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Ch17hw9 - 1 1 14 Plane M°II°" °I Rigid BOdIes Energy...

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Unformatted text preview: 1 1 14 Plane M°II°" °I Rigid BOdIes: Energy and 17.74 and 17.75 A 240-mm-radius cylinder of mass 8 kg rests 011;.‘a', 1. M°memum Melh°ds 3-kg carriage. The system is at rest when a force P of magnitude.__- 10 N is applied as shown for 1.2 5. Knowing that the cylinder rolls_, ' without sliding on the carriage and neglecting the mass of the -_ ' I wheels of the carriage, determine the resulting velocity of (a) the carriage, (b) the center of the cylinder. Fig. P1174 Fig. P17.75 17.76 In the gear arrangement shown, gears A and C are attached to rod ABC, which is free to rotate about B, while the inner gear B is ' fixed. Knowing that the system is at rest, determine the magnitude of the couple M which must be applied to rod ABC, if 2.5 5 later I the angular velocity of the rod is to be 240 rpm clockwise. Gears A and C weigh 2.5 lb each and may be considered as disks of Fig. P1176 radius 2 in.; rod ABC weighs 4 lb. 17.77 A sphere of radius r and mass m is placed on a horizontal floor I with no linear velocity but with a clockwise angular velocity coo. Denoting by M the coefficient of kinetic friction between the sphere and the floor, determine (a) the time t1 at which the sphere will start rolling without sliding, (b) the linear and angular veloci- I I ties of the sphere at time t1. | . L h - 17.78 A sphere of radius r and mass m is projected along a rough 1 F'g' P 17 '7 7 horizontal surface with the initial velocities shown. If the final I velocity of the sphere is to be zero, express (a) the required mag— I I nitude of (no in terms of 00 and r, (b) the time required for the I sphere to come to rest in terms of no and the coefficient of kinetic ‘ | | friction ,uk. Fig. Pl7.78 17.79 A 2.5-lb disk of radius 4 in. is attached to the yoke BCD by means of short shafts fitted in bearings at B and D. The 1.5-lb yoke has a H I radius of gyration of 3 in. about the x axis. Initially the assembly is rotating at 120 rpm with the disk in the plane of the yoke (0 = 0). If the disk is slightly disturbed and rotates with respect to the yoke | until 0 = 90°, where it is stopped by a small bar at D, determine Fig. P'I7.79 the final angular velocity of the assembly. 17.80 Two panels A and B are attached with hinges to a rectangular plate "05'6"“ 'I 'I 15 and held by a wire as shown. The plate and the panels are made i of the same material and have the same thickness. The entire assem- = - bly is rotating with an angular velocity (00 when the wire breaks. -' - Determine the angular velocity of the assembly after the panels have come to rest against the plate. 17.81 A 1.6—kg tube AB can slide freely on rod DE which in turn can rotate freely in a horizontal plane. Initially the assembly is rotating with an angular velocity (u = 5 rad/s and the tube is held in posi— tion by a cord. The moment of inertia of the rod and bracket about the vertical axis of rotation is 0.30 kg - m2 and the centroidal moment of inertia of the tube about a vertical axis is 0.0025 kg ‘ m2. If the cord suddenly breaks, determine (a) the angular velocity of the assembly after the tube has moved to end E, (b) the energy lost i during the plastic impact at E. Fig. ”7.80 I l I Fig. PI7.8I l 17.82 Two 0.8-lb balls are to be put successively into the center C of the '- ‘ slender 4-lb tube AB. Knowing that when the first ball is put into '_ the tube the initial angular velocity of the tube is 8 rad/s and - - . neglecting the effect of friction, determine the angular velocity of ' the tube just after (a) the first ball has left the tube, (19) the second I ball has left the tube. .. ‘ I | ‘ | | . I I l | Fig. P1182 '- l 17.87 The circular platform A is fitted with a rim of ZOO-mm inner radius Pmb'ems 'I 'I 17 '_ and can rotate freely about the vertical shaft. It is known that the .. l- platform-rim unit has a mass of 5 kgand a radius of gyration of _ l- 175 mm with respect to the shaft. At a time when the platform is rotating with an angular velocity of 50 rpm, a 3—kg disk B of radius I 80 mm is placed on the platform with no velocity. Knowing that disk !' B then slides until it comes to rest relative to the platform against : the rim, determine the final angular velocity of the platform. 200 mm Fig. P17.87 17.88 A small 2—kg collar C can slide freely on a thin ring of mass 3 kg and radius 250 mm. The ring is welded to a short vertical shaft, which can rotate freely in a fixed bearing. Initially the ring has an angular velocity of 35 rad/s and the collar is at the top of the ring (6 = 0) when it is given a slight nudge. Neglecting the effect of . friction, determine (a) the angular velocity of the ring as the collar passes through the position 0 = 90°, ([9) the corresponding velocity of the collar relative to the ring. Fig. P1288 ' I i 17.89 Collar C has a mass of 8 kg and can slide freely on rod AB, which in turn can rotate freely in a horizontal plane. The assembly is rotating with an angular velocity 0) of 1.5 rad/s when a spring located between A and C is released, projecting the collar along the rod with an initial relative speed or = 1.5 m/s. Knowing that the combined mass moment of inertia about B of the rod and spring is 1.2 kg - m2, determine (a) the minimum distance between the collar and point B in the ensuing motion, (19) the corresponding angular velocity of the assembly. Fig. P1189 17.90 In Prob. 17.89, determine