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Engineering Mechanics - DynamicsChapter 14Units Used:3kip 10 lbGiven:W 4000 lbv0 55g 32.2ftsft2sSolution:1 W 2 v0 Area 02 g Area 1 W 2 v02 g Area 187.888 kip ftWe must produce this much work with thebarrels.Assume that 5 ft x 15 ftArea ( 2 ft ) ( 9 kip) ( 3 ft) ( 18 kip) ( x 5 ft) ( 27 kip)x Area 72 kip ft 5 ft27 kipx 9.292 ftCheck that the assumption is corrrect!Problem 14-22The collar has a mass M and is supported on the rod having a coefficient of kinetic frictionμk. The attached spring has an unstretched length l and a stiffness k. Determine the speedof the collar after the applied force F causes it to be displaced a distance s s1 from pointA. When s 0 the collar is held at rest.Given:M 30 kga 0.5 mF 200 Nμ k 0.4θ 45 degs1 1.5 ml 0.2 mNk 50mg 9.81m2s255

Engineering Mechanics - DynamicsChapter 14Solution:NC 1 NGuessesv 1msGivenNC M g F sin ( θ ) 0F cos ( θ ) s1 μ k NC s1 NC Find ( NC , v) v 112 122k ( a l) k ( s1 a l) M v222NC 152.9 Nv 1.666msProblem 14-23The block of weight W is released from rest at A and slides down the smooth circular surfaceAB. It then continues to slide along the horizontal rough surface until it strikes the spring.Determine how far it compresses the spring before stopping.Given:W 5 lbμ k 0.2a 3 ftθ 90 degb 2 ftk 40lbftSolution:Guessd 1 ftGivenW a μ k W( b d) d Find ( d)1 2kd 02d 0.782 ft*Problem 14-24The block has a mass M and moves within the smooth vertical slot. If it starts from rest whenthe attached spring is in the unstretched position at A, determine the constant vertical force Fwhich must be applied to the cord so that the block attains a speed vB when it reaches sB.Neglect the size and mass of the pulley. Hint: The work of F can be determined by finding thedifference Δ l in cord lengths AC and BC and using UF F Δ l.Given:M 0.8 kgl 0.4 mmsb 0.3 mvB 2.5256

Engineering Mechanics - DynamicssB 0.15 mChapter 14k 100NmSolution:Δl Guess22l b (l sB)2 b2F 1NGivenF Δl M g sB F Find ( F)1122k sB M vB22F 43.9 NProblem 14-25The collar has a mass M and is moving at speed v1 when x 0 and a force of F is appliedto it. The direction θ of this force varies such that θ ax, where θ is clockwise, measuredin degrees. Determine the speed of the collar when x x1. The coefficient of kinetic frictionbetween the collar and the rod is μk.Given:M 5 kgF 60 Nμ k 0.3x1 3 mv1 8msg 9.81m2sa 10degmSolution:N F sin ( θ ) M gmsGuessv 5Given 112 2M v1 F cos ( a x) dx μ k F sin ( a x) M g dx M v22 0 0x1v Find ( v)x1v 10.47ms257

Engineering Mechanics - DynamicsChapter 14Problem 14-26Cylinder A has weight WA and block B has weight WB. Determine the distance A must descend fromrest before it obtains speed vA. Also, what is the tension in the cord supporting block A? Neglect themass of the cord and pulleys.Given:ftsWA 60 lbvA 8WB 10 lbg 32.2ft2sSolution:L 2sA sB0 2vA vBSystem0 WA d WB2d 1 WA 2 1 WB 2 vA ( 2vA)2 g 2 g WA 4WB 2 vA2g d WA 2WBd 2.484 ftBlock A alone1 WA 20 WA d T d vA2 g T WA WA vA2g d2T 36 lbProblem 14-27The conveyor belt delivers crate each of mass M to the ramp at A such that the crate’s velocity is vA,directed down along the ramp. If the coefficient of kinetic friction between each crate and the ramp isμk, determine the speed at which each crate slides off the ramp at B. Assume that no tipping occurs.Given:M 12 kgvA 2.5msμ k 0.3g 9.81m2sθ 30 dega 3m258

Engineering Mechanics - DynamicsChapter 14Solution:Nc M g cos ( θ )1122M vA ( M g a)sin ( θ ) μ k Nc a M vB22vB vA ( 2g a)sin ( θ ) ( 2μ k g) cos ( θ ) a2msvB 4.52*Problem 14-28When the skier of weight W is at point A he has a speed vA. Determine his speed when he reachespoint B on the smooth slope. For this distance the slope follows the cosine curve shown. Also, what isthe normal force on his skis at B and his rate of increase in speed? Neglect friction and air resistance.Given:W 150 lbvA 5ftsa 50 ftb 100 ftd 35 ftSolution: x b y ( x) ( a)cos πy'' ( x) dy' ( x)dxθ B atan ( y' ( d) )GuessesF N 1 lby' ( x) ρ ( x) dy ( x)dx(1 y' ( x) 2)3y'' ( x)ρ B ρ ( d)v' 1ftvB 12sGivenfts1 W 21 W 2 vA W( y ( 0 ft) y ( d) ) vB2 g 2 g F N W cos ( θ B)2 W vB g ρB W sin ( θ B) 259 W v' g

