chapter19

chapter19 - PROBLEMS 19—1 The rigid body(slab has a mass...

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Unformatted text preview: PROBLEMS 19—1. The rigid body (slab) has a mass m and is rotating with an angular velocity 1» about an axis passing through the fixed point 0. Show that the momenta of all the particles composing the body can be represented by a single vector having a magnitude va and acting through point P, called the center of percussion, which lies at a distance rp/G = ké/rG/O from the mass center G. Here k6 is the radius of gyration of the body, computed about an axis perpendicular to the plane of motion and passing through G. Prob. 19—1 19—2. At a given instant, the body has a linear momentum L = va and an angular momentum HG = [G to computed about its mass center. Show that the angular momentum of the body computed about the instantaneous center of zero velocity [C can be expressed as Hm = 1mm, where [1C represents the body‘s moment of inertia computed about the instantaneous axis of zero velocity. As shown, the IC is located at a distance rG/lc away from the mass center G. va \G) [ow Prob. 19—2 PROBLEMS 497 19—3. Show that if a slab is rotating about a fixed axis perpendicular to the slab and passing through its mass center G, the angular momentum is the same when computed about any other point P on the slab. “Y A? Prob. 19—3 *19—4. Gear A rotates along the inside of the circular gear rack R. IfA has a weight of 4 lb and a radius of gyration of k3 = 0.5 ft, determine its angular momentum about point C when mm = 30 rad/s and (3) (UR = 0, (b) wR = 20 rad/s. Prob. 19—4 498 CHAPTER 19 PLANAR KINETICS OF A RIGID BODY: IMPULSE AND MOMENTUM 19—5. Solve Prob. 17—55 using the principle of impulse and momentum. 19—6. Solve Prob. 17—54 using the principle of impulse and momentum. 19—7. Solve Prob. 17—69 using the principle of impulse and momentum. *19—8. Solve Prob. 17—80 using the principle of impulse and momentum. 19—9. Solve Prob. 17—73 using the principle of impulse and momentum. 19—10. A flywheel has a mass of 60 kg and a radius of gyration of kg 2 150 mm about an axis of rotation passing through its mass center. If a motor supplies a clockwise torque having a magnitude of M = (52‘) N - m, where t is in seconds, determine the flywheel’s angular velocity in t = 3 5. Initially the flywheel is rotating clockwise at w] = 2 rad/s. 19—11. A wire of negligible mass is wrapped around the outer surface of the 2-kg disk. If the disk is released from rest, determine its angular velocity in 3 s. Prob. 19—11 *19—12. The spool has a mass of 30 kg and a radius of gyration k0 = 0.25 m. Block A has a mass of 25 kg, and block B has a mass of 10 kg. If they are released from rest, determine the time required for block A to attain a speed of 2 m/s. Neglect the mass of the ropes. Prob. 19—12 19—13. The man pulls the rope off the reel with a constant force of 8 lb in the direction shown. If the reel has a weight of 250 lb and radius of gyration k6 = 0.8 ft about the trunnion (pin) at A, determine the angular velocity of the reel in 3 5 starting from rest. Neglect friction and the weight of rope that is removed. Prob. 19—13 19—14. Angular motion is transmitted from a driver wheel A to the driven wheel B by friction between the wheels at C. If A always rotates at a constant rate of 16 rad/s, and the coefficient of kinetic friction between the wheels is Mk = 0.2, determine the time required for B to reach a constant angular velocity once the wheels make contact with a normal force of 50 N. What is the final angular velocity of wheel B? Wheel B has a mass of 90 kg and a radius of gyration about its axis of rotation of kc = 120 mm. Prob. 19—14 19—15. The 4-kg slender rod rests on a smooth floor. If it is kicked so as to receive a horizontal impulse I = 8 N - s at point A as shown, determine its angular velocity and the speed of its mass center. PROBLEMS 499 *19—16. A cord of negligible mass is wrapped around the outer surface of the 50-lb cylinder and its end is subjected to a constant horizontal force of P = 2 lb. If the cylinder rolls without slipping at A, determine its angular velocity in 4 5 starting from rest. Neglect the thickness of the cord. P=21b A.—..N~—_..--..--_.