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Mechanics Engineering - Dynamics Chapter 18 Problem 18-1 At a given instant the body of mass m has an angular velocity and its mass center has a velocity vG. Show that its kinetic energy can be represented as T = 1/2 IIC 2, where IIC is the moment of inertia of the body computed about the instantaneous axis of zero velocity, located a distance rGIC from the mass center as shown. Solution: T 1 2 m vG 2 1 m 2 rGIC 1 2 2 IG 2 where vG = rGIC T 1 2 IG 2 IG 2 T 1 2 m rGIC 2 1 2 IIC 2 However m(rGIC)2 + IG = IIC T Problem 18-2 The wheel is made from a thin ring of mass mring and two slender rods each of mass mrod. If the torsional spring attached to the wheel's center has stiffness k, so that the torque on the center of the wheel is M = k , determine the maximum angular velocity of the wheel if it is rotated two revolutions and then released from rest. Given: mring mrod k r 2 5 kg 2 kg Nm rad 0.5 m Solution: IO IO T1 2 1 2 mrod ( 2r) 12 2 mring r 2 1.583 kg m U12 T2 591 Engineering Mechanics - Dynamics Chapter 18 0 0 4 k d 1 2 IO 2 k 4 IO 14.1 rad s Problem 18-3 At the instant shown, the disk of weight W has counterclockwise angular velocity center has velocity v. Determine the kinetic energy of the disk at this instant. Given: W 30 lb 5 rad s ft s when its v r g 20 2 ft 32.2 ft s 2 Solution: 1 1 W 2 r 2 2 g 1 W 2 v 2 g T 2 T 210 ft lb *Problem 18-4 The uniform rectangular plate has weight W. If the plate is pinned at A and has an angular velocity , determine the kinetic energy of the plate. Given: W 30 lb 3 a b Solution: T 1 2 m vG 2 1 2 IG 2 592 rad s 2 ft 1 ft Engineering Mechanics - Dynamics Chapter 18 T T 1 W 2 g 6.99 ft lb b 2 a 2 2 2 1 1 W 2 12 g b 2 a 2 2 Problem 18-5 At the instant shown, link AB has angular velocity AB. If each link is considered as a uniform slender bar with weight density , determine the total kinetic energy of the system. Given: AB rad s lb 0.5 in 2 45 deg a b c 3 in 4 in 5 in Solution: Guesses rad s in s g BC 1 CD 1 rad s T 1 lb ft vGx Given 0 0 1 vGy 1 in s 0 a 0 0 0 BC b 0 0 b 2 0 0 0 CD c cos c sin 0 0 AB 0 0 AB 3 0 a 0 0 0 BC 3 vGx vGy 0 3 0 2 BC T 1 2 a 3 2 AB 1 2 b 12 1 2 b vGx 2 vGy 2 1 2 c 3 2 CD 593 Engineering Mechanics - Dynamics Chapter 18 BC CD BC CD vGx vGy T Find BC CD vGx vGy T 1.5 1.697 rad s T vGx vGy 0.5 0.25 ft s 0.0188 ft lb Problem 18-6 Determine the kinetic energy of the system of three links. Links AB and CD each have weight W1, and link BC has weight W2. Given: W1 W2 AB 10 lb 20 lb 5 rad s rAB rBC rCD g 1 ft 2 ft 1 ft 32.2 ft s 2 Solution: BC 0 rad s 2 rAB CD AB rCD 1 W1 rCD 2 g 3 2 2 CD T 1 W1 2 g 10.4 ft lb rAB 3 2 AB 1 W2 2 g AB rAB 2 T Problem 18-7 The mechanism consists of two rods, AB and BC, which have weights W1 and W2, respectively, and a block at C of weight W3. Determine the kinetic energy of the system at the instant shown, when the block is moving at speed vC. 594 Engineering Mechanics - Dynamics Chapter 18 Given: W1 W2 W3 rAB rBC vC g 3 10 lb 20 lb 4 lb 2 ft 4 ft ft s ft s Solution: BC 2 32.2 0 rad s 2 vC AB rAB 1 W2 2 vC 2 g 1 W3 2 vC 2 g T 1 W1 2 g rAB 3 2 AB T 3.82 lb ft *Problem 18-8 The bar of weight W is pinned at its center O and connected to a torsional spring. The spring has a stiffness k, so that the torque developed is M = k If the bar is released from rest when it is vertical at = 90 , determine its angular velocity at the instant = 0 Use the prinicple of work and energy. Given: W k a g Solution: 0 10 lb 5 lb ft rad 1 ft 32.2 ft s 2 90 deg f 0 deg 595 Engineering Mechanics - Dynamics Chapter 18 Guess 1 2 k 0 2 1 rad s 1 2 k f 2 1 W ( 2a) 2 g 12 10.9 rad s 2 2 Given Find Problem 18-9 A force P is applied to the cable which causes the reel of mass M to turn since it is resting on the two rollers A and B of the dispenser. Determine the angular velocity of the reel after it