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Engineering Mechanics - DynamicsChapter 16Problem 16-1A wheel has an initial clockwise angular velocity ω and a constant angular acceleration α. Determinethe number of revolutions it must undergo to acquire a clockwise angular velocity ωf. What time isrequired?Units Used:rev 2π radGiven:ω 10radsradα 32s2Solution:22ωf 15ω f ω 2α θθ ωf ω α tt ωf ω2αωf ωαrads2θ 3.32 revt 1.67 sProblem 16-2A flywheel has its angular speed increased uniformly from ω1 to ω2 in time t. If the diameter of thewheel is D, determine the magnitudes of the normal and tangential components of acceleration of apoint on the rim of the wheel at time t, and the total distance the point travels during the time period.Given:ω1 15Solution:r radsω2 60D2ω2 ω1 α tα at α rradsω2 ω1tat 0.563t 80 sα 0.56D 2 ftrad2sft2s2an ω 2 ran 3600ft2s2ω2 ω1θ 2αd θr2θ 3000 radd 3000 ftProblem 16-3The angular velocity of the disk is defined by ω at2 b. Determine the magnitudes of the velocityand acceleration of point A on the disk when t t1.396
Engineering Mechanics - DynamicsChapter 16Given:rada 53sradsb 2r 0.8 mt1 0.5 sSolution: t t1ω at b2ω 3.25radsα 2a tα 5.00rad2sv ωr(α r)2 (ω2r)a 2v 2.60msa 9.35m2s*Problem 16-4The figure shows the internal gearing of a “spinner” used for drilling wells. With constantangular acceleration, the motor M rotates the shaft S to angular velocity ωM in time t startingfrom rest. Determine the angular acceleration of the drill-pipe connection D and the number ofrevolutions it makes during the start up at t.rev 2πUnits Used:Given:ωM 100revminrD 150 mmrM 60 mmt 2sSolution:ωM α M tαM ωMtα M 5.24rad2sα M rM α D rD397
Engineering Mechanics - DynamicsChapter 16 rM rD αD αM θ α D 2.09rad2s12αD t2θ 0.67 revProblem 16-5If gear A starts from rest and has a constant angular acceleration αA, determine the time needed forgear B to attain an angular velocity ωB.Given:radαA 2rB 0.5 ft2sradωB 50srA 0.2 ftSolution:The point in contact with both gearshas a speed ofvp ωB rBvp 25.00ftsThus,ωA So thatvpωA 125.00rAω αc tt radsωAαAt 62.50 sProblem 16-6If the armature A of the electric motor in the drillhas a constant angular acceleration αA, determineits angular velocity and angular displacement attime t. The motor starts from rest.Given:α A 20rad2t 3ssSolution:ω αc tω αA tω 60.00rads398
Engineering Mechanics - Dynamicsθ 12αA t2Chapter 16θ 90.00 radProblem 16-7The mechanism for a car window winder is shown in the figure. Here the handle turns the small cogC, which rotates the spur gear S, thereby rotating the fixed-connected lever AB which raises track D inwhich the window rests. The window is free to slide on the track. If the handle is wound with angularvelocity ωc, determine the speed of points A and E and the speed vw of the window at the instant θ.Given:radsωc 0.5rC 20 mmθ 30 degrs 50 mmrA 200 mmSolution:vC ωc rCmsvC 0.01ωs vCωs 0.20rsradsvA vE ωs rAvA ωs rAvA vE 40.00mmsPoints A and E move along circular paths. The vertical component closes the window.vw vA cos ( θ )vw 34.6mms*Problem 16-8The pinion gear A on the motor shaft is given a constant angular acceleration α. If the gears A andB have the dimensions shown, determine the angular velocity and angular displacement of theoutput shaft C, when t t1 starting from rest. The shaft is fixed to B and turns with it.Given:α 3rad2s399
