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chap_V_hwk - Chapter V: Rigid Body Kinetics Homework...

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Unformatted text preview: Chapter V: Rigid Body Kinetics Homework Chapter V Rigid Body Kinetics Homework V-1 V-2 Chapter V: Rigid Body Kinetics Homework Chapter V: Rigid Body Kinetics Homework V-3 Rigid Body Kinetics Homework Problems ME 274 Problem V.1 Problem V-1 Given: is traveling along a along a horizontal roadway with a of v .of v0 . A some point in time the A truck A truck is traveling horizontal roadway with a speed speed A some point in time truck driver starts braking with the truck experiencing a constant0acceleration a. A homogeneous, the truck driver starts braking with the truck experiencing a constant acceleration a. A triangular block of mass M rests on the bed of the truck. Assume that the block does not slip on homogeneous, triangular block of mass M rests on the bed of the truck. Determine the the truckbed during braking. normal reactions at supports A and B of the block during the braking. Assume that the block Determine the the truckbed during braking. Find: does not slip onnormal reactions at supports A and B of the block during the braking. b h A B Use the followingparameters in your analysis: M = 400kg , h = = mmb ter3m and a = 6m/sec2 . Use the following parameters in your analysis: M = 300 kg , h 5 3 , e = s , b = 2 meters and a = 4 m / sec 2 . V-4 Chapter V: Rigid Body Kinetics Homework Rigid Body Problem V.2 Kinetics Homework Problems ME 274 Given: A thin, homogeneous bar of mass M is supported by parallel cables AC and BD, as shown below Problem V-2The bar is released from rest with the cables at an angle of θ measured from in the figure. A thin, homogeneous bar of mass M is supported by parallel cables AC and BD, as the horizontal. shown below in the figure. The bar is released from rest with the cables at an angle of ! Find: measured from the horizontal. DetermineBD on release. cables AC and BD on release. Determine the tension in cables AC and the tension in g C ! D L L B A 3L/4 L/4 Use the following parameters in your analysis: M = 300kg , L = 2.5m and θ = 30◦ . Use the following parameters in your analysis: M = 250 kg , L = 3 meters and ! = 53.13° . Chapter V: Rigid Body Kinetics Homework mework Problems Problem V.3 V-5 ME 274 Given: A uniform slender rod is at rest upon a frictionless horizontal surface when a force F is applied in the direction shown. Assume that the rod has mass m and length of L. r rod is at rest upon a frictionless horizontal surface when a force F is ection shown. Assuming that the rod has mass m and length ofrods center of mass G, and b) the rod’s angular Find: Determine: a) the acceleration of the L, acceleration ofacceleration. the rod’s center of mass G and (b) the rod’s angular ◦ Use 5 lb, F = 15 parameters ft. : ! = 36.87°, mg =the followinglb and L = 2 in your analysis: θ = 30 , mg = 5lb, F = 20lb and L = 3f t. V-6 Chapter V: Rigid Body Kinetics Homework Rigid Body Kinetics Homework Problems ME 274 Problem V.4 Problem V-4 Given: The system shown below consists of a uniform slender bar of mass m and an attached The system shown m. Point G represents the midpoint (center of mass) of the bar. The particle of equal mass below consists of a uniform slender bar of mass m and an attached system particle from rest in the Point G represents the midpoint (center of mass) of the bar. If is released of equal mass m.position shown. this system is released from rest in the position shown, determine (a) the value of x for which the system’s angular acceleration is the system’s (b) the acceleration is maximum, and Find: Determine: a) the value of x for whichmaximum andangular corresponding angular b) acceleration. the corresponding angular acceleration. L/2 L/2 x O G m m A gid Body Kinetics HChapter V: Rigid omework Problems ME 274 Body Kinetics Homework V-7 roblem V-5 Problem V.5 pallet/winch/roller system is used to move crate B of mass M, where the pallet A has a Given: A ass of m. Determine the pallet/winch/roller system is used to move crate B of mass M, where the pallet A has a mass of m. a) maximum winch force that can be applied to the pallet such that the crate does not tip and Find: Determine: a) the maximum winch force that can be applied to the pallet such that the crate does not friction required between the crate and pallet to ensure b) the coefficient of statictip, and b) the coefficient of static friction required between the crate and pallet to ensure that tipping occurs that tipping occurs prior to slip. prior to slip. d1 d1 d2 G d2 B A Use = 4 kg, M parameters 0 your and d = m m. se the following: mthe1following= 65 kg, d1 =in .3 m, analysis: 0.6= 16kg , M = 80kg , d1 = 0.3m and d2 = 0.6m. 2 V-8 Chapter V: Rigid Body Kinetics Homework netics Homework Problems V-6 ME 274 Problem V.6 Given: The center of mass G of the disk shown below (mass m) is offset by a distance e from the of mass G of the disk shown below (mass position shown,aG is directly above O as the wheel rolls without slip geometric center O. At the m) is offset by distance e from the center O. If Gthrough the bottom of a as the wheel rolls without slip through is in the position shown circular path shown. of a circular path shown, find the normal force N that is exerted by the path on Find: For this position, determine the normal force N that is exerted by the path on the disk. C ! R e O G r Use the following e = 0.05 m, ! = 3 analysis: R = 1. m, llowing: R = 1.25 m, r = 0.25 m, parameters in yourrad/s, and m = 505kg. r = 0.2m, e = 0.1m, ω = 4rad/sec and m = 75kg . Chapter V: Rigid Body Kinetics Homework V-9 netics Homework Problems V-7 ME 274 Problem V.7 Given: The system shown below consists of a uniform slender bar of mass m and length L and an shown belowattached of a uniform slender barThemass m and length L and rest in the position shown with the consists particle of equal mass. of system is released from particle of equal mass. If the system is released from mass of the roller, and assume its radius is negligible. roller resting on the incline. Neglect the rest in the position h the roller resting on the incline, determine the initial acceleration of A. lect the mass of the roller and assume its radius is negligible. Find: Determine the initial acceleration of A. A ! L, m m V-10 Chapter V: Rigid Body Kinetics Homework Problem V.8 igid Body Kinetics Homework Problems ME 274 roblem V-8 Given: System shown with mA and mB being the masses of pallet A and homogeneous disk B, respectively. ind the maximum force T that can be applied to the pallet A by the winch such that the Find: not the maximum to roll. omogeneous disk willFindslip as it beginsforce T that can be applied to the pallet A by the winch such that the homogeneous disk will not slip as it begins to roll. µ s, µk B r O A Use the following parameters r your analysis: mA and kg 0.25. se the following: mA = 20 kg, mB = 150 kg,in = 0.15 m, µs = 0.3, = 25µk ,=mB = 175kg , r = 0.15m, µs = 0.3 and µk = 0.25. Problem V-9 A thin, homogeneous bar AB (having a length of L and mass m) is pinned to ground at pointChapter V: Rigid a distance of L/4 from end A). A homogeneous disk of mass M O (where O is at Body Kinetics Homework and radius r is welded to the bar AB with end B being at the center of the disk. A constant forceProblemend A in such a way that the line of action of F is always perpendicular to F acts at V.9 OA. At instant “1”, the system is released from rest with AB being horizontal. V-11 Given: A thin, homogeneous bar system made up of of L and mass m) is pinned to ground at a) Draw a free body diagram of theAB (having a lengthbar AB and the disk. List point O (where O is at a distance of L/4 from end A). A homogeneous disk of mass M and radius r which, if any, nonconservative forces do work on this system. is welded to the bar AB with end B being at the center of the disk. A constant force F acts at end b) Determine thethat the velocity of bar ABis always perpendicular to OA. The system is released A in such a way angular line of action of F when AB reaches its vertical position (instant “2”). Note that since the from rest with AB being horizontal.disk is welded to AB, the disk has the same angular velocity as bar AB. Find: For this problem: a) determine the angular disk as particle of mass M reaches c) Repeat your calculations in b) but here treat the velocityaof bar AB when AB (that its vertical position, and b) ! 0 is, consider r repeat your calculations in a) but here treat the disk as a particle of mass M. ). F M r m A L/4 O 3L/4 B g VERTICAL PLANE Use the following parameters in your analysis: L = rs r 1 1 e m = 40kg , , Use the following parameters in your analysis: L = 2.5 mete3m,, r == mmt,er , m = 40 kgM = 50kg and M = F = g and. F = 100 Newtons . 