prob. 17—42 17—43. The forklift and operator have a combined weight of 10 000 lb and center of mass at G. If the forklift is used to lift the 2000-lb concrete pipe, determine the maximum vertical acceleration it can give to the pipe so that it does not tip forward on its front wheels. *17—44. The forklift and operator have a combined weight of 10 000 lb and center of mass at G. If the forklift is used to lift the 2000—lb concrete pipe, determine the normal reactions on each of its four wheels if the pipe is given an upward acceleration of 4 ft/sz. Probs. 17—43/44 17—45. The van has a weight of 4500 lb and center of gravity at C”. It carries a fixed 800-1b load which has a center of gravity at G,. If the van is traveling at 40 ft/s, determine the distance it skids before stopping. The brakes cause ail the wheels to lock or skid. The coefficient of kinetic friction between the wheels and the pavement is pk = 0.3. Compare this distance with that of the van being empty. Neglect the mass of the wheels. l +21% s BLG Prob. 17—45 17—46. The “muscle car” is designed to do a “wheeley,” i.e., to be able to lift its front wheels off the ground in the manner shown when it accelerates. If the 1.35-Mg car has a center of mass at G, determine the minimum torque that must be developed at both rear wheels in order to do this. Also, what is the smallest necessary coefficient of static friction assuming the thick-walled rear wheels do not slip on the pavement? Neglect the mass of the wheels. Prob. 17—46 17—47. The bicycle and rider have a mass of 80 kg with center of mass located at G. If the coefficient of kinetic friction at the rear tire is M; = 0.8, determine the normal reactions at the tires A and B, and the deceleration of the rider, when the rear wheel locks for braking. What is the normal reaction at the rear wheel when the bicycle is traveling at constant velocity and the brakes are not applied? Neglect the mass of the wheels. PROBLEMS 41 3 =“17—48. The bicycle and rider have a mass of 80 kg with center of mass located at G. Determine the minimum coefficient of kinetic friction between the road and the wheels so that the rear wheel B starts to lift off the ground when the rider applies the brakes to the front wheel. Neglect the mass of the wheels. lr055 {11,le 0.4 m - Probs. 17-47148 17—49. The dresser has a weight of 80 lb and is pushed along the floor. If the coefficient of static friction at A and B is ILLS : 0.3 and the coefficient of kinetic friction is pk = 0.2, determine the smallest horizontal force P needed to cause motion. If this force is increased slightly, determine the acceleration of the dresser. Also, what are the normal reactions at A and B when it begins to move? 17—50. The dresser has a weight of 80 lb and is pushed along the floor. If the coefficient of static friction at A and B is MS = 0.3 and the coefficient of kinetic friction is ptk = 0.2. determine the maximum horizontal force P that can be applied without causing the dresser to tip over. Lnsnakusna Probs. 17—49/50 414 CHAPTER 17 17—51. The crate C has a weight of 150 lb and rests on the truck elevator for which the coefficient of static friction is M = 0.4. Determine the largest initial angular acceleration a, starting from rest, which the parallel links AB and DE can have without causing the crate to slip. No tipping occurs. Prob. 17—51 PLANAR KINETICS OF A RIGID BODY: FORCE AND ACCELERATION >"17—52. The two 3-lb rods EF and H] are fixed (welded) to the link AC at E. Determine the normal force NE, shear force V5, and moment ME, which the bar AC exerts on FE at E if at the instant 6 = 300 link AB has an angular velocity to = 5 rad/s and an angular acceleration a = 8 rad/s2 as shown. Prob. 17—52 422 CHAPTER 17 PROBLEMS 17——53. The 80-kg disk is supported by a pin at A. If it is released from rest from the position shown, determine the initial horizontal and vertical components of reaction at the pin. Prob. 17—53 17—54. The lO-kg wheel has a radius of gyration kA = 200 mm. If the wheel is subjected to a moment M = (St) N - m, where t is in seconds, determine its angular velocity when t = 3 5 starting from rest. Also, compute the reactions which the fixed pin A exerts on the wheel during the motion. Prob. 17-54 17—55. The fan blade has a mass of 2 kg and a moment of inertia IO = 0.18 kg - m2 about an axis passing through its center 0. If it is subjected to a moment of M = 3(1 — 6—0'2’) N - m, where tis in seconds, determine its angular velocity when I = 4 5 starting from rest. Prob. 