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

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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 nslate...
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