50 k 100N V-12 Chapter V: Rigid Body Kinetics Homework Rigid Body Kinetics Homework Problems ME 274 Problem V.10 Problem V-10 A A thin, homogeneous bar mass M is supported by p parallel cables AC and BD, as Given: thin, homogeneous bar of of mass M is supported byarallel cables AC and BD, as shown hown figure. The figure. The bar is r rest with the cables at an angle at θ angle of below sin the below in thebar is released fromeleased from rest with the cables of an= θ0 measured from the=horizontal. ! ! 0 measured from the horizontal. a D problem: a) the speed of the center of of the the bar ! = θ = Find: For )this etermine the speed of the center of massmass ofbar whenwhen90° . 90◦ , and b) the tension inbcables AC andthe tension θ = 90◦ . AC and BD when ! = 90° . ) Determine BD when in cables g C ! D L L B A 3L/4 L/4 Use the following parameters in your analysis: M = 120kg , L = 3.5m and θ0 = 30◦ . Use the following parameters in your analysis: M = 100 kg , L = 4 meters and ! 0 = 36.87° . Chapter V: Rigid Body Kinetics Homework V-13 Problem V.11 Given: ConsiderBody Kinetics Homework Problems Rigid the three systems (each of total mass m) shown below where: (I) 274 ME made up on a particle of mass m supported within a massless ring by massless spokes, (II) made up of identical, P total mass thin spokes of roblem V-11 m in a massless ring, and (III) a thin ring of mass m. Each system is Consider the three systems (each of total mass m) shown below where: (I) made up on a released from prest, with eachssystem rolling withoutring by massless spokes, (II) made up of article of mass m upported within a massless slipping on a rough incline. identical, thin spokes of total mass m in a massless ring, and (III) a thin ring of mass m. Each the velocity of the mass center O of each system when each system has made Find: Determine system is released from rest, with each system rolling without slipping on a rough ncline. Determine the velocity of the inclines. Rank the speeds of O system two complete irevolutions in rolling down the mass center O of each system when eachfor the three systems. has made two complete revolutions in rolling down the inclines. Rank the speeds of O for Explain in words why the speeds endwords why those rankings.with those rankings. up with the speeds end up the three systems. Explain in g O O R no slip ! (I) massless ring, massless spokes and particle of mass m O R R no slip ! (II) massless ring and identical spokes (with total mass of spokes m) no slip ! (III) thin ring of mass m V-14 Chapter V: Rigid Body Kinetics Homework Problem V.12 Given: A solid drum having a mass of 2m and centroidal radius of gyration kO is pinned to ground ME 274 at its center of mass O. A cable is wrapped around the outer radius (2R) of the drum supports block A (having a mass of 3m). A second cable is wrapped around the inner radius (R) of the drum supports block B (having a mass of m). Assume that the cables do not slip on the drum as the system moves. The system is released from rest. netics Homework Problems V-12 m having a mass of 2m and centroidal radius of gyration kO is pinned to ts center of mass O. A cable is wrapped around theof the drum after the the has rotated through an angle of θ. Find: Determine the angular speed outer radius (2R) of drum orts block A (having a mass of 3m). A second cable is wrapped around the s (R) of the drum supports block B ass of m). Assume that the cables 2m on the drum as the system moves. is released from rest at instant “1’. 2R w a free body diagram (FBD) of the O tem made up of the drum and g ticles A and B. List which, if any, R conservative forces do work on this tem. termine the angular speed of the m after the drum has rotated through angle of !. A m B 3m lowing parameters theyour analysis: m = 50 kg ,your analysis:ter ,= 60kg , R = 0.5m, k = 0.6m and θ = 180◦ . Use in following parameters in R = 0.75 me m 0 eters and ! = 180° . Rigid Body Kinetics Homework Problems ME 274 Chapter V:V-13 Body Kinetics Homework Problem Rigid V-15 One end of an inextensible cable is attached to a rigid wall at B. This cable is wrapped around a homogeneous disk of mass m and outer radius R with the free end of the cable Problem V.13 remaining horizontal at all time. A constant horizontal force F acts at the free end A of the cable. A spring of stiffness k is attached between the center of the disk O and ground. Given: One end of an inextensible cable is attached to a rigid wall at B. This cable is wrapped As the disk moves to the left, it does not slip on the cable. The system is released from around a homogeneous disk of mass m and outer radius R with the free end of the cable remaining rest at instant “1’ with the spring unstretched. horizontal at all time. A constant horizontal force F acts at the free end A of the cable. A spring of stiffness k is attached between the center thethe disk O which, if any, As the disk moves to the a) Draw a free body diagram (FBD) of of disk. List and ground. nonconservative left, it does not slip on the this system. system is released from rest with the spring unstretched. forces do work on cable. The b) Determine the speed of the center of the disk O after O has moved through a Find: Determine the speed of the center of the disk O after O has moved through a distance d to distance d to the right. the right. A R k F no slip O no slip B C Use the following parameters in your analysis: m = 50kg , R = 0.6m , e s Use the following parameters in your analysis: m = 60 kg , R = 0.5 metkr=, 250N/m, F = 2200N k = 300 N / m and d = 1.75m., F = 2000 N and d = 2 meters . Rigid Body Kinetics Homework Problems ME 274 V-16 Chapter V: Rigid Body Kinetics Homework Problem V-14 A thin homogeneous bar OA having a length of L and mass m is pinned to ground at O. A Problem sV.14 thin homogeneous bar AB (having a length of 2L and mass 2m) is pinned to bar econd OA at A, and end B of the bar is constrained to move within a smooth, horizontal track. Given: A thin homogeneous bar OA having a length of L and mass m is pinned to ground at O. A constant force F acts horizontally at end B. The bar is released from rest at instant “1” A second thin homogeneous bar AB (having a length of 2L and mass 2m) is pinned to bar OA at with link OA being horizontal and with pin B being directly below pin O. A, and end B of the bar is constrained to move within a smooth, horizontal track. A constant force a) D at end B. body diagram of the system with up OA being horizontal List F acts horizontally raw a freeThe bar is released from rest madelink of links OA and AB. and with which, if pin nonconservative forces do pin B being directly below any,O, as shown in the figure. work on this system. b) Determine the speed of end B of bar AB when the system has reached a position Find: Determine the speed of end B of bar AB when the system has reached a position of link OA of link OA being vertical (instant “2”). being vertical. m L O g A 2m 2L F B VERTICAL PLANE Use the following parameters in your analysis: F = 600lb, L = 3f t and mg = 200lb. Use the following parameters in your analysis: F = 500 lb , L = 4 ft and mg = 200 lb . Chapter V: Rigid Body Kinetics Homework V-17 Rigid Body KineticsProblemProblems Homework V.15 ME 274 Given: The center of the homogeneous disk of mass m shown below has an initial velocity of v0 Problem V-15 up the slope. The center of the homogeneous disk of mass m shown below has an initial velocity of v0 Find: Find the of the of the disk after an elapsed time of ∆t, assuming that up the slope. Find the velocityvelocitydisk after an elapsed time of !t, assuming that the the disk rolls without slip. disk rolls without slip. O r ! ◦ Use m 8 kg, r = 0.3 m, ! = in your analysis: / = and Use the following: the=following parameters 36.87°, v0 = 36 m msec 10kg ,!rt = 105secs . = 30 , v0 = 30m/sec and = 0. m , θ ∆t = 12sec. V-18 etics Homework Problems Problem V.16 Chapter V: Rigid Body Kinetics Homework ME 274 -16 Given: A homogeneous, slender bar of mass M length L moves is sliding across a frictionless eous, slender horizontal surface with L moves vG slidingit is suddenly struck at its end by a particle of mass bar of mass M length a speed is when across a frictionless m moving with velocity vm in the direction shown. Assume that the particle sticks to the bar surface with a speed vG when it is suddenly struck at its end by a particle of immediately on impact. ving with velocity vm in the direction shown. Assuming that the particle e bar, determine the velocity of the bar’s center of mass G and the angular Find: Determine the velocity of the bar’s center of mass G and the angular velocity of the bar the bar post-impact. post-impact. lowing: M = 3 Ug, L = 0.5 m, vG = 4 m/s, m = 0.3 kg and vm = 20 m/s.kg , L = 0.75m, vG = 5m/sec, m = 0.4kg k se the