Engineering Mechanics - DynamicsChapter 16t1 2 sr1 35 mmr2 125 mmSolution:αA α r1 αA r2 αC r1 α A r2 α Cω C α C t1θC 12ωC 1.68α C t12radsθ C 1.68 radProblem 16-9The motor M begins rotating at an angular rate ω a(1 ebt). If the pulleys and fan have theradii shown, determine the magnitudes of the velocity and acceleration of point P on the fanblade when t t1 . Also, what is the maximum speed of this point?Given:a 4radb 1r1 1 ins1r2 4 inst1 0.5 sr3 16 inSolution:r1 ω1 r2 ω2t t1(btω1 a 1 evP r3 ω2) r1 ω1 r2 ω2 vP 6.30ins400
Engineering Mechanics - Dynamicsbtα 1 a b eChapter 16 r1 α1 r2 α2 (α 2 r3)2 (ω22r3)aP 2aP 10.02in2sAs t approaches ω1 aωf r1r2ω1vf r3 ωfvf 16.00insProblem 16-10The disk is originally rotating at angular velocity ω0. If it is subjected to a constant angularacceleration α, determine the magnitudes of the velocity and the n and t components ofacceleration of point A at the instant t.Given:ω0 8α 6radsrad2st 0.5 sr 2 ftSolution:ω ω0 α tvA rωvA 22.00ftsat rαat 12.00ft2san rω2an 242.00ft2s401
Engineering Mechanics - DynamicsChapter 16Problem 16-11The disk is originally rotating at angular velocity ω0. If it is subjectedto a constant angular acceleration α, determine the magnitudes ofthe velocity and the n and t components of acceleration of point Bjust after the wheel undergoes a rotation θ.Given:rev 2π radα 6rad2r 1.5 ftsω0 8radsθ 2 revSolution:ω 2ω0 2 α θω 14.66vB rωvB 22ftsaBt rαaBt 9ftrads2saBn rω2aBn 322ft2s*Problem 16-12The anemometer measures the speed ofthe wind due to the rotation of the threecups. If during a time period t1 a wind gustcauses the cups to have an angular velocityω (At2 B ), determine (a) the speed ofthe cups when t t2, (b) the total distancetraveled by each cup during the time periodt1, and (c) the angular acceleration of thecups when t t2. Neglect the size of thecups for the calculation.Given:t1 3 sA 2t2 2 s13sB 3r 1.5 ft1s402
Engineering Mechanics - DynamicsChapter 16Solution:2ω 2 A t2 Bv2 rω2ftv2 16.50st 1 2d r A t B dt 0α dω 2d 40.50 ftα 2A t2dtα 8.00rad2sProblem 16-13A motor gives disk A a clockwise angular acceleration αA at2 b. If the initial angular velocity ofthe disk is ω0, determine the magnitudes of the velocity and acceleration of block B when t t1.Given:a 0.6rad4ω0 6sb 0.75rad2radr 0.15 mst1 2 ssSolution:2α A a t1 bωA a 3t1 b t1 ω 03vB ωA rvB 1.365maB α A raB 0.472ms2sProblem 16-14The turntable T is driven by the frictional idler wheel A, which simultaneously bears againstthe inner rim of the turntable and the motor-shaft spindle B. Determine the required diameterd of the spindle if the motor turns it with angular velocity ωB and it is required that theturntable rotate with angular velocity ωT .403
Engineering Mechanics - DynamicsChapter 16Given:ωB 25radsradsωT 2a 9 inSolution: a d d2ω B ω A 2 2 ωA ωB da a d 2ω A ωT a 2 ωB d2d2 ωT ad 2ω T aωBd 1.44 inProblem 16-15Gear A is in mesh with gear B as shown. If A starts from rest and has constant angularacceleration αΑ , determine the time needed for B to attain an angular velocity ωB.Given:αA 2rad2sradωB 50srA 25 mmrB 100 mmSolution: rA αA rB α A rA α B rBαB ωB α B tt ωBαBt 100.0 s404