Engineering Mechanics - DynamicsChapter 14 vB FN Find ( vB , FN , v' ) v' vB 42.2ftsF N 50.6 lbv' 26.2ft2sProblem 14-29When the block A of weight W1 is released from rest it lifts the two weights B and C each ofweight W2. Determine the maximum distance A will fall before its motion is momentarilystopped. Neglect the weight of the cord and the size of the pulleys.Given:W1 12 lbW2 15 lba 4 ftSolution:Guessy 10 ftGivenW1 y 2W2(22)a y a 0y Find ( y)y 3.81 ftProblem 14-30The catapulting mechanism is used to propel slider A of mass M to the right along the smooth track.The propelling action is obtained by drawing the pulley attached to rod BC rapidly to the left bymeans of a piston P. If the piston applies constant force F to rod BC such that it moves it a distanced, determine the speed attained by the slider if it was originally at rest. Neglect the mass of thepulleys, cable, piston, and rod BC.Units Used:3kN 10 NGiven:M 10 kg F 20 kN d 0.2 mSolution:0 Fd 12Mv2v 2F dMv 28.284260ms

Engineering Mechanics - DynamicsChapter 14Problem 14-31The collar has mass M and slides along the smooth rod. Two springs are attached to it and the endsof the rod as shown. If each spring has an uncompressed length L and the collar has speed v0when s 0, determine the maximum compression of each spring due to the back-and-forth(oscillating) motion of the collar.Given:M 20 kga 0.25 mL 1mkA 50v0 2NmNkB 100mmsSolution:12 12M v0 ( kA kB) d 022Mv0kA kBd d 0.73 m*Problem 14-32The cyclist travels to point A, pedaling until he reaches speed vA. He then coasts freely up the curvedsurface. Determine the normal force he exerts on the surface when he reaches point B. The total massof the bike and man is M. Neglect friction, the mass of the wheels, and the size of the bicycle.Units Used:3kN 10 NGiven:vA 8msM 75 kga 4mSolution:Wheny x2y ay 1122M vA M g y M vB22Now find the radius of curvaturea4vB x y 1m2vA 2g yy avB 6.66212 x261dx 12 ymsdy 0

Engineering Mechanics - DynamicsChapter 14y xyy' xρ Thusdydxy'' xy22x(1 y' 2)Wheny xρ y''8y vB2 NB M g cos ( 45 deg) M ρ The cyclist travels to point A, pedaling untilhe reaches speed vA. He then coasts freely upthe curved surface. Determine how high hereaches up the surface before he comes to astop. Also, what are the resultant normalforce on the surface at this point and hisacceleration? The total mass of the bike andman is M. Neglect friction, the mass of thewheels, and the size of the bicycle.Given:M 75 kga 4mSolution:12M vA M g y 02x (y' a y)2yx1yρ 2.828 mProblem 14-33msy'' 3 vB2 NB M g cos ( 45 deg) M ρ vA 4y' 1y vA2y 0.815 m2gx 1.203 mθ atan ( y')θ 39.462 degNB M g cos ( θ ) 0NB M g cos ( θ )NB 568.03 NM g sin ( θ ) M atat g sin ( θ )at 6.235m2s262NB 1.697 kN

Engineering Mechanics - DynamicsChapter 14Problem 14-34The block of weight W is pressed against the spring so as to compress it a distance δ whenit is at A. If the plane is smooth, determine the distance d, measured from the wall, towhere the block strikes the ground. Neglect the size of the block.Given:W 10 lbe 4 ftδ 2 ftf 3 ftk 100lbftm2sfθ atan Solution:Guessesg 9.81 e vB 1ftst 1sd 1 ftGiven1 W 21 2kδ W f vB2 g 2 vB t Find ( vB , t , d) d d vB cos ( θ ) tvB 33.08fts0 f vB sin ( θ ) t t 1.369 s g t2 2 d 36.2 ftProblem 14-35The man at the window A wishes to throw a sack of mass M onto the ground. To do this he allows itto swing from rest at B to point C, when he releases the cord at θ θ1. Determine the speed at whichit strikes the ground and the distance R.Given:θ 1 30 degh 16 mL 8mg 9.81m2sM 30 kgSolution:0 M g L cos ( θ 1 ) 12M vC2vC 2g L cos ( θ 1 )263vC 11.659ms