following parameters in your analysis: M = 4 and vm = 25m/sec. Chapter V: Rigid Body Kinetics Homework omework Problems V-19 ME 274 Problem V.17 Given: The skater begins to spin in place with his arms extended, and then increases his rate of rotation his arms extended, and then increases his ns to spin in place with by bringing his arms closer to his body. rate of ging his arms closer to his body. Estimate the skater’s rotational speed his arms, if heFind: Estimate the skaters rotational speed afterahe retracts his arms, if he started rotating, with started rotating, with his arms fully extended, at rate of his arms fully extended, atas uniformω1 = 1rev/s. tInitially, you can model the extended arms as itially, you can model the extended arms a rate of slender rods hat are uniform slender rods that are 28 inches long inches in nd weigh 24 lb each and the torso as a 124 lb cylinder, 18and weigh 24 lb each and the torso as a 124 lb cylinder, 18 inches in diameter. be modeled as are retracted, the his arms are retracted, the skater can After his arms a solid cylinder 18skater can be modeled as a solid cylinder 18 inches ter and weighing 172 lb. in diameter and weighing 172 lb. V-20 Body Kinetics Homework Problems Problem V.18 Chapter V: Rigid Body Kinetics Homework ME 274 blem V-18 Given: The slender bar, shown below, of mass m and length L has a downward velocity vG at the instant shown. Assume that the pivot at the end of bar becomes attached to the support after slender bar, shown below, of mass m and length L has a downward velocity at the falling h. nt shown. Assuming that the pivot at the end of bar becomes attached to the support falling h, determine the angular the angular the bar immediately after impact. Find: Determine velocity of velocity of the bar immediately after impact. Use the following parameters in your analysis: m = 4kg , L = 0.4m, vG = 3m/sec and h = 4m. the following: m = 3 kg, L = 0.3 m, vG = 4 m/s and h = 3 m. Chapter V: Rigid Body Kinetics Homework V-21 Problem V.19 Given: Homogneous bar AB of mass m is supported by a base AC that is translating to the left with a constant acceleration of a. Cable AC restrains the rotation of the bar as it translates to the s Homework Problems ME 274 left. 9 (6/8) Find: Determine: a) the tension in the cable, and b) the reaction force acting on bar AB at A. ar AB of mass m is supported by a s translating to the left with a constant a. Cable AC restrains the rotation of nslates to the left. Determine sion in the cable. ction force acting on bar AB at A. Use the following parameters in your analysis: m = 5kg and a = 0.6g . ing parameters in your analysis: m = 5 kg and a = 0.6 g . V-22 Chapter V: Rigid Body Kinetics Homework Problem V.20 mework Problems /40) Given: A constant frictional torque from the bearings at O resist the motion of the pulley system ME 274 shown. The mass and radius of gyration of the pulley are known to be m and kO , respectively. Find: Determine the angular acceleration of the pulley on release. al torque M f from the bearings at O f the pulley system shown. The mass tion of the pulley are known to be m ely. Determine the angular acceleration lease. Use the following parameters in your analysis: Mf = 2N m, m = 8kg and kO = 225mm. parameters in your analysis: M f = 2 N ! m , m = 8 kg and Chapter V: Rigid Body Kinetics Homework Kinetics Homework Problems Problem V.21 V-21 (6/94) V-23 ME 274 Given: A couple M acts on a slender, uniform bar of mass m. Assume all surfaces to be smooth. M acts on a slender, uniform bar of mass m. Determine the reaction forces Find: Determine the reaction forces acting on the bar at A and B on release. the bar at A and B on release. Assume all surfaces to be smooth. V-24 etics Homework Problems Problem V.22 -22 (6/109) Chapter V: Rigid Body Kinetics Homework ME 274 Given: A slender, homogeneous bar of mass m is released from rest. omogeneous bar of mass m is released from rest. Determine the contact force Find: Determine the contact force on the bar due to the smooth horizontal surface on release. ue to the smooth horizontal surface. Chapter V: Rigid Body Kinetics Homework V-25 Problem V.23 Given: The inhomogeneous disk shown has a mass of m, its center of mass at G and a centroidal radius of gyration of kG . At the position shown, G is on the same horizontal line as the geometric center O of the disk and the disk is rolling without slipping with a CCW angular speed of ω . omework Problems (6/110) ME 274 Find: Determine: a) the normal force acting on the disk by the horizontal surface, and b) the friction force acting on the disk by the horizontal surface. ous disk shown has a mass of m, its G and a centroidal radius of gyration of ion shown, G is on the same horizontal etric center O of the disk and the disk is lipping with a CCW angular speed of ! . etermine: al force acting on the disk by the l surface. n force acting on the disk by the l surface. Use the following parameters in your analysis: m = 10kg , ω = 2rad/sec and kG = 64mm. g parameters in your analysis: m = 10 kg , ! = 2 rad / sec and V-26 Chapter V: Rigid Body Kinetics Homework Problem V.24 mework Problems 6/117) Given: A constant frictional torque Mf from the bearings at O resist the motion of the pulley ME downward with a speed of v . system shown. The system is released with A traveling 274 A1 Find: Determine the speed of block A after it drops through a distance of d. nal torque M f from the bearings at O of the pulley system shown. If the with A traveling downward with a ermine the speed of block A after it istance of d. m, kO A Use the following parameters in your analysis: m = 12kg , Mf = 3N m, vA1 = 0.3m/sec, d = 1.5m and kO = 210mm. parameters in your analysis: m = 12 kg , M f = 3 N ! m , d = 1.5 meters and kO = 210 mm . Chapter V: Rigid Body Kinetics Homework V-27 Problem V.25 omework ProblemsGiven: A wheel is made up of a thin ring (mass M ) ME 274 spokes (each of mass m). The wheel and four is initially at rest. Assume that the wheel does not slip as it rolls. (6/121) Find: Determine the speed of the wheel’s center after it has moved a distance d. up of a thin ring (mass M) and four ass m). If the wheel is initially at ed of the wheel’s center after it has d. Assume that the wheel does not Use the following parameters in your analysis: d = 3m, m = 2kg and M = 10kg . g parameters in your analysis: d = 3 meters , m = 2 kg and V-28 Chapter V: Rigid Body Kinetics Homework Problem V.26 mework Problems Given: The pulley has a mass of m and a radius ofME 274 gyration about its center O of kO . The spring is initially stretched by an amount of ∆1 . The system is released from rest. /140) Find: Determine the speed of O after O has dropped through a distance of d. ass of m and a radius of gyration about its the spring is initially stretched by an amount d from rest, find the speed of O after O has distance of d. Use the following parameters in your analysis: m = 50kg , d = 50mm, k = 1500N/m, ∆1 = 100m parameters in and kanalysis: m . 50 kg , d = 50 mm , your O = 300mm= 1 = 100 mm and kO = 300 mm . Chapter V: Rigid Body Kinetics Homework V-29 Problem V.27 work Problems Given: A bullet having a weight of mg strikes a stationary slender, homogeneous bar with a weight ME 274 of M g . The bullet becomes embedded in the bar after impact. 83) Find: Determine the angular velocity of the bar immediately after impact. ight of mg strikes a stationary slender, ith a weight of Mg. The bullet becomes after impact. Determine the angular mediately after impact. Use the following parameters in your analysis: M g = 20lb and mg = 2ounces. arameters in your analysis: Mg = 20 lb and mg = 2 ounces . V-30 Chapter V: Rigid Body Kinetics Homework Problem V.28 Given: A homogeneous sphere of mass m is projected with an initial angular speed of ω0 and with an initial speed of its center G of v0 . The coefficient of kinetic friction between the sphere and the rough incline is µk . Homework Problems ME 274 8 (6/196) Find: Determine: a) the time during which sphere slips on the incline, and b) the angular velocity of the sphere and the speed of G immediately after slipping has ceased. s sphere of mass m is projected with ar speed of ! 