Engineering Mechanics - DynamicsChapter 16*Problem 16-16The blade on the horizontal-axis windmill is turningwith an angular velocity ω0. Determine the distancepoint P on the tip of the blade has traveled if the bladeattains an angular velocity ω in time t. The angularacceleration is constant. Also, what is the magnitudeof the acceleration of this point at time t?Given:ω0 2radst 3sω 5radsrp 15 ftSolution:α ω ω0tt dp rp ω0 α t dt 0an rp ωap 2 an at dp 157.50 ftat rp αap 375.30ft2sProblem 16-17The blade on the horizontal-axis windmill is turning with an angular velocity ω0. If it is given anangular acceleration α, determine the angular velocity and the magnitude of acceleration ofpoint P on the tip of the blade at time t.405
Engineering Mechanics - DynamicsChapter 16Given:radsω0 2α 0.6rad2r 15 ftt 3ssSolution:radsω ω0 α tω 3.80apt α rapt 9.00ft2s2apn ω rapn 216.60ft2sap 22apt apnap 217ft2sProblem 16-18Starting from rest when s 0, pulley A is given anangular acceleration αΑ kθ. Determine the speedof block B when it has risen to s s1. The pulley hasan inner hub D which is fixed to C and turns with it.Given: 2k 6srC 150 mms1 6 mrD 75 mmrA 50 mmSolution:θ A rA θ C rCα A kθωA rA ωC rCθ C rD s1 θ A2 k 2 2 ωA rC s1 rA rDθA 2 rA ωA rC ωC ωA k θAvB ωC rD406vB 14.70ms
Engineering Mechanics - DynamicsChapter 16Problem 16-19Starting from rest when s 0, pulley A is given a constant angular acceleration αΑ . Determinethe speed of block B when it has risen to s s1 .The pulley has an inner hub D which is fixed toC and turns with it.Given:αA 6rad2ss1 6 mrA 50 mmrC 150 mmrD 75 mmSolution:θ A rA θ C rCωA2 rC s1 rA rD θA θ C rD s12 α Aθ AωA rA ωC rCωA ωC 2α A θ ArArCωAvB ωC rDvB 1.34ms*Problem 16-20Initially the motor on the circular saw turns its drive shaft at ω kt2/3. If the radii of gears A and B arerA and rB respectively, determine the magnitudes of the velocity and acceleration of a tooth C on thesaw blade after the drive shaft rotates through angle θ θ1 starting from rest.407
Engineering Mechanics - DynamicsChapter 16Given:rA 0.25 inrB 1 inrC 2.5 inθ 1 5 radk 20rad5s3Solution:253ωA k tθA 35kt33 5θ 1 t1 3k 5t1 0.59 s2ω A k t1ωB rArB3ωA 14.09ωAωB 3.52radsrads 1αA 23k t13α A 15.88(rCα B)2 (rCωB2)αB 2svC rCωBaC rad2rArBαAvC 8.81inaC 32.6inα B 3.97rad2ss2sProblem 16-21Due to the screw at E, the actuator provides linear motion to the arm at F when the motor turnsthe gear at A. If the gears have the radii listed, and the screw at E has pitch p, determine thespeed at F when the motor turns A with angular velocity ωA. Hint: The screw pitch indicates theamount of advance of the screw for each full revolution.408
Engineering Mechanics - DynamicsChapter 16Given:rev 2π radp 2mmrevωA 20radsrA 10 mmrB 50 mmrC 15 mmrD 60 mmSolution:ωA rA ωB rB rA rC ωA rB rD ωB rC ωD rDradωD ωD 1vF ωD pvF 0.318smmsProblem 16-22A motor gives gear A angular acceleration αA aθ 3 b. Ifthis gear is initially turning with angular velocity ωA0,determine the angular velocity of gear B after A undergoes anangular displacement θ1.Given:rev 2π radrada 0.252sb 0.5rad2sωA0 20rads409