--__-~ Prob. 19—16 19—17. The drum has a mass of 70 kg, a radius of 300 mm, and radius of gyration k0 = 125 mm. If the coefficients of static and kinetic friction at A are ,us = 0.4 and ,uk = 0.3, respectively, determine the drum’s angular velocity 2 s after it is released from rest.Take 0 = 30°. Prob. 19—15 Prob. 19—17 500 CHAPTER 19 19—18. The double pulley consists of two wheels which are attached to one another and turn at the same rate. The pulley has a mass of 15 kg and a radius of gyration k0 = 110mm. If the block at A has a mass of 40 kg, determine the speed of the block in 3 s after a constant force F = 2 kN is applied to the rope wrapped around the inner hub of the pulley. The block is originally at rest. Neglect the mass of the rope. Prob. 19-18 19—19. The spool has a weight of 30 lb and a radius of gyration k0 : 0.45 ft. A cord is wrapped around its inner hub and the end subjected to a horizontal force P = 5 lb. Determine the spool’s angular velocity in 4 3 starting from rest. Assume the spool rolls without slipping. Prob. 19—19 PLANAR KINETICS OF A RIGID BODY: IMPULSE AND MOMENTUM *19——20. The two gears A and B have weights and radii of gyration of WA = 15 lb, k4 = 0.5 ft and W8 = 101b, k3 = 0.35 ft, respectively. If a motor transmits a couple moment to gear .8 of M = 2(1 — 605’) lb ~ fti where {is in seconds, determine the angular velocity of gear A in I = 5 5, starting from rest. Prob. 19—20 19-21. Spool B is at rest and spool A is rotating at 6rad/s when the slack in the cord connecting them is taken up. Determine the angular velocity of each spool immediately after the cord is jerked tight by the spinning of spool A. The weights and radii of gyration of A and B are WA = 301b,kA = 0.8 ft and W3 = 151b,k3 = 0.6 ft, respectively. Prob. 19—21 19—22. A 4-kg disk A is mounted on arm BC, which has a negligible mass. If a torque of M = (5e0'5’) N - m, where tis in seconds, is applied to the arm at C, determine the angular velocity of BC in 2 5 starting from rest. Solve the problem assuming that (a) the disk is set in a smooth bearing at B so that it rotates with curvilinear translation, (b) the disk is fixed to the shaft BC, and (c) the disk is given an initial freely spinning angular velocity of (up = {—80k} rad/s prior to application of the torque. Prob. 19—22 19—23. The inner hub of the wheel rests on the horizontal track. If it does not slip at A, determine the speed of the 10-1b block in 2 s after the block is released from rest. The wheel has a weight of 30 lb and a radius of gyration kc = 1.30 ft. Neglect the mass of the pulley and cord. Prob. 19—23 PROBLEMS 5 01 *19—24. If the hoop has a weight W and radius r and is thrown onto a rough surface with a velocity VG parallel to the surface, determine the amount of backspin, (no, it must be given so that it stops spinning at the same instant that its forward velocity is zero. It is not necessary to know the coefficient of kinetic friction at A for the calculation. Prob. 19—24 19—25. The 10-lb rectangular plate is at rest on a smooth horizontal floor. If it is given the horizontal impulses shown, determine its angular velocity and the velocity of the mass center. y Slb-s Prob. 19—25 502 CHAPTER 19 PLANAR KINETICS OF A RIGID BODY: IMPULSE AND MOMENTUM 19-26. The ball of mass m and radius r rolls along an *19—28. The slender rod has a mass m and is suspended inclined plane for which the coefficient of static friction is n. at its end A by a cord. If the rod receives a horizontal If the ball is released from rest, determine the maximum blow giving it an impulse I at its bottom B, determine the angle 0 for the incline so that it rolls without slipping at A. location y of the point P about which the rod appears to rotate during the impact. Prob. 19—26 Prob. 19—28 19—27. The spool has a weight of 75 lb and a radius of 19—29. Athin rod havingamass of 4 kg is balanced vertically gyration k0 = 1.20 ft. If the block B weighs 60 lb, and a as shown. Determine the height h at which it can be struck force F = 25 lb is applied to the cord, determine the speed with a horizontal force F and not slip on the floor. This requires of the block in 5 5 starting from rest. Neglect the mass of that the frictional force atA be essentially zero. the cord. Prob. 19—27 Prob. 19—29 19—30. The square plate has a mass m and is suspended at its corner A by a cord. If it receives a horizontal impulse I at corner B, determine the location y of the point P about which the plate appears to rotate during the impact. Prob. 19—30 19—31. Determine the height h of the bumper of the pool table, so that when the pool ball of mass m strikes it, no frictional force will be developed between the ball and the table at A. Assume the bumper exerts only a horizontal force on the ball. Prob. 19—31 PROBLEMS 503 *19—32. The double pulley consists of two wheels which are attached to one another and turn at the same rate. The pulley has a mass of 15 kg and a radius of gyration of k0 = 110 mm. If the block at A has a mass of 40 kg and the container at B has a mass of 85 kg, including its contents, determine the speed of the container when t = 3 s after it is released from rest. Prob. 19—32 19—33. The crate has a mass m6. Determine the constant speed 120 it acquires as it moves down the conveyor. The rollers each have a radius of r, mass m, and are spaced d apart. Note that friction causes each roller to rotate when the crate comes in contact with it. Prob. 19—33 512 CHAPTER 19 PLANAR KINETICS OF A RIGID BODY: IMPULSE AND MOMENTUM PROBLEMS 19—34. Two wheels A and B have masses mA and m3, and radii of gyration about their central vertical axes of k A and k3, respectively. If they are freely rotating in the same direction at 00A and tag about the same vertical axis, determine their common angular velocity after they are brought into contact and slipping between them stops. 