0 and with an initial ter G of v0 . The coefficient of between the sphere and the rough etermine: e during which sphere slips on the ular velocity of the sphere and the f G immediately after slipping has Rigid Body Kinetics Homework Problems ME 274 Chapter V: Rigid Body Kinetics Homework V-31 Problem V-29 A crate of mass m is being pulled to the right by a cable attached to a winch. The winch is pulling in Problem V.29 the cable with the angular acceleration of the winch pulley being ! . The crate moves along a rough, horizontal surface with the coefficient of kinetic friction Given: A crate ofsurface andbeing pulled to the .right by athat the crate slides without The winch between this mass m is the crate being µ k Assume cable attached to a winch. tipping. is pulling in the cable with the angular acceleration of the winch pulley being α. The crate moves a) Determine the maximum angular acceleration ! of the winch pulley such that the along a rough, horizontal surface with the coefficient of kinetic friction between this surface and crate does not tip. the crate being µk . Assume that the crate slides without tipping. b) Determine the tension in the cable corresponding to the value of ! found above in a). Find: Determine: a) the maximum angular acceleration α of the winch pulley such that the crate does not tip, and b) the tension in the cable corresponding to the value of α found above in a). b/2 b/2 ! r h d µk A B Use the following parameters in your analysis: r = 0.6f t, mg = 120lb, d = 2.5f t, h = 3f t, b = 4f t and µk = 0.2. Use the following parameters in your analysis: r = 0.5 ft , mg = 100 lb , d = 2.5 ft , h = 3 ft , b = 4 ft and µ k = 0.2 . V-32 Chapter V: Rigid Body Kinetics Homework Rigid Body Kinetics Homework Problems ME 274 Problem V.30 Problem V-30 Given: A homogeneous bar of mass M and length L L is pinned to ground O. TheThe other of A homogeneous of mass M and length is pinned to ground at at O. other end end of the bar, pointpoint O, is supportedaby a cable thatpulled over a smooth pulley and attached to the bar, O, is supported by cable that is is pulled over a smooth pulley and attached a block Boof block m.of mass B isBlockto slide on a slide on ahorizontal surface. At the instant of t a mass B Block m. able B is able to smooth, smooth, horizontal surface. At the release with barof release withat an angle of θ with the horizontal, the bar is stationary and the instant OB oriented bar OB oriented at an angle of ! with the horizontal, the bar is cable is taut with the two free sections with the two oriented horizontally and vertically, as shown stationary and the cable is taut of the cable free sections of the cable oriented in the figure. Assumeand vertically, aspulley to be negligible compared to the of theof the bar be horizontally the mass of the shown in the figure. Assume the mass mass pulley to and block. negligible compared to the mass of the bar and block. Determine the acceleration block B on release. Find: Determine the acceleration ofof block B on release. B O m smooth ! M, L g A Use the followingfollowing parameters in your analysis: M m = 20g , ,m = 25m g , θ = 60◦ . Use the parameters in your analysis: M = 50kg , = 40 k kg L = 2 k and L = 1.5 meters and ! = 53.13° . Chapter V: Rigid Body Kinetics Homework V-33 Rigid Body Kinetics Homework Problems ME 274 Problem V.31 Problem V-31 A thin homogeneous bar OA (of negligible mass mass len of L) is L) is to ground at Given: A thin homogeneous bar OA (of negligible and ofand gthlengthpinnedpinned to ground at O.O. The other end of the bar pinned to tosecond thin homogeneous barbar AB (of massand length The other end of the bar is is pinned a a second thin homogeneous AB (of mass m m 2Land length 2L)of bar AB is ablebar slideis able to slidehorizontal surface. At thesurface. of release ) at A. End B at A. End B of to AB on a smooth, on a smooth, horizontal instant At the instant of horizontal and end B of bar AB is located B of bar AB is located directly from rest, OA isrelease from rest, OA is horizontal and end directly below O. below O. Find: Determine the angular acceleration of bar AB on release. Write your answer as a vector. Determine the angular acceleration of bar AB on release. Write your answer as a vector. O L A 2L g smooth B Use se the following parameters your analysis: m = 50kg and L = 3m. meters . U the following parameters in in your analysis: m = 40 kg and L = 2.5 Rigid Body Kinetics Homework Problems ME 274 V-34 Chapter V: Rigid Body Kinetics Homework Problem V-32 Particle P (having a mass of m) is able to slide within a smooth slot cut in bar OA where Problem rotates about a vertical axis passing through end O. A torque T acts on OA. At a OA V.32 particular instant, P is at a distance of R from O and is moving outward from O in the slot Given:t a rate of P! (having ato the arm asis able to slide within a smooth of !cut in OA has a a Particle R relative mass of m) arm OA is rotating with a rate slot . Bar bar OA where OA rotates of M andvertical axis gyration through bout O. A torque T acts on OA. At a particular mass about a a radius of passing of kO a end point O. ˙ instant, P is at a distance of R from O and is moving outward from O in the slot at a rate of R relative to a) Determine the angular accelerationratebar ω . Bar this instant. the arm as arm OA is rotating with a of of OA at OA has a mass of M and a radius of gyration of kO about point O. b) Determine the acceleration of particle P. Find:Write your answer as a vector. Determine: a) the angular acceleration of bar OA at this instant, and b) the acceleration of particle P. ! T R smooth O P A HORIZONTAL PLANE ˙ Use the following parameters in your analysis: m = 30kg , M = 40kg , R = 0.5m, R = 4m/sec, Use the, following parameters and kO = 1.5m. m = 40 kg , M = 50 kg , in your analysis: ω = 5rad/sec T = 175N m (CCW) ! R = 0.6 meters , R = !3m / sec , ! = 4 rad / sec , T = 150 N ! m ( CCW ) and kO = 2 meters . Chapter V: Rigid Body Kinetics Homework V-35 Problem V.33 Rigid Body Kinetics Homework Problems ME 274 Given: A T-shaped bar consisting of a bar AB (of mass m and length L) is welded to the end of an Problem V-33 bar that is pinned at to ground at O. Tip A is pressed against a spring with a identical second spring constant bar consisting of a bar (m, L) welded to the end 1). a second bar (m, L) is A T-shaped k and the spring is compressed by ∆ (Position of pinned at O. Tip A is pressed against a spring with a spring constant k and the spring is Find: Determine the(Position velocity of the T-bar at position 2. of the T-bar at position 2. compressed by ! angular 1). Determine the angular velocity Position 1 Position 2 B O O G1 G1 G2 A g A G2 B Use following parameters in your analysis: m = 0 = 0.2 = L = 0 k = k = 80 N/m, ∆ = Use thethe following parameters for your analysis: m.4kg , Lkg, 0.8m, .6 m,100N/m and ! = 0.6m. 0.5 m. V-36 Chapter V: Rigid Body Kinetics Homework Problem V.34 Rigid Body Kinetics Homework Problems ME 274 Given: A homogeneous disk (mass M and radius R) is attached to a homogeneous, thin rod OA (mass m) atVits center O. A constant force F is applied at O and point A is confined so that it Problem -34 moves along a smooth, verticalis attached disk rolls without m) at its center O. A constant A disk (mass M) of radius R slot. The to rod OA (mass slipping. If the system is released from rest with is = θ1 . at O and point A is confined so that it moves along a smooth vertical force F θ applied slot. The disk rolls without slipping. If the system is released from rest when !1 = 60°, find the velocity of point A when !2 = 0. Find: Determine the velocity of point A when θ = θ2 . smooth A M, R g m, L ! F O no slip C UUse the following parameters in your analysis: M = 10 ,kg, m 6kg ,kg, = 1m, m,= 2=52 m, F 80N , se the following parameters in your analysis: M = 12kg m = = 5 R R = 1 L L . m, F = == 0 N. ◦ and θ2 = 0. θ1 5 53.13 Chapter V: Rigid Body Kinetics Homework V-37 Rigid Body Kinetics Homework Problems ME 274 Problem V.35 Problem V-35 Given: Bars OA and AB each have a mass of m and a length of L. B is constrained to move in Bthe horizontal direction only. mass of mMO is appliedof L. B OA at O. The system isin ars OA and AB each have a A couple and a length to link is constrained to move released from rest when thedangle between A couple AB is applied. at O. Theall surfacesreleased from the horizontal irection only. OA and M is θ = θ1 Assume system is are smooth. rest when the angle between OA and AB is !1. Assume all surfaces are smooth. Find: Determine the of point B when ! when Determine the velocity velocity of point B = !2. θ = θ2 . B O ! MO L L g A Use the following parameters for your analysis: mm 10 12kgL = 2 m2.!m= θ 80°, 36.87◦ ,0° = 0 and Use the following parameters in your analysis: = = kg, , L = , 5 1 , 11 = !2 = 3 θ2 and MO 6040m. M = = N Nm. O V-38 Chapter V: Rigid Body Kinetics Homework Rigid Body Kinetics Homework Problems ME 274 Problem V.36 Problem V-36 Given: A uniform, disk (of mass m and radius r) is pinned at is applied to amass O. Force F is A uniform disk (m r) is pinned at its center of mass O. Force F its center of rope applied toaround the disk. The diskthe released from rest.released from rest. Assume that the rope wrapped a rope wrapped around is disk. The disk is does not seconds, After 2 slip on the disk. a) determine the angular velocity of the disk. Find: At time t, determine: a) the angular velocity of the disk, and b) the reaction force components on)theeterminepin O. b d disk at the reaction force components at the pin. r O F Use the following parameters in your analysis: m = 15kg , r = 0.4m, F = 200N and t = 2sec. Use the following parameters for your analysis: m = 10 kg, r = 0.3 m, F = 250 N. Chapter V: Rigid Body Kinetics Homework V-39 Problem V.37 Homework Problems Rigid Body Kinetics ME 274 Given: A disk-37 mass m and radius r rolls without slipping. Problem V with A disk with mass m and radius r rolls without slip. Find the minimum velocity of G for Find: disk to the minimum velocity of G for the d. Assume that the disk does not bounce the Find roll over an obstruction with height disk to roll over an obstruction with height d. Assume that the disk obstruction and remains in contact with the obstruction as it rolls over with when it strikes the does not bounce when it strikes the obstruction and remains in contact thehe obstruction. it rolls over the obstruction. t obstruction as r G vG d U the following parameters in for your analysis: m = 2 , kg, 0 4 0 and d = 0 03 m. Use se the following parameters your analysis: m = 18kg0 r = r .