has made two revolutions starting from rest. Neglect the mass of the rollers and the mass of the cable. The radius of gyration of the reel about its center axis is kG. Given: P M kG 20 N 175 kg 0.42 m 30 deg ri r0 a Solution: 0 P2 2 ri 8 P ri M kG Problem 18-10 The rotary screen S is used to wash limestone. When empty it has a mass M1 and a radius of gyration kG. Rotation is achieved by applying a torque M about the drive wheel A. If no slipping occurs at A and the supporting wheel at B is free to roll, determine the angular velocity of the screen after it has rotated n revolutions. Neglect the mass of A and B. 596 250 mm 500 mm 400 mm 1 2 2 M kG 2 2.02 rad s 2 Engineering Mechanics - Dynamics Chapter 18 Unit Used: rev Given: M1 kG M rS rA n Solution: rS rA 1 2 2 M1 kG 2 800 kg 1.75 m 280 N m 2m 0.3 m 5 rev 2 rad M n 2M rS n rA M1 kG 2 6.92 rad s Problem 18-11 A yo-yo has weight W and radius of gyration kO. If it is released from rest, determine how far it must descend in order to attain angular velocity . Neglect the mass of the string and assume that the string is wound around the central peg such that the mean radius at which it unravels is r. Given: W kO 0.3 lb 0.06 ft rad s 70 r g 0.02 ft 32.2 ft s 2 597 Engineering Mechanics - Dynamics Chapter 18 Solution: 1 W 2 g kO 2g 2 2 2 0 Wh 2 r 1 W 2 kO 2 g 2 r h h 0.304 ft *Problem 18-12 The soap-box car has weight Wc including the passenger but excluding its four wheels. Each wheel has weight Ww, radius r, and radius of gyration k, computed about an axis passing through the wheel's axle. Determine the car's speed after it has traveled a distance d starting from rest. The wheels roll without slipping. Neglect air resistance. Given: Wc Ww r k d 110 lb 5 lb 0.5 ft 0.3 ft 100 ft 30 deg g 32.2 ft s Solution: Wc 4Ww d sin 1 Wc 4Ww 2 v g 2 g k r 2 2 1 Ww 2 4 k 2 g ft s v r 2 v 2 Wc Wc 4Ww d sin 4Ww 4Ww v 55.2 2 598 Engineering Mechanics - Dynamics Chapter 18 Problem 18-13 The pendulum of the Charpy impact machine has mass M and radius of gyration kA. If it is released from rest when = 0 determine its angular velocity just before it strikes the specimen S, = 90 . Given: M kA d g 50 kg 1.75 m 1.25 m 9.81 m s Solution: 0 Mgd 1 2 2 M kA 2 2 2.83 rad s 2 2g d 2 kA 2 2 Problem 18-14 The pulley of mass Mp has a radius of gyration about O of kO. If a motor M supplies a force to the cable of P = a (b ce dx), where x is the amount of cable wound up, determine the speed of the crate of mass Mc when it has been hoisted a distance h starting from rest. Neglect the mass of the cable and assume the cable does not slip on the pulley. Given: Mp Mc kO r h Solution: Guesses vc 1 m s 10 kg 50 kg 0.21 m 0.3 m 2m a b c d 800 N 3 2 1 m 599 Engineering Mechanics - Dynamics Chapter 18 h Given 0 ab ce dx dx 1 2 2 vc Mp kO 2 1 2 r Mc vc 2 Mc g h vc Find vc vc 9.419 m s Problem 18-15 The uniform pipe has a mass M and radius of gyration about the z axis of kG. If the worker pushes on it with a horizontal force F, applied perpendicular to the pipe, determine the pipe's angular velocity when it has rotated through angle about the z axis, starting from rest. Assume the pipe does not swing. Units Used: Mg Given: M kG F Solution: 0 Fl 1 2 M kG 2 2 10 kg 3 16 Mg 2.7 m 50 N r l 90 deg 0.75 m 3m 1 M MFl kG 0.0636 rad s *Problem 18-16 The slender rod of mass mrod is subjected to the force and couple moment. When it is in the position shown it has angular velocity 1. Determine its angular velocity at the instant it has rotated downward 90. The force is always applied perpendicular to the axis of the rod. Motion occurs in the vertical plane. 