19-35. The Hubble Space Telescope is powered by two solar panels as shown. The body of the telescope has a mass of 11 Mg and radii of gyration k x = 1.64 m and k\. = 3.85 m, whereas the solar panels can be considered as thin plates, each having a mass of 54 kg. Due to an internal drive, the panels are given an angular velocity of {0.6j} rad/s, measured relative to the telescope. Determine the angular velocity of the telescope due to the rotation of the panels. Prior to rotating the panels, the telescope was originally traveling at VG = {—400i + 250j + 175k} m/s. Neglect its orbital rotation. Prob. 19-35 *19—36. The platform swing consists of a ZOO-lb flat plate suspended by four rods of negligible weight. When the swing is at rest, the 150-lb man jumps off the platform when his center of gravity G is 10 ft from the pin at A. This is done with a horizontal velocity of 5 ft/s, measured relative to the swing at the level of G. Determine the angular velocity he imparts to the swing just after jumping off. Prob. 19—36 19—37. Each of the two slender rods and the disk have the same mass m. Also, the length of each rod is equal to the diameter d of the disk. If the assembly is rotating with an angular velocity 001 when the rods are directed outward, determine the angular velocity of the assembly if by internal means the rods are brought to an upright vertical position. Prob. 19—37 19—38. The rod has a length L and mass m. A smooth collar having a negligible size and one-fourth the mass of the rod is placed on the rod at its midpoint. If the rod is freely rotating at u) about its end and the collar is released, determine the rods angular velocity just before the collar flies off the rod. Also. what is the speed of the collar as it leaves the rod? Prob. 19—38 19—39. A man has a moment of inertia I: about the z axis. He is originally at rest and standing on a small platform which can turn freely. If he is handed a wheel which is rotating at w and has a moment of inertia I about its spinning axis, determine his angular velocity if (a) he holds the wheel upright as shown, (b) turns the wheel out, 0 = 90°, and (c) turns the wheel downward, 0 = 180°. PROBLEMS 51 3 *19—40. The space satellite has a mass of 125 kg and a moment of inertia IZ = 0.940 kg-mz, excluding the four solar panels A, B, C, and D. Each solar panel has a mass of 20 kg and can be approximated as a thin plate. If the satellite is originally spinning about the z axis at a constant rate a)z = 0.5 rad/s when 6 = 90°, determine the rate of spin if all the panels are raised and reach the upward position, 6 = 0°, at the same instant Prob. 19—40 19—41. The 2—kg rod ACB supports the two 4—kg disks at its ends. If both disks are given a clockwise angular velocity (cu/4)] = (wB)1 = 5 rad/s while the rod is held stationary and then released, determine the angular velocity of the rod after both disks have stopped spinning relative to the rod due to frictional resistance at the pins A and 8. Motion is in the horizontal plane. Neglect friction at pin C. Prob. 19—39 Prob. 19-41 514 CHAPTER 19 PLANAR KINETICS OF A RIGID BODY: IMPULSE AND MOMENTUM 19—42. Disk A has a weight of 20 lb. An inextensible cable is attached to the 10-lb weight and wrapped around the disk. The weight is dropped 2 ft before the slack is taken up. If the impact is perfectly elastic, i.e., e = 1, determine the angular velocity of the disk just after impact. Prob. 19—42 19—43. A thin disk of mass m has an angular velocity wl while rotating on a smooth surface. Determine its new angular velocity just after the hook at its edge strikes the peg P and the disk starts to rotate about P without rebounding. Prob. 19—43 *19—44. The pendulum consists of a 5-lb slender rod AB and a 10-lb wooden block. A projectile weighing 0.2 lb is fired into the center of the block with a velocity of 1000 ft/s. If the pendulum is initially at rest, and the projectile embeds itself into the block, determine the angular velocity of the pendulum just after the impact. 