Engineering Mechanics - DynamicsChapter 16rA 0.05 mrB 0.15 mθ 1 10 revSolution:θ 13ωA ωA0 2 aθ b dθ 32α A aθ b(2)0θ 1 3 ωA0 2aθ b dθ 2ωA ωA 1395.940ωB rArBωAωB 465radsradsProblem 16-23A motor gives gear A angular acceleration αA kt3. If this gear is initially turning with angularvelocity ωA0, determine the angular velocity of gear B when t t1.Given:k 4t1 2 srad5srA 0.05 mωA0 20srB 0.15 mt t1Solution:αA k tωB rad3rArBk 4ωA t ωA0ωA 36.00 4 ωAωB 12.00radsrads*Problem 16-24For a short time a motor of the random-orbit sander drives the gear A with an angular velocityωA A(t3 Bt). This gear is connected to gear B, which is fixed connected to the shaft CD.The end of this shaft is connected to the eccentric spindle EF and pad P, which causes the padto orbit around shaft CD at a radius rE. Determine the magnitudes of the velocity and the410
Engineering Mechanics - DynamicsChapter 16tangential and normal components of acceleration of the spindle EF at time t after starting fromrest.Given:rA 10 mm rB 40 mm rE 15 mmA 40rad2B 6s4t 2ssSolution:ωA A t B t(3)ωB ()αB 2α A A 3t BrArBrArBv ωB rEv 3.00at α B rEat 2.70ωAαAmsm2s2an ωB rEan 600.00m2sProblem 16-25For a short time the motor of the random-orbit sander drives the gear A with an angularvelocity ωA kθ 2. This gear is connected to gear B, which is fixed connected to the shaftCD. The end of this shaft is connected to the eccentric spindle EF and pad P, which causesthe pad to orbit around shaft CD at a radius rE. Determine the magnitudes of the velocity andthe tangential and normal components of acceleration of the spindle EF when θ θ1 startingfrom rest.Units Used:rev 2π rad411
Engineering Mechanics - DynamicsChapter 16Given:k 5rA 10 mmradsrB 40 mmθ 1 0.5 revrE 15 mmSolution:ωA kθ 12( 2)(2kθ1)α A kθ 1ωB rArBωAαB rArBv ωB rEv 0.19at α B rEat 5.81αAmsm2s2an ωB rEan 2.28m2sProblem 16-26The engine shaft S on the lawnmowerrotates at a constant angular rate ωA.Determine the magnitudes of the velocityand acceleration of point P on the bladeand the distance P travels in time t. Theshaft S is connected to the driver pulley A,and the motion is transmitted to the beltthat passes over the idler pulleys at B andC and to the pulley at D. This pulley isconnected to the blade and to another beltthat drives the other blade.Given:ωA 40radsrP 200 mmrA 75 mmαA 0rD 50 mmt 3s412
Engineering Mechanics - DynamicsChapter 16Solution:ωD rArDωAvP ωD rPvP 12.002aP ωD rPmaP 720.00sm2s ωA t rA rD sP rP sP 36.00 mProblem 16-27The operation of “reverse” for a three-speed automotive transmission is illustrated schematicallyin the figure. If the crank shaft G is turning with angular speed ωG , determine the angular speedof the drive shaft H. Each of the gears rotates about a fixed axis. Note that gears A and B, C andD, and E and F are in mesh. The radii of each of these gears are listed.Given:ωG 60radsrA 90 mmrB 30 mmrC 30 mmrD 50 mmrE 70 mmrF 60 mmSolution:ωB ωD ωH rArBωGωB 180.00radωBωD 108.00radωDωH 126.00radrCrDrErFsss413
Engineering Mechanics - DynamicsChapter 16*Problem 16-28Morse Industrial manufactures the speed reducer shown. If a motor drives the gear shaft S with anangular acceleration α kebt, determine the angular velocity of shaft E at time t after starting fromrest. The radius of each gear is listed. Note that gears B and C are fixed connected to the same shaft.Given:rA 20 mmrB 80 mmrC 30 mmrD 120 mmradk 0.42s 1b 1st 2sSolution:t btω k e dt 0 rA rC ω rB rD ωE ω 2.56radsωE 0.160radsProblem 16-29Morse Industrial manufactures thespeed reducer shown. If a motordrives the gear shaft S with anangular acceleration α kω-3,determine the angular velocity ofshaft E at time t1 after gear Sstarts from an angular velocity ω0when t 0. The radius of eachgear is listed. Note that gears Band C are fixed connected to thesame shaft.Given:rA 20 mmrB 80 mm414