600 Engineering Mechanics - Dynamics Chapter 18 Given: mrod 1 4 kg 6 rad s F M a g 15 N 40 N m 3m 9.81 m s 2 Solution: Guess 2 1 rad s Given 2 2 1 2 2 2 1 mrod a 2 3 Find Fa 2 8.25 mrod g rad s a 2 M 1 mrod a 2 2 3 2 2 2 Problem 18-17 The slender rod of mass M is subjected to the force and couple moment. When the rod is in the position shown it has angular velocity 1. Determine its angular velocity at the instant it has rotated 360. The force is always applied perpendicular to the axis of the rod and motion occurs in the vertical plane. Given: mrod 1 4 kg 6 rad s M a g 40 N m 3m 9.81 m s 2 F Solution: 15 N Guess 2 1 rad s 601 Engineering Mechanics - Dynamics Chapter 18 Given 1 mrod a 2 3 2 2 1 F a2 M2 1 mrod a 2 rad s 3 2 2 2 2 Find 2 2 11.2 Problem 18-18 The elevator car E has mass mE and the counterweight C has mass mC. If a motor turns the driving sheave A with constant torque M, determine the speed of the elevator when it has ascended a distance d starting from rest. Each sheave A and B has mass mS and radius of gyration k about its mass center or pinned axis. Neglect the mass of the cable and assume the cable does not slip on the sheaves. Units Used: Given: mE mC mS M Solution: d r Mg 1000 kg 1.80 Mg 2.30 Mg 150 kg 100 N m d r k g 10 m 0.35 m 0.2 m 9.81 m s 2 M mE d r mC g d 1 2 mE mC 2mS k r 2 2 v 2 2M v mE mC mC g d 2mS k r 2 v 4.973 m s mE 2 Problem 18-19 The elevator car E has mass mE and the counterweight C has mass mC. If a motor turns the driving sheave A with torque a 2 + b, determine the speed of the elevator when it has ascended a distance d starting from rest. Each sheave A and B has mass mS and radius of gyration k about its mass center or pinned axis. Neglect the mass of the cable and assume the cable does not slip on the sheaves. 602 Engineering Mechanics - Dynamics Chapter 18 Units Used: Mg Given: mE mC mS a b d r k g Solution: Guess v d 1000 kg 1.80 Mg 2.30 Mg 150 kg 0.06 N m 7.5 N m 12 m 0.35 m 0.2 m 9.81 m s 2 1 m s 2 Given r a 0 2 bd mE m s mC g d 1 2 mE mC 2mS k r 2 v 2 v Find ( v) v 5.343 *Problem 18-20 The wheel has a mass M1 and a radius of gyration kO. A motor supplies a torque M = (a + b), about the drive shaft at O. Determine the speed of the loading car, which has a mass M2, after it travels a distance s = d. Initially the car is at rest when s = 0 and = 0. Neglect the mass of the attached cable and the mass of the car's wheels. 603 Engineering Mechanics - Dynamics Chapter 18 Given: M1 M2 d a b r kO 100 kg 300 kg 4m 40 N m 900 N m 0.3 m 0.2 m 30 deg Solution: Guess v d r 1 m s 2 Given 0 a b d m s 1 2 M2 v 2 1 2 M1 kO 2 v r M2 g d sin v Find ( v) v 7.49 Problem 18-21 The gear has a weight W and a radius of gyration kG. If the spring is unstretched when the torque M is applied, determine the gear's angular velocity after its mass center G has moved to the left a distance d. Given: W M ro ri d k 15 lb 6 lb ft 0.5 ft 0.4 ft 2 ft 3 lb ft 604 Engineering Mechanics - Dynamics Chapter 18 kG Solution: Guess 0.375 ft 1 rad s 1 W 2 g 2 Given M d ro ro 1 W 2 g 2 2 kG 1 2 k ri ro ro 2 d Find 7.08 rad s Problem 18-22 The disk of mass md is originally at rest, and the spring holds it in equilibrium. A couple moment M is then applied to the disk as shown. Determine its angular velocity at the instant its mass center G has moved distance d down along the inclined plane. The disk rolls without slipping. Given: md M d k 20 kg 30 N m 0.8 m 150 N m Guess 1 rad s md g sin 0.654 m r g 30 deg 0.2 m 9.81 m s 2 Solution: Initial stretch in the spring md g sin k k d0 d0 d0 Given M d r md g d sin k 2 d d0 2 d0 2 1 2 md r 2 1 1 2 2 md r 2 2 605 Engineering Mechanics - Dynamics Chapter 18 Find 11.0 rad s Problem 18-23 The disk of mass md is originally at rest, and the spring holds it in equilibrium. A couple moment M is then applied to the disk as shown. Determine how far the center of mass of the disk travels down along the incline, measured from the equilibrium position, before it stops. The disk rolls without slipping. Given: md M k 20 kg 30 N m 150 N m 30 deg r g 0.2 m 9.81 m s Solution: 2 Guess d 3m Initial stretch in the spring md g sin k d r