2) =1000ft/s ' Prob. 19—44 19—45. The pendulum consists of a slender 2-kg rod AB and 5-kg disk. It is released from rest without rotating. When it falls 0.3 m, the end A strikes the hook S, which provides a permanent connection. Determine the angular velocity of the pendulum after it has rotated 90°. Treat the pendulum’s weight during impact as a nonimpulsive force. Prob. 19—45 PROBLEMS 51 5 19—46. A horizontal circular platform has a weight of *19~48. Two children A and B, each having a mass of 30 300 lb and a radius of gyration kz = 8 ft about the z axis kg, sit at the edge of the merry-go-round which is rotating at passing through its center 0. The platform is free to rotate w = 2 rad/s. Excluding the children, the merry—go-round about the z axis and is initially at rest. A man having a has a mass of 180 kg and a radius of gyration kZ = 0.6 m. weight of 150 lb begins to run along the edge in a circular Determine the angular velocity of the merry-go-round if A path of radius 10ft. If he has a speed of 4 ft/s and maintains jumps off horizontally in the —n direction with a speed of this speed relative to the platform, determine the angular 2 m/s, measured with respect to the merry-go—round. What velocity of the platform. Neglect friction. is the merry-go—round’s angular velocity if B then jumps off horizontally in the +1? direction with a speed of 2m/s, measured with respect to the merry-go-round? Neglect friction and the size of each child. Prob. 19—46 a) = 2 rad /s Prob. 19—48 19-47. The square plate has a weight W and is rotating on 19—49. A 7-g bullet having a velocity of 800 m/s is fired the smooth surface with a constant angular velocity wo. into the edge of the S-kg disk as shown. Determine the Determine the new angular velocity of the plate just after angular velocity of the disk just after the bullet becomes its corner strikes the peg P and the plate starts to rotate embedded in it. Also, calculate how far 9 the disk will swing about P without rebounding. until it stops. The disk is originally at rest. Prob. 19—47 Prob. 19—49 516 CHAPTER 19 19—50. The two disks each weigh 10 lb. If they are released from rest when 6 = 30°, determine 0 after they collide and rebound from each other. The coefficient of restitution is e = 0.75. When 0 = 0°, the disks hang so that they just touch one another. PLANAR KINETICS OF A RIGID BODY: IMPULSE AND MOMENTUM *19—52. The pendulum consists of a 10-lb solid ball and 4—lb rod. If it is released from rest when 61 = 0°, determine the angle (92 after the ball strikes the wall, rebounds, and the pendulum swings up to the point of momentary rest. Take 6 = 0.6. Prob. 19-50 19—51. The 15-lb rod AB is released from rest in the vertical position. If the coefficient of restitution between the floor and the cushion at B is e = 0.7, determine how high the end of the rod rebounds after impact with the floor. 2ft Prob. 19—51 Prob. 19—52 19-53. The plank has a weight of 30 lb, center of gravity at G, and it rests on the two sawhorses at A and B. If the end D is raised 2 ft above the top of the sawhorses and is released from rest, determine how high end C will rise from the top of the sawhorses after the plank falls so that it rotates clockwise about A, strikes and pivots on the sawhorses at B, and rotates clockwise off the sawhorse at A. i l % 3 ft 4.1.5 ft++l.5 MW 3 ft—J Prob. 19—53 19—54. Tests of impact on the fixed crash dummy are conducted using the SOD-lb ram that is released from rest at 6 = 30°, and allowed to fall and strike the dummy at 6 = 90°. If the coefficient of restitution between the dummy and the ram is e = 0.4‘ determine the angle 9 to which the ram will rebound before momentarily coming to rest. Prob. 19—54 19—55. The solid ball of mass m is dropped with a velocity v1 onto the edge of the rough step. If it rebounds horizontally off the step with a velocity v2, determine the angle 6 at which contact occurs. Assume no slipping when the ball strikes the step. The coefficient of restitution is e. Prob. 19—55 PROBLEMS 51 7 *19—56. A solid ball with a mass m is thrown on the ground such that at the instant of contact it has an angular velocity ml and velocity components (v(;)x1 and 06)“ as shown. If the ground is rough so no slipping occurs. determine the components of the velocity of its mass center just after impact. The coefficient of restitution is e. Prob. 19—56 ...
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This note was uploaded on 04/16/2008 for the course CE 325 taught by Professor Docwong during the Spring '08 term at USC.

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chapter19 - PROBLEMS 19—1 The rigid body(slab has a mass...

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