=m.3 m, d = 0..02m. V-40 Chapter V: Rigid Body Kinetics Homework dy Kinetics Homework Problems ME 274 Problem V.38 em V-38 Given: After traveling at a constant forward speed of vG1 for some distance, the driver of the car shown forward speed applied for some distance, all four of the raveling at a constant below suddenlyof 12 m/sthe brakes, causing the driver of thecars wheels to instantaneously lock up. The car skidded to rest in a distance all with a constant wn below (m = 2500 kg) suddenly applied the brakes, causingof Lfour of the car’s acceleration. The combined mass of the car and driver is m. to instantaneously lock up. The car skidded to rest in 35 m. Determine the tude of the normal reactions and friction forces at each wheel as the car skidded to Find: Determine the assumed, 1 the normal = 2.25 m, and h = ote that constant deceleration can bemagnitudedof= 1.75 m, d2 reactions and friction forces at each wheel as the car skids to rest. . G h A B d1 d2 Use the following parameters in your analysis: m = 2600kg , L = 35m, d1 = 1.7m, d2 = 2.1m and h = 1.15m. Chapter V: Rigid Body Kinetics Homework V-41 ody Kinetics Homework Problems ME 274 Problem V.39 lem V-39 Given: A pulley of mass M is connected to two blocks as shown. Assume that the axle at C is ley of mass 10frictionless.1 = 0.08 m and d2 = 0.2 m is connected to two blocks as kg with d . Assuming that the axle at C is frictionless, mA = 12 kg, and mB = 8 kg, determine Find: Determine the angular acceleration of block. gular acceleration of the pulley and the acceleration of each the pulley and the acceleration of each block. d1 C B A mA d2 mB Use the following parameters in your analysis: M = 10kg , mA = 12kg , mB = 8kg , d1 = 0.08m and d2 = 0.2m. Problem V-40 Given: A homogeneous wheel having a mass of M and outer radius of r is able to roll without slipping on a horizontal surface. A cable that is attached to the center V-42 O of the wheel is hung over an idealChapterand Rigid Body Kinetics particle B pulley V: is also attached to a Homework having a mass of m. A second cable, attached to point E on the wheel, is also Problem V.40 hung over an ideal pulley and is attached to particle A having a mass of m. The system is released from rest with point E directly below point O. Given: A homogeneous wheel having a mass of M and outer radius of r is able to roll without slipping on a horizontal surface. A cable that is attached to the center O of the wheel is hung over an ideal pulley and is also attached to a particle B having a mass of m. A second cable, attached to point E on the wheel, is also hung over an ideal pulley and is attached to particle A having a mass of m. The system is released from rest withangular acceleration of the wheel. Express your Find: At the time of release, find the point E directly below point O. final answer as a vector. For your numerical answer, use: Find: At the time of release, find the angular acceleration of the wheel. Express your final answer as a vector. m = 10 kg M = 20 kg r = 2 meters y x ideal pulley r A ideal pulley r/2 E m M O no slip B Use the following parameters in your analysis: m = 10kg , M = 20kg and r = 2m. m Problem V-41 Given: A thin, homogeneous bar having a mass of M and length L is pinned to ground at its mass center G. Chapter V: Rigid Body Kinetics Homework Particle B, having a mass of m, is attached to the V-43 right end of the bar. A spring, having a stiffness of k, is attached between end A of the bar and pin C on a wall. The pin G is a distance of b from the wall. Problem V.41 hen the bar is horizontal (Position 1 shown below), the spring is W unstretched. Given: A thin, homogeneous bar having a mass of M and length L is pinned to ground at its mass center G. Particle B, having a mass of m, is rigidly attached to the right end of the bar. A spring, having a stiffness of k , is attached between end A of the bar and pin C on a wall. The pin G is Find: If the bar is released from rest in Position 1 above, find the angular velocity of a distance of b from the wall. When the bar is horizontal (Position 1 shown below), the spring is the bar in Position 2 when the bar is in a vertical position. Use the following unstretched. parameters for your final numerical solution: Find: If the bar is 15 kg from m = 25 kg rest in Position 1 above,00 N/meter M = released k = 1 find the angular velocity of the bar in Position 2 when the bar is in a vertical position. L = 3 meters b = 2.5 meters L M Position 1 C G m A k B g b A G Position 2 C B Use the following parameters in your analysis: M = 15kg , m = 25kg , k = 100N/m, L = 3m and b = 2.5m. Rigid Body Kinetics Homework Problems ME 274 Problem V-42 A thin homogeneous bar having a mass of M = 100 kg and length of 5 meters is pinned to V-44 Chapter V: Rigid Body Kinetics Homework ground at point O in such a way that the bar can rotate about O in a horizontal plane. Puck A, with a mass of m = 50 kg, strikes the bar at point B (located at a distance of 3 Problem V.42 meters from the pin at O) with a speed of vA1 = 30 m / sec , with A initially moving in Given: A the direction shown below. The mass of M and length of Lstruck by the groundAssume thin homogeneous bar having a bar is at rest before being is pinned to puck. at point O in such that the puck sticks to the bar after impact. a horizontal plane. Puck A, with a mass of a way that the bar can rotate about O in m, strikes the bar at point B (located at a distance of d from the pin at O) with a speed of vA1 , Determine the angular velocity of the bar after the puck strikes the bar. Assume all with A initially moving in the direction shown below. The bar is at rest before being struck by the surfaces to be smooth. Treat the puck as a particle. puck. Assume that the puck sticks to the bar after impact. HINT: Treat the puck and the bar as a single system in your analysis. You will need to s angular velocity of the diagram of puck strikes order to receive full credit Find: Determine thehow a correct free bodybar after the the system inthe bar. Assume all surfacesfor to be smooth. Treatthe problem. a particle. the puck as O O ! d vA1 G B ! vA1 A M m Top View Use the following parameters in your analysis: M = 100kg , m = 50kg , L = 5m, d = 3m, vA1 = 30m/sec and θ = 30◦ . L Rigid Body Kinetics Homework Problems ME 274 Problem V-43 Chapter traight, rigid bar is welded to a wheel with a radius of R = 1.5 meters and center at O. V-45 A s V: Rigid Body Kinetics Homework The combined bar-wheel system has a total mass of m = 60 kg, its mass center at point G (at a distance of d = 0.5 meters from O), and a centroidal radius of gyration of Problem V.43 kG = 0.8 meters . The bar-wheel system is released from rest with points O and G lying Given: a horizontalrigid bar is welded to a horizontal surface that R and center at O. The combined on A straight, line. Assume that the wheel with a radius of supports the system is bar-wheel system has a total mass of m, its mass center at point G (at a distance of d from O), SMOOTH. and a centroidal radius of gyration of kG . The bar-wheel system is released from rest with points F G lying on the same system is line. Assume that the O andor the instant after thehorizontal released, determine: horizontal surface that supports the system isa) the angular acceleration of the wheel. SMOOTH. b) the acceleration of the system center of mass G. Find: For the instant when the system is released, determine a) the angular acceleration of the wheel, xpress your acceleration VECTORS. center of mass G. E and b) the answers as of the system R g d O G smooth Use the following parameters in your analysis: R = 1.5m, m = 60kg , d = 0.5m and kG = 0.8m. V-46 