k d0 d0 md g sin 0.654 m d0 Given M md g d sin Find ( d) d k 2 d d0 2 d0 2 0 d 2m *Problem 18-24 The linkage consists of two rods AB and CD each of weight W1 and bar AD of weight W2. When = 0, rod AB is rotating with angular velocity O. If rod CD is subjected to a couple moment M 606 Engineering Mechanics - Dynamics Chapter 18 and bar AD is subjected to a horizontal force P as shown, determine Given: W1 W2 8 lb 10 lb rad s 1 AB at the instant = 1. a b 2 ft 3 ft 0 2 90 deg 15 lb ft P 20 lb ft s 2 M g 32.2 Solution: U P a sin 1 1 M 1 rad s 2W1 a 2 1 cos 1 W2 a 1 cos 1 Guess Given 1 2 W 1 a2 g 3 2 2 0 1 W2 2 g a 0 2 U 1 2 2 W 1 a2 g 3 2 1 W2 2 g a 2 Find 5.739 rad s Problem 18-25 The linkage consists of two rods AB and CD each of weight W1 and bar AD of weight W2. When = 0, rod AB is rotating with angular velocity O. If rod CD is subjected to a couple moment M and bar AD is subjected to a horizontal force P as shown, determine AB at the instant = 1. 607 Engineering Mechanics - Dynamics Chapter 18 Given: W1 W2 8 lb 10 lb rad s 1 a b 2 ft 3 ft 0 2 45 deg 20 lb M 15 lb ft ft s 2 P g 32.2 Solution: U P a sin 1 1 M 1 rad s 2 0 2W1 a 2 1 cos 1 W2 a 1 cos 1 Guess Given 1 2 2 W 1 a2 g 3 1 W2 2 rad s g a 0 2 U 1 2 2 W 1 a2 g 3 2 1 W2 2 g a 2 Find 5.916 Problem 18-26 The spool has weight W and radius of gyration kG. A horizontal force P is applied to a cable wrapped around its inner core. If the spool is originally at rest, determine its angular velocity after the mass center G has moved distance d to the left. The spool rolls without slipping. Neglect the mass of the cable. Given: W 500 lb d 6 ft 608 Engineering Mechanics - Dynamics Chapter 18 kG P g 1.75 ft 15 lb 32.2 ft s 2 ri ro 0.8 ft 2.4 ft Solution: Guess ro ro ri 1 rad s 1 W 2 g ro 2 Given P d 1 W 2 2 kG 2 g Find 1.324 rad s Problem 18-27 The double pulley consists of two parts that are attached to one another. It has a weight Wp and a centroidal radius of gyration kO and is turning with an angular velocity clockwise. Determine the kinetic energy of the system. Assume that neither cable slips on the pulley. Given: WP WA WB 50 lb 20 lb 30 lb 20 rad s r1 r2 kO 0.5 ft 1 ft 0.6 ft Solution: KE 1 2 I 2 1 2 WA vA 2 1 2 WB vB 2 1 WA 2 g 2 KE KE 1 WP 2 2 kO 2 g 283 ft lb r2 1 WB 2 g r1 2 609 Engineering Mechanics - Dynamics Chapter 18 *Problem 18-28 The system consists of disk A of weight WA, slender rod BC of weight WBC, and smooth collar C of weight WC. If the disk rolls without slipping, determine the velocity of the collar at the instant the rod becomes horizontal, i.e. = 0 The system is released from rest when = Given: WA WBC WC 0 0. 20 lb 4 lb 1 lb 45 deg L r g 3 ft 0.8 ft 32.2 ft s 2 Solution: Guess Given L WBC cos 2 1 WC 2 vC 2 g ft s 2 1 WBC L 2 g 3 vC 1 ft s 0 WC L cos vC L 2 0 vC Find vC vC 13.3 Problem 18-29 The cement bucket of weight W1 is hoisted using a motor that supplies a torque M to the axle of the wheel. If the wheel has a weight W2 and a radius of gyration about O of kO, determine the speed of the bucket when it has been hoisted a distance h starting from rest. 610 Engineering Mechanics - Dynamics Chapter 18 Given: W1 W2 M kO h r Solution: Guess v h r 1 ft s 1 W1 2 g v 2 1500 lb 115 lb 2000 lb ft 0.95 ft 10 ft 1.25 ft Given M 1 W2 2 6.41 g ft s kO 2 v 2 r W1 h v Find ( v) v Problem 18-30 The assembly consists of two slender rods each of weight Wr and a disk of weight Wd. If the spring is unstretched when = 1 and the assembly is released from rest at this position, determine the angular velocity of rod AB at the instant = 0. The disk rolls without slipping. Given: Wr Wd 1 15 lb 20 lb 45 deg 4 lb ft k L r Solution: Guess 3 ft 1 ft 1 rad s 611 Engineering Mechanics - Dynamics Chapter 18 Given 2Wr L 2 sin 1 1 2 rad s k 2L 2L cos 2 1 2 1 1 Wr 2 L 2 3 g 2 Find 4.284 Problem 18-31 The uniform door has mass M and can be treated as a thin plate