the required magnitude of the initial relative speed or if during the ensuing motion the minimum distance between collar C and point B is to be 300 mm. 17.96 A bullet weighing 0.08 lb is fired with a horizontal velocity of 1800 ft/s'i into the lower end of a slender 15-lb bar of length L = 30 in. Know:- ing that h = 12 in. and that the bar is initially at rest, determine... (a) the angular velocity of the bar immediately after the bullé'tl. becomes embedded, (b) the impulsive reaction at C, assuming" that the bullet becomes embedded in 0.001 s. ""'_ Fig. P1796 In Prob. 17.96, determine (a) the required distance h if the impu_1_-_,: - sive reaction at C is to be zero, (19) the corresponding angular}. velocity of the bar immediately after the bullet becomes; embedded. A 45—g bullet is fired with a velocity of 400 m/s at 0 = 30° into a 9-k'g7 square panel of side 19 = 200 mm. Knowing that h = 150 mm and: that the panel is initially at rest, determine (a) the velocity of thef center of the panel immediately after the bullet becomes embedded; (b) the impulsive reaction at A, assuming that the bullet become; embedded in 2 ms. Fig. P1798 and P1799 A 45—g bullet is fired with a velocity of 400 m/s at 9 = 5° into a 9-kg- square panel of side I) = 200 mm. Knowing that h = 150 mm and that the panel is initially at rest, determine (a) the required distance h if the horizontal component of the impulsive reaction at A is to be;- zero, (19) the corresponding velocity of the center of the panel imme=' diately after the bullet becomes embedded. Plane Motion of Rigid Bodies: Energy and _ I ‘28 Momentum Methods Fig. P17.106 17. 104 17.107 A uniform slender bar of length L and mass m is supported by a frictionless horizontal table. Initially the bar is spinning about its mass center G with a constant angular velocity to}. Suddenly latch D is moved to the right and is struck by end A of the bar. Assum- ing that the impact of A and D is perfectly plastic, determine the angular velocity of the bar and the velocity of its mass center immediately after the impact. Fig. “7.104 Solve Prob. 17.104, assuming that the impact of A and D is per- fectly elastic. A uniform slender rod of length L is dropped onto rigid supports at A and B. Since support B is slightly lower than support A, the rod strikes A with a velocity 31 before it strikes B. Assuming per- fectly elastic impact at both A and B, determine the angular veloc- ity of the rod and the velocity of its mass center immediately after the rod ((1) strikes support A, (b) strikes support B, (0) again strikes support A. A uniform slender rod AB is at rest on a frictionless horizontal table when end A of the rod is struck by a hammer which delivers an impulse that is perpendicular to the rod. In the subsequent motion, determine the distance I) through which the rod will move each time it completes'a full revolution. Fig. 917.107 , 17.108 rim 17.108 A uniform sphere of radius r rolls down the incline shown without slipping. It hits a horizontal surface and, after Slipping for a while, it starts rolling again. Assuming that the sphere does not bounce as it hits the horizontal surface, determine its angular velocity and the velocity of its mass center after it has resumed rolling. Fig. “7.108 17.109 The slender rod AB of length L forms an angle B with the vertical 17.110 17.111 as it strikes the frictionless surface shown with a vertical velocity 371 and no angular velocity. Assuming that the impact is perfectly elastic, derive an expression for the angular velocity of the rod immediately after the impact. Fig. ”7.109 Solve Prob. 17.109, assuming that the impact between rod AB and the frictionless surface is perfectly plastic. A uniformly loaded rectangular crate is released from rest in the position shown. Assuming that the floor is sufficiently rough to prevent slipping and that the impact at B is perfectly plastic, deter- mine the smallest value of the ratio (1/19 for which corner A will remain in contact with the floor. Fig. ”7.1" Problems 1129 I 'l 132 Plane Motion of Rigid Bodies: Energy and 17.123 Momentum Methods 17.124 1 17.125 L—+ L—+l Fig. P17.125 17.126 17. 127 Fig. P17. 127 A slender rod AB is released from rest in the position 5110an swings down to a vertical position and strikes a second and-ide-t1 cal rod CD which is resting on a frictionless surface. Assn-m111 that the coefficient of restitution between the rods is 0.5, d titer mine the velocity of rod CD immediately after the impact. ‘ ' Fig. P17.123 Solve Prob. 17.123, assuming that the impact between the rods.1s perfectly elastic. ‘ The plank CDE has a mass of 15 kg and rests on a small pivot at 12. The 55-kg gymnast A is standing on the plank at C when the 70_-'kg' gymnast B jumps from a height of 2.5 m and strikes the plank at E. Assuming perfectly plastic impact and that gymnast A is stand-I ing absolutely straight, determine the height to which gymnast ... will rise. Solve Prob. 17.125, assuming that the gymnasts change places: so that gymnast A jumps onto the plank while gymnast B stands at C. and 17.128 Member ABC has a mass of 2.4 kg and is attached to a pin support at B. An 800-g sphere D strikes the end of member ABC with a vertical velocity V] of 3 m/s. Knowing that L = 750 mm? and that the coefficient of restitution between the sphere and-. member ABC is 0.5, determine immediately after the impact (a) the angular velocity of member ABC, (1)) the velocity of the- sphere. Fig. PI7.'I28 ...
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