Engineering Mechanics - Dynamics0 Mgh Free FlightGiven12M vD2vD t 2sGuess0 Chapter 14vD 17.7182g hR 1m g t2 v sin ( θ ) t h L cos ( θ ) C11 2 t Find ( t , R) R mst 2.078 sR vC cos ( θ 1 ) t L( 1 sin ( θ 1 ) )R 33.0 m*Problem 14-36A block of weight W rests on the smooth semicylindrical surface. An elastic cord having astiffness k is attached to the block at B and to the base of the semicylinder at point C. If theblock is released from rest at A(θ 0 ), determine the unstretched length of the cord so theblock begins to leave the semicylinder at the instant θ θ1. Neglect the size of the block.Given:W 2 lbk 2lbftθ 1 45 dega 1.5 ftg 9.81m2sSolution:Guessδ 1 ftv1 1ftsGivenW sin ( θ 1 ) W v1 g a212 12k ( π a δ ) k ( π θ 1 ) a δ W a sin ( θ 1 ) 22 v1 Find ( v1 , δ ) δ v1 5.843fts W v1 g 2δ 2.77 ft2642

Engineering Mechanics - DynamicsChapter 14Problem 14-37A rocket of mass m is fired vertically from the surface of the earth, i.e., at r r1. Assuming thatno mass is lost as it travels upward, determine the work it must do against gravity to reach adistance r2. The force of gravity is F GMem/r2 (Eq. 13-1), where Me is the mass of the earthand r the distance between the rocket and the center of the earth.Solution: Me m 2 r F G r 2 1drU12 F dr G Me m r2 r1 1 1 r2 r1 U12 G Me m Problem 14-38The spring has a stiffness k and an unstretched length l0. As shown, it is confined by theplate and wall using cables so that its length is l. A block of weight W is given a speed vA whenit is at A, and it slides down the incline having a coefficient of kinetic friction μk. If it strikes theplate and pushes it forward a distance l1 before stopping, determine its speed at A. Neglect themass of the plate and spring.Given:W 4 lbd 3 ftl0 2 ftk 50l 1.5 ftμ k 0.2l1 0.25 fta 3Solution:aθ atan GuessvA 1lbftb 4 b fts265

Engineering Mechanics - DynamicsChapter 14Given11 W 222 vA W( sin ( θ ) μ k cos ( θ ) ) ( d l1 ) k ( l0 l l1 ) ( l0 l) 022 g vA Find ( vA)vA 5.80ftsProblem 14-39The “flying car” is a ride at an amusement park which consists of a car having wheels that roll alonga track mounted inside a rotating drum. By design the car cannot fall off the track, however motionof the car is developed by applying the car’s brake, thereby gripping the car to the track and allowingit to move with a constant speed of the track, vt. If the rider applies the brake when going from B toA and then releases it at the top of the drum, A, so that the car coasts freely down along the track toB (θ π rad), determine the speed of the car at B and the normal reaction which the drum exerts onthe car at B. Neglect friction during the motion from A to B. The rider and car have a total mass Mand the center of mass of the car and rider moves along a circular path having a radius r.Units Used:3kN 10 NGiven:M 250 kgr 8mvt 3msSolution:1122M vt Mg2r M vB22vB 2vt 4 g rvB 18.0ms vB2 NB M g M r vB2 NB M g r NB 12.5 kN266

Engineering Mechanics - DynamicsChapter 14*Problem 14-40The skier starts from rest at A and travels down the ramp. If friction and air resistance can beneglected, determine his speed vB when he reaches B. Also, find the distance d to where he strikesthe ground at C, if he makes the jump traveling horizontally at B. Neglect the skier’s size. He has amass M.Given:M 70 kgh1 50 mθ 30 degh2 4 mSolution:vB 1GuessesGivenmst 1sM g( h1 h2 ) d 1mvB t d cos ( θ )12M vB2 vB t Find ( vB , t , d) d t 3.753 svB 30.0 h2 d sin ( θ ) ms 1 2gt2d 130.2 mProblem 14-41A spring having a stiffness k is compressed a distance δ. The stored energy in the spring is usedto drive a machine which requires power P. Determine how long the spring can supply energy atthe required rate.3Units Used:kN 10 NGiven:k 5Solution:1 2U12 kδ P t2kNmδ 400 mm2 1 δt k 2 P P 90 Wt 4.44 sProblem 14-42Determine the power input for a motor necessary to lift a weight W at a constant rate v.The efficiency of the motor is ε.Given:W 300 lbfv 5ftsε 0.65267