Chapter V: Rigid Body Kinetics Homework Rigid Body Kinetics Homework Problems ME 274 Problem V.44 Problem V-44 Given: A homogeneous bar ABbar AB mass of M andof M andis supported by cables AD and A thin, homogeneous with a with a mass length L length L is supported by cables G, where G is the cecenter mass of the bar. At At the position shown below, the bar is AD andEEG, where G is the nter of of mass of the bar. the position shown below, the bar is horizontal, point A is directly D, and D, and end B is directly below E.bar is released from horizontal, point A is directly below below end B is directly below E. The rest in thethe bar isshown. from rest in the position shown, determine the angular acceleration If position released of the bar immediately after release. In your calculations, use M = 20 kg, L = 4 meters Find: Determine the angular acceleration of the bar immediately after release. and h = 1.5 meters. Ignore the mass of cables AD and EG. D E h g A G B L Use the following parameters in your analysis: M = 20kg , L = 4m and h = 1.5m. Chapter V: Rigid Body Kinetics Homework Rigid Body Kinetics Homework Problems ME 274 V-47 Problem VV.45 Problem -45 A thin homogeneous bar of length L = 1.5 meters and mass m = 30 kg is pinned to ground at point O. homogeneous bardof lengtha L andof M = m is kg and radius R = 0.6 point O. A Given: A thin A homogeneous isk with mass mass 100 pinned to ground at mhomogeneous diskto enda mass of bar.and radius R iswithout slipping on thethe bar.of a disk rolls eters is PINNED with A of the M The disk rolls PINNED to end A of inside The circular surface. The system is rof a circular surface. The= 90°. is released from rest with θ = 90◦ . without slipping on the inside eleased from rest with ! system Find: angular angular velocity of the ! = 0. Find the Find thevelocity of the bar whenbar when θ = 0. O L no slip ! g R A Use the following parameters in your analysis: L = 1.5m, R = 0.6m, m = 30kg and M = 100kg . V-48 in terms of the position x of: 3 2 aO = x 2 (x in feet and aO in ft/sec ) 2 At position 1 (x = 0), point O has a velocity Chapter V:to the right. Kinetics Homework of 1 ft/sec Rigid Body FProblem At position 2, where x = 2 ft, ind: V.46 a) determine the velocity of point O. Given: A cylinder with an outer radius of R rolls without slipping on a horizontal surface. The ) d O has an acceleration that is known in terms of the position center of thebdisk etermine the angular velocity and angular acceleration of the x of: aO = 3x2 /2 c in f t/sec2 ). At position 1 ( = 0), point (x in feet and aO ylinder. Write your answersxas vectors. O has a velocity of vO1 to the right. c) determine the velocity and acceleration of point A (at this position A is Find: For a Position 2 same horizontal line as O). Write your answers asb) determine the angular on the of point O: a) determine the velocity of point O, vectors. velocity and d) sketch the velocity vector for A (from c) velocityin the acceleration of point A (at angular acceleration of the cylinder, c) the above) and Position 2 figure this position A is elow. same horizontal line as O), and d) sketch the velocity vector for A (from b on the c) above. x O aO no slip Position 1 A O R Position 2 Use the following parameters in your analysis: R = 3f t, vO1 = 1f t/sec and x = 2f t. aO length L = 4 meters with the end of the bar located at the center O of the disk. The disk is able to roll without slipping on a horizontal surface. The system is released from rest at position 1 with the bar being horizontal at this position. On Body Kinetics Homework Chapter V: Rigid release, the system rolls to the right. At position 2 shown below the bar is V-49 vertical. Problem V.47 Find: a) Draw a free body diagram (FBD) of the system of the bar and disk Given: A homogeneous disk (having a mass of M and outer radius of R) is welded to a thin, together. homogeneous bar having a mass of m and length L with the end of the bar located at the center O b) is able all of the forces in your FBD in a) as surface. The system is released of the disk. The diskClassifyto roll without slipping on a horizontal to whether they do work from rest at Positionor with be included inhorizontal energy.position. On release, the system moves 1 will the bar being potential at this to the right. At Position 2 shown below the point G when the system is at position 2. c) Determine the speed of bar is vertical. Find: Determine the speed of point G when the system is at Position 2. Position 1 Position 2 L R O O G C G Use the following parameters in your analysis: M = 10kg , m = 20kg , R = 1m and L = 4m. Problem V-48 Given: A system is made up of two identical homogeneous disks (with each disk Chapter V: Rigid Body Kinetics Homework having a mass of M = 20 kg and outer radius of R = 0.5 m) and an inextensible cable. One end of the cable is attached to point O of disk 1 and Problem V.48other end of the cable is wrapped around the outer radius of disk 2. Disk 1 the sits on a SMOOTH horizontal surface, whereas disk 2 is on a rough incline Given: A system is made up ofon the incline.homogeneousis released from rest. having a mass of and does NOT slip two identical The system disks (with each disk V-50 M and outer radius of R) and an inextensible cable. One end of the cable is attached to point O of ind: 1 and the other end of the cable is wrapped around the outer radius of disk 2. Disk 1 sits F disk on a SMOOTH horizontal surface, whereas disk 2 is on a rough incline and does NOT slip on the a) Draw a free body diagram of each disk individually. incline. The system is released from rest. b) Determine the acceleration of the center A of disk 2 immediately after release. Find: Determine the acceleration of the center A of disk 2 immediately after release. 1 O smooth R ideal pulley B R 2 A C ! no slip Use the following parameters in your analysis: M = 20kg and R = 0.5m. Rigid Body Kinetics Homework Problems ME 274 Problem V-49 Chapter V: Rigid Body Kinetics Homework V-51 Given: An inextensible cable connects block B with point A on a rigid, homogeneous disk. At the instant shown, point A is directly above the disk’s center O. Block Problem V.49 B has a mass of m = 40 kg, and the disk has a mass of M = 20 kg. The disk is able to roll without slipping on a rough horizontal surface with C being the Given: An inextensible cable connects block B with point A on a rigid, homogeneous disk. At the contact oint instant shown, pointpA is .directly above the disks center O. Block B has a mass of m, and the disk has a mass of M . The disk is able to roll without slipping on a rough horizontal surface with C Find: If the system is released is released the rest. being the contact point. The system from rest atfromposition shown, find the angular acceleration of the disk. Write your answer as a vector. Find: Determine the angular acceleration of the disk immediately after release. M A R/2 O no slip R C B m Use the following parameters in your analysis: M = 20kg and m = 40kg . inetics Homework Problems ME 274 V-50 V-52 Chapter V: Rigid Body Kinetics Homework Particle B (having a mass of m = 10 kg) initially travels on a smooth HORIZONTAL surfaceV.50 a speed of vB1 = 20 m/sec. This particle strikes Problem with point E on a stationary thin homogeneous bar (having a mass of M = 15 kg), Given: being pinned to ground. The coefficient of restitution with end A of the bar Particle B (having a mass of m) initially travels on a smooth HORIZONTAL surface with a speed of v the This point E is known to be e a stationary thin homogeneous bar (having a mass between the particle and B 1 . bar atparticle strikes point E on = 0.6. Furthermore,of Mcontact surface of the bar beingpoint E is smooth. The coefficient of restitution between the the ), with end A between B and pinned to ground. particle and the bar at point E is known to be e. Furthermore, the contact surface between B and point E is smooth. Determine the angular velocity of the bar immediately after being impacted by the particle. Write your answer as a vector. Find: Determine the angular velocity of the bar immediately after being impacted by the particle. A B d m M L vB1 E ! Use the following parameters in your analysis: L = 3m, d = 2m, m = 10kg , M = 15kg , e = 0.6, θ = 60◦ and vB 1 = 20m/sec. Problem V-51 Given: A homogeneous, thin bar AB (of length L = 3 meters and mass m = 50 kg) is Chapter V: Rigid Body Kinetics Homework pinned to a block at A. The block has a mass of M = 100 kg and is constrained to move along a smooth horizontal surface. A horizontal force F = 200 N is Problem V.51 the bar’s center of mass G at an instant when the bar is vertical and applied to at rest. V-53 Given: A homogeneous, thin bar AB (of length L and mass m) is pinned to a block at A. The block has a this problem, is constrained to move along a smooth