having the dimensions If shown. it is connected to a torsional spring at A, which has stiffness k, determine the required initial twist of the spring in radians so that the door has an angular velocity when it closes at = 0 after being opened at = 90 and released from rest. Hint: For a torsional spring M = k where k is the stiffness and is the angle of twist. Given: M k 20 kg 80 Nm rad rad s a b 0.8 m 0.1 m 12 P c 2m 0N Solution: Guess Given 0 0 1 rad k d 0 90 deg 1 1 2 3 Ma 2 2 0 Find 0 0 1.659 rad *Problem 18-32 The uniform slender bar has a mass m and a length L. It is subjected to a uniform distributed load w0 which is always directed perpendicular to the axis of the bar. If it is released from the position shown, determine its angular velocity at the instant it has rotated 90. Solve the problem for rotation in (a) the horizontal plane, and (b) the vertical plane. 612 Engineering Mechanics - Dynamics Chapter 18 Solution: 2 0 L x w0 dx d 0 1 2 L w0 4 (a) 1 2 L w0 4 1 1 2 mL 2 3 2 3 w0 2m (b) 1 2 L w0 4 1 1 2 mL 2 3 2 mg L 2 3 w0 2m 3g L Problem 18-33 A ball of mass m and radius r is cast onto the horizontal surface such that it rolls without slipping. Determine the minimum speed vG of its mass center G so that it rolls completely around the loop of radius R + r without leaving the track. Solution: v R 2 2 mg m v 2 gR 2 1 2 2 mr 2 5 1 2 vG 5 vG r 1 2 m vG 2 1 gR 5 m g2R 1 gR 2 1 2 2 mr 2 5 v r vG 1 2 mv 2 3 3 gR 7 1 2 vG 2 2g R 613 Engineering Mechanics - Dynamics Chapter 18 Problem 18-34 The beam has weight W and is being raised to a vertical position by pulling very slowly on its bottom end A. If the cord fails when = 1 and the beam is essentially at rest, determine the speed of A at the instant cord BC becomes vertical. Neglect friction and the mass of the cords, and treat the beam as a slender rod. Given: W 1 1500 lb 60 deg 13 ft 12 ft 7 ft 32.2 ft s 2 L h a g Solution: W L sin 1 2 h 2 a 1 W 2 vA 2 g h 2 a ft s vA 2g L sin 1 2 vA 14.2 Problem 18-35 The pendulum of the Charpy impact machine has mass M and radius of gyration kA. If it is released from rest when = 0, determine its angular velocity just before it strikes the specimen S, = 90 using the conservation of energy equation. Given: M kA d g 50 kg 1.75 m 1.25 m 9.81 m s 2 614 Engineering Mechanics - Dynamics Chapter 18 Solution: 0 Mgd 0 rad s 1 2 2 M kA 2 2 2g d 2 kA 2 2 2.83 *Problem 18-36 The soap-box car has weight Wc including the passenger but excluding its four wheels. Each wheel has weight Ww radius r, and radius of gyration k, computed about an axis passing through the wheel's axle. Determine the car's speed after it has traveled distance d starting from rest. The wheels roll without slipping. Neglect air resistance. Solve using conservation of energy. Given: Wc Ww r k Solution: 1 Wc 4Ww 2 v 2 g g k r 2 110 lb 5 lb 0.5 ft 0.3 ft d 100 ft 30 deg ft s 2 g 32.2 0 Wc 4Ww d sin 0 1 Ww 2 4 k 2 g ft s v r 2 v 2 Wc Wc 4Ww d sin 4Ww 4Ww v 55.2 2 Problem 18-37 The assembly consists of two slender rods each of weight Wr and a disk of weight Wd. If the spring is unstretched when = 1 and the assembly is released from rest at this position, determine the angular velocity of rod AB at the instant Solve using the conservation of energy. = 0. The disk rolls without slipping. 615 Engineering Mechanics - Dynamics Chapter 18 Given: Wr Wd 1 15 lb 20 lb 45 deg k L r 4 lb ft 3 ft 1 ft rad s Solution: Given Guess 1 0 2Wr L sin 1 2 2 1 1 Wr 2 L 2 3 g rad s 2 1 k 2L 2 2L cos 2 1 Find 4.284 Problem 18-38 A yo-yo has weight W and radius of gyration kO. If it is released from rest, determine how far it must descend in order to attain angular velocity . Neglect the mass of the string and assume that the string is wound around the central peg such that the mean radius at which it unravels is r. Solve using the conservation of energy. Given: W kO 0.3 lb 0.06 ft 70 