17—55 PLANAR KINETICS OF A RIGID BODY: FORCE AND ACCELERATION *17—56. The 10-lb rod is pin connected to its support at A and has an angular velocity a) = 4 rad/s when it is in the horizontal position shown. Determine its angular acceleration and the horizontal and vertical components of reaction which the pin exerts on the rod at this instant. a): 4rad/s >3: m l a l+ 6 ft » Prob. 17—56 17—57. The pendulum consists of a 15-lb disk and a 10-1b slender rod. Determine the horizontal and vertical components of reaction that the pin 0 exerts on the rod just as it passes the horizontal position, at which time its angular velocity is w = 8 rad/s. w = 8 rad/s ie3ftfii Prob. 17—57 0.75 ft ‘ 17-58. The pendulum consists of a uniform 5-kg plate and a 2-kg slender rod. Determine the horizontal and vertical components of reaction that the pin 0 exerts on the rod at the instant 0 = 30", at which time its angular velocity is w = 3 rad/s. Prob. 17—58 17—59. The 10-lb bar is pinned at its center 0 and connected to a torsional spring. The spring has a stiffness k = 5 lb- ft/rad, so that the torque developed is M = (56) lb - ft, where 0 is in radians. If the bar is released from rest when it is vertical at 6 = 90°, determine its angular velocity at the instant 0 = 0°. *17—60. The 10-lb bar is pinned at its center 0 and connected to a torsional spring. The spring has a stiffness k = 5 lb -ft/rad, so that the torque developed is M = (50) lb - ft, where 0 is in radians. If the bar is released from rest when it is vertical at 0 = 90°, determine its angular velocity at the instant 6 = 45°. Probs. 17—59/60 17—61. The 20—kg roll of paper has a radius of gyration kA = 90 mm about an axis passing through point A. It is pin—supported at both ends by two brackets AB. If the roll rests against a wall for which the coefficient of kinetic friction is Mk = 0.2 and a vertical force F = 30 N is applied to the end of the paper, determine the angular acceleration of the roll as the paper unrolls. Prob. 17-61 PROBLEMS 423 17—62. The cylinder has a radius r and mass m and rests in the trough for which the coefficient of kinetic friction at A and B is ,uk. If a horizontal force P is applied to the cylinder. determine the cylinder‘s angular acceleration when it begins to spin. P Prob. 17-62 17—63. The uniform slender rod has a mass of 5 kg. If the cord at A is cut. determine the reaction at the pin 0. (a) when the rod is still in the horizontal position, and (b) when the rod swings to the vertical position. Prob. 17—63 *17—64. The bar has a mass m and length I. If it is released from rest from the position 0 = 30°. determine its angular acceleration and the horizontal and vertical components of reaction at the pin 0. Prob. 17—64 424 CHAPTER 17 17—65. The kinetic diagram representing the general rotational motion of a rigid body about a fixed axis at 0 is shown in the figure. Show that [Ga may be eliminated by moving the vectors m(aG), and m(aG),, to point P, located a distance rap = ké/rOG from the center of mass G of the body. Here kc represents the radius of gyration of the body about G. The point P is called the center of percussion of the body. N P, / // meg), \ I a G G r ,/m(ac). 0” 0 ’06 Prob. 17—65 17—66. Determine the position rP of the center of percussion P of the 10—lb slender bar. (See Prob. 17—65.) What is the horizontal force Ax at the pin when the bar is struck at P with a force of F = 20 lb? 4ft Prob. 17—66 PLANAR KINETICS OF A RIGID BODY: FORCE AND ACCELERATION 17—67. The 4-kg slender rod is supported horizontally by a spring at A and a cord at B. Determine the angular acceleration of the rod and the acceleration of the rods mass center at the instant the cord at B is cut. Him: The stiffness of the spring is not needed for the calculation. Prob. 17—67 *17—68. In order to experimentally determine the moment of inertia [G of a 4-kg connecting rod, the rod is suspended horizontally at A by a cord and at B by a bearing and piezoelectric sensor. an instrument used for measuring force. Under these equilibrium conditions, the force at B is measured as 14.6 N. If, at the instant the cord is released, the reaction at B is measured as 9.3 N, determine the value of [0. The support at B does not move when the measurement is taken. For the calculation. the horizontal location of G must be determined. Prob. 17—68 17—69. The lO-lb disk D is subjected to a counterclockwise moment of M = (10t) 1b - ft, where t is in seconds. D...
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