horizontal surface. A horizontal Find: For mass of M and force F is) applied to the bar’s centerdiagramsG atthe bar and the block below.vertical and at rest. a Complete the free body of mass for an instant when the bar is b) Determine the angular acceleration of the bar for the instant when the Find: Determine F is angular acceleration of the bar for the instant when the force F is applied to force the applied to the bar. the bar. M g A F G L m B Use the following parameters in your analysis: L = 3m, m = 50kg , M = 100kg and F = 200N . Rigid Body Kinetics Homework Problems ME 274 Problem V-52 V-54 Chapter V: Rigid Body Kinetics Homework Given: A homogeneous disk having a mass of m = 100 kg and radius r = 0.5 meters rolls without slipping on a rough, inclined plane. Pin C at the center of the Problem isk is constrained to move within a smooth slot in an arm that is pinned to d V.52 ground at O. The arm has a mass of M = 300 kg with its center of mass at Given: point G (with d = 0.6 having ) and with mradius of gyration about point O of on a rough, A homogeneous disk meters a mass of a and radius r rolls without slipping inclined plane.0.4 meters .the center of is released from rest when move .within a smooth slot in an kO = Pin C at The system the disk is constrained to ! = 0 arm that is pinned to ground at O. The arm has a mass of M with its center of mass at point G and with a radius of gyration about point O of kO . The system is released from rest when θ = 0. Find: Determine the angular speed of the arm when ! = 90°. At this position, C is at a distance of h = 1.5 meters directly below O. ◦ Find: Determine the angular speed of the arm when θ = 90 . O smooth ! r g h d G C no slip " Use the following parameters in your analysis: m = 100kg , M = 300kg , r = 0.5m, d = 0.6m, h = 1.5m, kO = 0.4 and φ = 36.87◦ . Problem V-53 CGiven:V: Block Body Kinetics Homework hapter Rigid B with mass M is being transported up the slope by using two cylindrical V-55 rollers A and D, each having a mass m and radius r, as shown. A constant force F is Problem V.53 applied on the block and the system starts at rest. Assume that there is no slip at all surfaces.! Given: Block B with mass M is being transported up the slope by using two cylindrical rollers A and D,: eachThe speed mass m andwhen the,block has moved a distance d up the hill. on the block having a of block B radius r as shown. A constant force F is applied Find and the system starts at rest. Assume that there is no slip at all surfaces. Use the following parameters: d = 2 m, m = 5 kg, M = 50 kg, r = 0.5m, ! = 30o, F = 400 Find: Determine the speed of block B when the block has moved a distance d up the hill. N. M B F A m, r D no slip m, r no slip ! Use the following parameters in your analysis: d = 2m, m = 5kg , M = 50kg , r = 0.5m, θ = 30◦ and F = 400N . Rigid Body Kinetics Homework Problems ME 274 Problem V-54 Chapter V: Rigid Body Kinetics Homework Given: A drum has a mass of M = 30 kg, outer radius of R = 0.5 m, inner radius of r = 0.25 m, a centroidal radius of gyration of kG = 0.6 m and centroid at the Problem V.54 geometric center G. A cable is wrapped around the inner radius of the drum and is connected to particle A having a mass of m = 10 kg. The cable is also Given: A drum has aover an ideal pulley.radius of R, stiffness K = 100 ,N/m is attached wrapped mass of M , outer A spring of inner radius of r a centroidal radius of gyration of kGbetween G of at the geometric center G. A cable is released from rest with theradius and centroid the drum and ground. The system is wrapped around the inner of the drum and is connected to particle A having a mass of m. The cable is also wrapped over an spring unstretched. V-56 ideal pulley. A spring of stiffness K is attached between G of the drum and ground. The system is released from rest with the spring unstretched. Find: Determine the speed of A after it has dropped 2 meters. Find: Determine the speed of A after it has dropped through a distance of d. pulley has negligible mass M R K r G no slip m A Use the following parameters in your analysis: M = 30kg , m = 10kg , R = 0.5m, r = 0.25m, kG = 0.3m, K = 100N/m and d = 2m. Rigid Body Kinetics Homework Problems ME 274 Chapter V: Rigid Body Kinetics Homework V-57 Problem V-55 Given: The homogeneous thin bar AB shown below has a mass of M = 100 kg and a Problem V.55 length of L = 3 meters. The bar is supported by lightweight, inextensible cable BO at end B and by a smooth, inclined wall at end A, where ! = 53.13°. The Given: The homogeneous thin bar AB shown below has a mass of M and a length of L. The bar bar is released from rest with the bar in a horizontal orientation and with cable is supported by lightweight, inextensible cable BO at end B and by a smooth, inclined wall at end BO being vertical. A. The bar is released from rest with the bar in a horizontal orientation and with cable BO being vertical. Find: Determine the tension in cable BO immediately after the bar is released. Find: Determine the tension in cable BO immediately after the bar is released. O smooth wall A L B ! Use the following parameters in your analysis: M = 100kg , L = 3m and θ = 53.13◦ . Rigid Body Kinetics Homework Problems ME 274 V-58 Chapter V: Rigid Body Kinetics Homework Problem V-56 Problem V.56 Given: A thin, homogeneous bar OA (having a length of 4 meters and a mass of M = 100 kg), is pinned to ground at O. An inextensible cable is attached to end A Given: Aothin, homogeneous bar OA (having a to block B, with a mass of M ), is mass ofto ground f the bar. This cable is also attached length of L and B also having a pinned at O. An inextensible cable is attachedsmall pulley the bar. Thispulley is also attached to block B, M. The cable is hung over a to end A of C where the cable has negligible with B having a The system Theeleased from rest wa small= 90°. Assume that the cable negligible mass. mass of m. is r cable is hung over hen ! pulley C where the pulley has mass. The emains taut for all time. rest when θ = 90◦ . Assume that the cable remains taut for all r system is released from time. FFind: Determine thethe velocity of block B when bar OA has reached theposition of θ!= 0. Write ind: Determine velocity of block B when bar OA has reached the position of = as a vector. your answer 0. Write your answer as a vector. d A L/2 M g C small, lightweight pulley ! L/2 O m B Use the following parameters in your analysis: m = 20kg , M = 100kg , L = 3m and d = 4m. the center of the disk A is moving to the right with a speed of vA1 = 5 m/sec and has a counterclockwise rotation rate of !1 = 8 rad/sec, as shown in Figure 1 below. In Figure 2 below is shown the instant at which the disk ceases to slip as it continues to move on the horizontal surface. Chapter V: Rigid Body Kinetics Homework V-59 Find: For this problem: a) Determine the speed of A, vA2, at the instant in Figure 2 when the disk ceases to slip on the h outer radius of R Given: A homogeneous disk (with mass M andorizontal surface. ) is placed on a rough surface. b) Determine the e the disk A is moving motion as the disk speed from When placed on this surface, the center oflapsed time during the to the right with a moves of vA1 t rotation rate of ω1 , 1 shown in Figure Figure 2. and has a counterclockwise he position in Figureas to the position in 1 below. In Figure 2 below is Problem V.57 shown the instant at which the disk ceases to slip as it continues to move on the horizontal surface. You must provide an accurate free body diagram of the disk in order to receive full credit f this problem: Find: Foror your work. a) determine the speed of A, vA2 , at the instant in Figure 2 when the disk ceases to slip on the horizontal surface, and b) determine the elapsed time during the motion HINT: from the using both the linear impulse-momentum equation and the angular as the disk moves Consider position in Figure 1 to the position in Figure 2. impulse-momentum equation on the disk in solving this problem. !1 M FIGURE 1 A R vA1 µk slipping !2 FIGURE 2 A vA2 no slip Use the following parameters in your analysis: M = 50kg , R = 0.5m, µk = 0.3, vA1 = 5m/sec and ω1 = 8rad/sec. Rigid Body Kinetics Homework Problems ME 274 V-60 Chapter V: Rigid Body Kinetics Homework Problem V-58 Given: Link AB (having a mass of m with uniform mass distribution and length L) is Problem V.58 supported by a vertical cable OA (of length 0.6L and negligible mass) at A, and end B is constrained to move within a smooth slot, as shown below. The Given: Link AB (having a mass of m with uniform mass distribution and length L) is supported bar is released from rest when it is horizontal. by a vertical cable OA (of length 0.6L and negligible mass) at A, and end B is constrained to move within a smooth slot, as shown below. The bar is released from rest when it is