r g Solution: 0 Ws 2 rad s 0.02 ft 32.2 ft s 2 1 W 2 g kO 2g 2 r 2 1 W 2 kO 2 g 2 0 r s 2 s 0.304 ft 616 Engineering Mechanics - Dynamics Chapter 18 Problem 18-39 The beam has weight W and is being raised to a vertical position by pulling very slowly on its bottom end A. If the cord fails when = 1 and the beam is essentially at rest, determine the speed of A at the instant cord BC becomes vertical. Neglect friction and the mass of the cords, and treat the beam as a slender rod. Solve using the conservation of energy. Given: W 1 1500 lb 60 deg 13 ft 12 ft 7 ft 32.2 ft s 2 L h a g Solution: 0 W L sin 1 2 L sin 1 2 1 W 2 vA 2 g h 2 a W h 2 a vA 2g vA 14.2 ft s *Problem 18-40 The system consists of disk A of weight WA, slender rod BC of weight WBC, and smooth collar C of weight WC. If the disk rolls without slipping, determine the velocity of the collar at the instant the rod becomes horizontal, i.e. = 0 The system is released from rest when = 0. Solve using the conservation of energy. Given: WA WBC 20 lb 4 lb L r 3 ft 0.8 ft 617 Engineering Mechanics - Dynamics Chapter 18 WC 0 1 lb 45 deg g 32.2 ft s 2 Solution: Guess Given L 2 1 WC 0 2 2 1 WBC L vC 1 ft s 0 WBC cos 0 WC L cos ft s 2 g vC vC L 2 2 g 3 0 vC Find vC vC 13.3 Problem 18-41 The spool has mass mS and radius of gyration kO. If block A of mass mA is released from rest, determine the distance the block must fall in order for the spool to have angular velocity . Also, what is the tension in the cord while the block is in motion? Neglect the mass of the cord. Given: ms mA 5 Solution: Guesses Given 0 0 1 2 ms kO 1 2 2 2 50 kg 20 kg rad s ri ro kO 0.2 m 0.3 m 0.280 m g 9.81 m s 2 d 1m T 1N 1 2 mA ri 2 mA g d 0 d T 0 Td mA ri 2 mA g d Find ( d T) d 0.301 m T 163 N 618 Engineering Mechanics - Dynamics Chapter 18 Problem 18-42 When slender bar AB of mass M is horizontal it is at rest and the spring is unstretched. Determine the stiffness k of the spring so that the motion of the bar is momentarily stopped when it has rotated downward 90. Given: M a b g Solution: 0 0 0 1 2 k (a b) 2 10 kg 1.5 m 1.5 m 9.81 m s 2 a 2 2 b Mg a 2 N m k (a Mga b) 2 a 2 2 k 42.8 b Problem 18-43 The disk of weight W is rotating about pin A in the vertical plane with an angular velocity when = 0. Determine its angular velocity at the instant shown, 90 deg. Also, compute the horizontal and vertical components of reaction at A at this instant. Given: W 1 15 lb 2 rad s 90 deg 0.5 ft 32.2 ft s 2 r g Solution: Guesses Ax 1 lb Ay 1 lb 2 1 rad s 1 rad s 2 619 Engineering Mechanics - Dynamics Chapter 18 Given 1 3 W 2 r 2 2 g Ax W g 2 1 2 Wr 1 3 W 2 r 2 2 g Ay W W g 2 2 r 2 r Wr 3 W 2 g r 2 Ax Ay 2 Find A x A y 2 42.9 rad s 2 2 9.48 rad s Ax Ay 20.9 5.0 lb *Problem 18-44 The door is made from one piece, whose ends move along the horizontal and vertical tracks. If the door is in the open position = 0 , and then released, determine the speed at which its end A strikes the stop at C. Assume the door is a thin plate of weight W having width c. Given: W a b c g Solution: Guesses vA 1 ft s 1 rad s 180 lb 3 ft 5 ft 10 ft 32.2 ft s 2 Given 1 W (a 2 g b) 12 2 0 0 b a 2 b 2 2 W a 2 b vA b vA Find vA 6.378 rad s vA 31.9 ft s 620 Engineering Mechanics - Dynamics Chapter 18 Problem 18-45 The overhead door BC is pushed slightly from its open position and then rotates downward about the pin at A. Determine its angular velocity just before its end B strikes the floor. Assume the door is a thin plate having a mass M and length l. Neglect the mass of the supporting frame AB and AC. Given: M l h Solution: 180 kg 6m 5m 2 d h l 2 2 Guess 1 rad s 1 Ml 2 12 2 Given Mgd Md 2 2 Find 2.03 rad s Problem 18-46 The cylinder of weight W1 is attached to the slender rod of weight W2 which is pinned at point A. At the instant = the rod has an angular velocity 0 as shown. Determine the angle f to which the rod swings