Engineering Mechanics - DynamicsP Solution:Chapter 14WvP 4.20 hpεProblem 14-43An electrically powered train car draws a power P. If the car has weight W and starts fromrest, determine the maximum speed it attains in time t. The mechanical efficiency is ε.Given:P 30 kWSolution:εP F v W 40000 lbft 30 sε 0.8W d v vg dt tv εP g v dv dt W 0 v 2ε P g tWv 29.2fts0*Problem 14-44A truck has a weight W and an engine which transmits a power P to all the wheels.Assuming that the wheels do not slip on the ground, determine the angle θ of the largestincline the truck can climb at a constant speed v.Given:W 25000 lbfv 50ftsP 350 hpSolution:F W sin ( θ )P W v θ asin P W sin ( θ ) vθ 8.86 degProblem 14-45An automobile having mass M travels up a slope at constant speed v. If mechanical friction andwind resistance are neglected, determine the power developed by the engine if the automobile hasefficiency ε.Units Used:3Mg 10 kg268

Engineering Mechanics - DynamicsChapter 14Given:M 2 Mgθ 7 degv 100kmhrε 0.65Solution:P M g sin ( θ ) vP eng PεP 66.419 kWP eng 102.2 kWProblem 14-46The escalator steps move with a constant speedv. If the steps are of height h and length l,determine the power of a motor needed to liftan average mass M per step.There are n steps.Given:M 150 kgh 125 mmn 32l 250 mmv 0.6msd nhSolution:hθ atan l P n Mgv sin ( θ )P 12.63 kWProblem 14-47If the escalator in Prob. 14 46 is notmoving, determine the constant speedat which a man having a mass M mustwalk up the steps to generate powerP—the same amount that is needed topower a standard light bulb.Given:M 80 kgh 125 mmn 32l 250 mmv 0.6msP 100 W269

Engineering Mechanics - DynamicsChapter 14Solution:hθ atan P F v sin ( θ ) l v PM g sin ( θ )v 0.285ms*Problem 14-48An electric streetcar has a weight W and accelerates along a horizontal straight road from restsuch that the power is always P. Determine how far it must travel to reach a speed of v.Given:W 15000 lbfv 40ftsP 100 hpSolution:P Fv W a v g Guessd 1 ftGivend P dsc 0 v W v2 d v g dsc W v2 dv g d Find ( d)0Problem 14-49The crate of weight W is given speed v in time t1 startingfrom rest. If the acceleration is constant, determine thepower that must be supplied to the motor when t t2.The motor has an efficiency ε. Neglect the mass of thepulley and cable.Given:W 50 lbfv 10ftst1 4 st2 2 sε 0.76g 32.2ft2sSolution:a vt1v2 a t2a 2.5ft2sftv2 5s270d 180.8 ft

Engineering Mechanics - DynamicsF W W a g Chapter 14F W P F v2 W a g P 0.49 hpF 53.882 lbfP motor PεP motor 0.645 hpProblem 14-50A car has a mass M and accelerates along a horizontal straight road from rest such that thepower is always a constant amount P. Determine how far it must travel to reach a speed of v.Solution:Power: Since the power output is constant, then the traction force F varieswith v. Applying Eq. 14-10, we haveP FvF Equation of Motion:PvP Mava Kinematics: Applying equation ds vdvaPMv, we havev s2 Mvdv 1 ds P 0 3Mvs 3P0Problem 14-51To dramatize the loss of energy in an automobile, consider a car having a weight Wcar that istraveling at velocity v. If the car is brought to a stop, determine how long a light bulb withpower Pbulb must burn to expend the same amount of energy.Given:Wcar 5000 lbfv 35mihrP bulb 100 Wg 32.2ft2sSolution:1 Wcar 2 v Pbulb t2 g 2t Wcar vt 46.2 min2g Pbulb271

Engineering Mechanics - DynamicsChapter 14*Problem 14-52Determine the power output of the draw-works motor M necessary to lift the drill pipe of weight Wupward with a constant speed v. The cable is tied to the top of the oil rig, wraps around the lowerpulley, then around the top pulley, and then to the motor.Given:W 600 lbfftsv 4Solution:P WvP 4.36 hpProblem 14-53The elevator of mass mel starts from rest and travels upward witha constant acceleration ac. Determine the power output of themotor M when t t1. Neglect the mass of the pulleys and cable.Given:mel 500 kgac 2m2st1 3 sg 9.81m2sSolution:F mel g mel acF mel( g ac)3F 5.905 10 N272

Engineering Mechanics - DynamicsChapter 14msv1 ac t1v1 6P F v1P 35.4 kWProblem 14-54The crate has mass mc and rests on a surface for which the coefficients of static and kineticfriction are μs and μk respectively. If the motor M supplies a cable force of F at2 b,determine the power output de

Engineering Mechanics - Dynamics Chapter 14 The deformation is T1 U12 T2 1 2 Mv 2 Favgx 0 x 1 2 M v 2 Favg x 38.2mm Problem 14-6 The crate of mass M is subjected to forces F1 and F2, as shown. If it is originally at rest, determine the distance it slides in order to attain a speed v. The coefficient of kinetic .