horizontal. Find: Determine the angular acceleration of AB on release. Use the following parameters: m = acceleration of meters. Find: Determine the angular40 kg and L = 2AB on release. g O 0.6 L massless cable m, L B A ! smooth Use the following parameters in your analysis: m = 40kg and L = 2m. Chapter V: Rigid Body Kinetics Homework V-61 Problem V.59 Given: An inhomogeneous disk (having a mass of m, outer radius R, mass center at G and centroidal radius of gyration of kG ) moves along a SMOOTH horizontal surface. The distance from omework Problems the geometric center O of the disk to the center of mass G is given by e. At the instant shown, G ME 274 is at the same vertical height as O as the disk translates to the right with a speed of vO . Find: Determine the normal force of the ground acting at the contact point C at this instant. mogeneous disk (having a mass of m, dius R, mass center at G and centroidal f gyration of kG) moves along a H horizontal surface. The distance e geometric center O of the disk to the f mass G is given by e. At the instant G is at the same vertical height as O as translates to the right with a speed of m g R e G vO O SMOOTH ine the normal force of the ground t the contact point C at this instant. In C alysis, use the following parameter m = 10 kg, R = 0.5 meters, e = 0.3 kG = 0.4 meters,se nd vO = 20 m/sec. U a the following parameters in your analysis: m = 10kg , R = 0.5m, e = 0.3m, kG = 0.4m and vO = 20m/sec. ct point C slips on the smooth horizontal surface. Rigid Body Kinetics Homework Problems V-62 Problem V-60 M Chapter V: Rigid Body Kinetics Homework Given: The compound wheel shown rolls without slipping up the incline on its hub and is pulled by a constant force T = 100 N applied to a cord wrapped arou Given: The compound wheel shown rollsouter rim. The wheel starts from rest, hubs and is of m = 40 kg, and has its without slipping up the incline on its has a mass pulled by a constant force T applied to a cord wrappedgaround its outer rim. The mass O of kOfrom.15 meters. Assume radius of yration about its center of wheel starts = 0 rest, has a mass of m, and has a radius of gyration about not center of mass O of kO . Assume that the the cable does its slip on the wheel. Problem V.60 cable does not slip on the wheel. F velocity The angular after its center wheel after itsacenter O hasd up 2 meters up Find: Determine the angularind: of the wheel velocity of the O has moved distance of moved incline. the incline. F no slip R/2 O R ! Use the following parameters in your analysis: m = 40kg , R = 0.2m, d = 2m, T = 100N and kO = 0.15m. Problem V-61 Given: A uniform rectangular plate of mass m is able to move on a smooth Chapter V: HORIZONTAL plane with corner A of the plate being constrained to move Rigid Body Kinetics Homework V-63 within a smooth guide in the plane of motion. The plate is initially at rest with edge AB aligned with the guide for A. A force P is applied at corner D. Problem V.61 Find Determine the initial angular acceleration to move on smooth HORIZONTAL Given:: A uniform rectangular plate of mass m is able of the plate.aWrite your answer as a plane with corner vector. plate being constrained to move within a smooth guide in the plane of motion. A of the The plate is initially at rest with edge AB aligned with the guide for A. A force P is applied at corner the following parameters in your analysis: m = 200 kg , b = 4 meters , h = 3 meters , Use D. ! = 36.87° and P = 800 N . Find: Determine the initial angular acceleration of the plate. A E b P h ! B D smooth guide for pin A HORIZONTAL PLANE Use the following parameters in your analysis: m = 200kg , b = 4m, h = 3m, θ = 36.87◦ and P = 800N . Rigid Body Kinetics Homework Problems ME 274 V-64 Problem V-62 Given: Chapter V: Rigid Body Kinetics Homework A smooth, massless roller is attached to end A of a thin, homogeneous bar of Problem V.62 length L = 2 meters and mass m = 10 kg. The roller is allowed to move on a horizontal guide. The bar is released from rest with the bar being at an angle Given: A smooth, masslessrom the vertical, asto end A of a thin, homogeneous bar of length L of ! = 53.13° f roller is attached shown below. meters and mass m. The roller is allowed to move on a horizontal guide. The bar is released from rest at the position shown below. Find: Determine the acceleration of the center of mass of the bar at the instant of release. acceleration of the center of mass Find: Determine the State your answer in vector form. of the bar at the instant of release. State your answer in vector form. A smooth, massless roller m, L g ! Use the following parameters in your analysis: m = 10kg , L = 2m and θ = 53.13◦ . cs Homework Problems Chapter ME 274 V: Rigid Body Kinetics Homework V-65 63 Problem V.63 gear of mass m and outer radius R meshes with rack A (having a mass of ). Rack A is constrained to move along a smooth vertical guide. Treat the Given: A gear of mass m and outer radius R meshes with rack A (having a mass of 2m). Rack ar as a homogeneous disk of radius R. Assume that the rack does not slip on A is constrained to move along a smooth vertical guide. Treat the gear as a homogeneous disk of e gear. radius R. Assume that the rack does not slip on the gear. etermine the acceleration of rack A when the system is released from rest. Find: Determine the acceleration of rack A when the system is released from rest. Write your rite your answer as a vector. answer as a vector. A no slip m C O g R 2m smooth Rigid Body Kinetics Homework Problems ME 274 V-66 Chapter V: Rigid Body Kinetics Homework Problem V-64 Given: A crate of mass 2m rests on block C of mass m that, in turn, rests on a smooth ! Problem V.64 horizontal surface. A horizontal force F acts to the right on block C. Given: A crate of mass 2m rests on block C of mass m that, in turn, rests on a smooth horizontal ! ￿ Find: Determine the acts to the right f block C. surface. A horizontal force F maximum force Fonor which the crate does not tip on C. Assume that sufficient friction exists between the crate and C such that the crate does Find: Determine the maximum C. Use the which the crate does not tip on analysis: that sufficient not slip on block force F for following parameters in your C. Assume friction exists between the ,crate 3 meC rs and b = the meters . not slip on block C. m = 100 kg h = and te such that 1.5 crate does b b 2m G no slip C A B h m ! F smooth Use the following parameters in your analysis: m = 100kg , h = 3m and b = 1.5m. Rigid Body Kinetics Homework Problems ME 274 Chapter V Rigid V-67 Problem V:-65 Body Kinetics Homework Given: A cart is made up of a body (of mass M) and FOUR wheels (each wheel of mV.65 Problem ass m and outer radius R). The rear wheels are solid and homogeneous, whereas the front wheels are spoked rims (with the spokes being massless). T cart is made up moving to the right with a FOUR v. At (each wheel Given: Ahe cart is initiallyof a body (of mass M ) andspeed ofwheelssome point, aof mass m and spring of stiffness k attached to the front of the cart whereas fixed wall. outer radius R). The rear wheels are solid and homogeneous,contacts athe front wheels are spoked rims (with the spokes being massless). The cart is initially moving to the right with a speed of v . At : point, a spring of stiffness k attached of the spring the contacting the fixed Findsome Determine the maximum compressionto the front ofafter cart contacts a wall. wall. Use the following parameters in your analysis: M = 14 kg , m = 3 kg , R = 0.15 meters , k = 800 N / m and v = 0.5 m / sec . Find: Determine the maximum compression of the spring after contacting the wall. v k M R A no slip two wheels (each of mass m ) B no slip R two wheels (each of mass m ) Use the following parameters in your analysis: M = 14kg , m = 3kg , R = 0.15m, k = 800N/m and v = 0.5m/sec. Rigid Body Kinetics Homework Problems ME 274 Problem V-66 V-68 Chapter V: Rigid Body Kinetics Homework Given: Particle P (of mass m) strikes end A of a stationary homogeneous L-shaped bar ( V.66 Problemof mass M) with a speed of v. The coefficient of restitution for the impact of P with end A of the bar is known to be e. Given: Particle P (of mass m) strikes end A of a stationary homogeneous L-shaped bar (of mass M) a speed of he angular speed of restitution for the impact of impact. Use the Find:with Determine tv . The coefficient of bar OA immediately after the P with end A of the bar is known to ollowing parameters in your analysis: m = 10kg , M = 30 kg , L = 2 meters , f be e. v = 40 m / sec and e = 0.5 . Find: Determine the angular speed of bar OA immediately after the impact. O L P m v A L HORIZONTAL PLANE Use the following parameters in your analysis: M = 30kg , m = 10kg , L = 2m, v = 40m/sec and e = 0.5. ...
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