before it momentarily stops. Given: W1 W2 0 0 80 lb 10 lb 1 rad s a b l g 1 ft 2 ft 5 ft 32.2 ft s 2 30 deg 621 Engineering Mechanics - Dynamics Chapter 18 Solution: W1 1 a 2 g 4 2 b 2 W1 f IA b 12 l 2 2 W1 g l b 2 2 W 2 l2 g 3 W1 l d W2 W2 Guess Given 1 deg W1 W2 d cos 0 1 2 IA 0 2 f f W1 W2 d cos f f Find 39.3 deg Problem 18-47 The compound disk pulley consists of a hub and attached outer rim. If it has mass mP and radius of gyration kG, determine the speed of block A after A descends distance d from rest. Blocks A and B each have a mass mb. Neglect the mass of the cords. Given: mp kG d Solution: Guess Given 0 0 1 2 mb vA 2 ri 1 mb vA 2 ro 2 3 kg 45 mm 0.2 m ri ro g 30 mm 100 mm 9.81 m s 2 mb 2 kg vA 1 m s 1 2 mp kG 2 vA ro 2 mb g d mb g ri ro d vA Find vA vA 1.404 m s 622 Engineering Mechanics - Dynamics Chapter 18 *Problem 18-48 The semicircular segment of mass M is released from rest in the position shown. Determine the velocity of point A when it has rotated counterclockwise 90. Assume that the segment rolls without slipping on the surface. The moment of inertia about its mass center is IG. Given: M IG Solution: Guesses 1 rad s 1 2 M vG 2 vG 1 m s M g( d r) m s vG d 2 r 15 kg 0.25 kg m 2 r d 0.15 m 0.4 m Given Mgd 1 2 IG 2 12.4 rad s vG Find vG d 0 2 d 2 0 vG 0.62 2.48 vA 2.48 0.00 m s vA 3.50 m s vA 0 Problem 18-49 The uniform stone (rectangular block) of weight W is being turned over on its side by pulling the vertical cable slowly upward until the stone begins to tip. If it then falls freely ( T = 0) from an essentially balanced at-rest position, determine the speed at which the corner A strikes the pad at B. The stone does not slip at its corner C as it falls. Given: W a b g Solution: Guess 1 rad s 623 150 lb 0.5 ft 2 ft 32.2 ft s 2 Engineering Mechanics - Dynamics Chapter 18 Given W a 2 b 2 2 b vA 1 W a b 2 g 3 11.9 ft s 2 2 2 W a 2 Find vA Problem 18-50 The assembly consists of pulley A of mass mA and pulley B of mass mB. If a block of mass mb is suspended from the cord, determine the block's speed after it descends a distance d starting from rest. Neglect the mass of the cord and treat the pulleys as thin disks. No slipping occurs. Given: mA mB mb d r R g 3 kg 10 kg 2 kg 0.5 m 30 mm 100 mm 9.81 m s Solution: Given 1 mA r 2 2 Find vb 2 2 Guess vb 1 m s 0 0 vb r vb 2 1 mB R 2 2 1.519 m s 2 vb R 2 1 2 mb vb 2 mb g d vb Problem 18-51 A uniform ladder having weight W is released from rest when it is in the vertical position. If it is allowed to fall freely, determine the angle at which the bottom end A starts to lift off the ground. For the calculation, assume the ladder to be a slender rod and neglect friction at A. 624 Engineering Mechanics - Dynamics Chapter 18 Given: W L g Solution: (a) The rod will rotate around point A until it loses contact with the horizontal constraint ( A x = 0). We will find this point first Guesses 1 30 lb 10 ft 32.2 ft s 2 30 deg 1 1 rad s 1 1 rad s 2 Given L 2 1 1 W 2 L 2 3 g 1 W 2 L 3 g 2 L 1 1 2 1 0 W W L cos 2 1 W L sin 1 2 L cos 2 1 1 2 sin 1 0 1 1 1 Find 1 1 1 1 48.19 deg 1 1.794 rad s 1 3.6 rad s 2 (b) Now the rod moves without any horizontal constraint. If we look for the point at which it loses contact with the floor (Ay = 0 ) we will find that this condition never occurs. *Problem 18-52 The slender rod AB of weight W is attached to a spring BC which, has unstretched length L. If the rod is released from rest when = 1,determine its angular velocity at the instant = 2. Given: W 25 lb 625 Engineering Mechanics - Dynamics Chapter 18 L k 1 2 4 ft 5 lb ft 30 deg 90 deg 32.2 ft s 2 g Solution: Guess Given 0 L sin 1 W 2 1 2 kL 2 21 cos 1 1 rad s 2 1 2 1 W L L 2 W sin 2 2 g 2 3 2 1 2 kL 2 1 cos 2 1 2 Find 1.178 rad s Problem 18-53 The slender rod AB of weight w is attached to a spring BC which has an unstretched length L. If the rod is released from rest when = 1, determine the angular velocity of the rod the instant the spring becomes unstretched. Given: W L k 25 lb 4 ft lb 5 ft g 1 30 deg 32.2 ft s 2 Solution: When the spring is unstretched rad s 626 2 120 deg Guess 1 Engineering Mechanics - Dynamics Chapter 18 Given L sin 1 2 1 2 kL 2 2 0 W 21 cos 1 1 1 W L L 2 W sin 2 2 g 2 3 2 1 2 kL 2 1 cos 2 1 2 2 Find 2.817 rad s Problem 18-54 A chain that has a negligible mass is draped over the sprocket which has mass ms and radius of gyration kO. If block A of mass mA is released from rest in the position s = s1, determine the angular velocity of the sprocket at the instant s = s2. Given: ms kO mA s1 s2 r g 2 kg 50 mm 4 kg 1m 2m 0.1 m 9.81 m s Solution: Guess 1 rad s 2 Given 0 mA g s1 1 mA r 2 2 1 2 2 ms kO 2 41.8 rad s mA g s2 Find 627 Engineering Mechanics - Dynamics Chapter 18 Problem 18-55 A chain that has a mass density is draped over the sprocket which has mass ms and radius of gyration kO. If block A of mass mA is released from rest in the position s = s1, determine the angular velocity of the sprocket at the instant s = s2. When released there is an equal amount of chain on each side. Neglect the portion of the chain that wraps over the sprocket. Given: ms kO mA g 2 kg 50 mm 4 kg 9.81 m s Solution: Guess T1 V1 Given T1 V1 T2 0 mA g s1 1 mA r 2 mA g s2 V1 T2 2 2 s1 s2 r 1m 2m 0.1 m 0.8 kg m 1Nm 1Nm T2 V2 1Nm 1Nm 10 rad s 2 s1 g s1 2 1 2 2s1 2s1 r 2 1 2 2 ms kO 2 s2 g V2 s2 2 V2 T1 T1 V1 T2 V2 s2 g 2s1 2 s2 Find T1 V1 T2 V 2 39.3 rad s 628 Engineering Mechanics - Dynamics Chapter 18 *Problem 18-56 Pulley A has weight WA and centroidal radius of gyration kB. Determine the speed of the crate C of weight WC at the instant s = s2. Initially, the crate is released from rest when s = s1. The pulley at P "rolls" downward on the cord without slipping. For the calculation, neglect the mass of this pulley and the cord as it unwinds from the inner and outer hubs of pulley A. Given: WA WC kB s1 Solution: Guess Given WC s1 1 WA 2 2 kB 2 g 1 WC 2 vC 2 g rad s WC s2 vC rA 2 rB 1 rad s vC 1 ft s 30 lb 20 lb 0.6 ft 5 ft rA rB rP s2 0.4 ft 0.8 ft rB 2 10 ft rA vC Find vC 18.9 vC 11.3 ft s Problem 18-57 The assembly consists of two bars of weight W1 which are pinconnected to the two disks of weight W2. If the bars are released from rest at = 0 , determine their angular velocities at the instant = 0. Assume the disks roll without slipping. Given: W1 W2 r l 0 8 lb 10 lb 0.5 ft 3 ft 60 deg 629 Engineering Mechanics - Dynamics Chapter 18 Solution: Guess 1 rad s 2 1 W1 2 g l 3 rad s 2 2 Given 2W1 l sin 0 2 Find 5.28 Problem 18-58 The assembly consists of two bars of weight W1 which are pin-connected to the two disks of weight W2. If the bars are released from rest at = 1, determine their angular velocities at the instant = 2. Assume the disks roll without slipping. Given: W1 W2 1 2 8 lb 10 lb 60 deg 30 deg 0.5 ft 3 ft r l Solution: Guesses 1 rad s vA 1 ft s Given l 2W1 sin 1 2 vA l sin 2 630 l 2W1 sin 2 2 1 W1 2 2 g l 3 2 2 1 3 W2 2 2 r 2 2 g vA r 2 Engineering Mechanics - Dynamics Chapter 18 vA Find vA vA 3.32 ft s 2.21 rad s Problem 18-59 The end A of the garage door AB travels along the horizontal track, and the end of member BC is attached to a spring at C. If the spring is originally unstretched, determine the stiffness k so that when the door falls downward from rest in the position shown, it will have zero angular velocity the moment it closes, i.e., when it and BC become vertical. Neglect the mass of member BC and assume the door is a thin plate having weight W and a width and height of length L. There is a similar connection and spring on the other side of the door. Given: W L a Solution: Guess Given 2 200 lb 12 ft 1 ft b 2 ft 15 deg k 1 lb ft d 1 ft b L 2 2 d 2 2d L cos 2 2 0 W L 2 1 L 2 k 2 2 b d k d Find ( k d) d 4.535 ft k 100.0 lb ft 631 ... View Full Document