Engineering Mechanics - DynamicsChapter 16rC 30 mmrD 120 mmradsω0 1k 4rad5st1 2 sSolution:ω1 1GuessradsGivent1ω 1 3 k dt ω dω 0 ω0ω1 2.40radsω1 Find ( ω1 ) rA rC ω1 rB rD ωE ωE 0.150Problem 16-30A tape having a thickness s wraps around thewheel which is turning at a constant rate ω.Assuming the unwrapped portion of tape remainshorizontal, determine the acceleration of point P ofthe unwrapped tape when the radius of thewrapped tape is r. Hint: Since vP ω r, take thetime derivative and note that dr/dt ω (s/2π).Solution:vP ωrap sincedvpdωdtdt 0,drdωr ωdtdt dr dt ap ω In one revolution r is increased by s, so that2πΔθ sΔr415rads
Engineering Mechanics - DynamicsChapter 16Hence,Δr ap s2πs2πΔθωdrs ωdt2π2Problem 16-31The sphere starts from rest at θ 0 and rotates with an angular acceleration α kθ.Determine the magnitudes of the velocity and acceleration of point P on the sphere at theinstant θ θ1.Given:θ 1 6 radr 8 inφ 30 degk 4rad2sSolution:α kθ 1 θ 12 k 2 2 ω2vP ωr cos ( φ )aP ω k θ1vP 6.93fts(α r cos (φ ) )2 (ω2r cos ( φ ))2aP 84.3ft2s*Problem 16-32The rod assembly is supported byball-and-socket joints at A and B. At theinstant shown it is rotating about the yaxis with angular velocity ω and hasangular acceleration α. Determine themagnitudes of the velocity andacceleration of point C at this instant.Solve the problem using Cartesianvectors and Eqs. 16-9 and 16-13.416
Engineering Mechanics - DynamicsChapter 16Given:ω 5radsα 8radb 0.4 m2c 0.3 msa 0.4 mSolution: 0 j 1 0 rAC a 0 c 1.50 mvC 0.00 s 2.00 vC ( ωj) rACaC ( α j) rAC ( ωj) vC 2.50ms 12.40 maC 0.00 2 4.30 s ( ωj) rAC aC 13.12m2sProblem 16-33The bar DC rotates uniformly about the shaft at D with a constant angular velocity ω. Determine thevelocity and acceleration of the bar AB, which is confined by the guides to move vertically.Solution:θ' ωθ'' α 0y l sin ( θ )y' vy l cos ( θ ) θ'(y'' ay l cos ( θ ) θ'' sin ( θ ) θ'vAB ωl cos ( θ )2)aAB ω l sin ( θ )2417
Engineering Mechanics - DynamicsChapter 16Problem 16-34At the instant shown, θ is given, and rod AB is subjected to a deceleration a when thevelocity is v. Determine the angular velocity and angular acceleration of link CD at thisinstant.Given:v 10msa 16m2sθ 60 degr 300 mmSolution:x 2r cos ( θ )x 0.30 mx' 2r sin ( θ ) θ'ω v2r sin ( θ )ω 19.2radsα 183radx'' 2r cos ( θ ) θ' 2r sin ( θ ) θ''2α a 2r cos ( θ ) ω22r sin ( θ )2sProblem 16-35The mechanism is used to convert theconstant circular motion ω of rod ABinto translating motion of rod CD.Determine the velocity and accelerationof CD for any angle θ of AB.Solution:x l cos ( θ )x' vx l sin ( θ ) θ'(x'' ax l sin ( θ ) θ'' cos ( θ ) θ'vx vCDax aCDvCD ω l sin ( θ )2)θ' ωandaCD ω l cos ( θ )2418θ'' α 0
Engineering Mechanics - DynamicsChapter 16*Problem 16-36Determine the angular velocity of rod AB for the given θ. The shaft and the center of the rollerC move forward at a constant rate v.Given:v 5msθ 30 degr 100 mmSolution:r x sin ( θ )x rsin ( θ )0 x' sin ( θ ) x cos ( θ ) θ' v sin ( θ ) x cos ( θ ) ωvω tan ( θ ) x ω 14.43radsProblem 16-37Determine the velocity of rod R for any angle θ of the cam C if the cam rotates with a constantangular velocity ω. The pin connection at O does not cause an interference with the motion ofA on C.Solution:Position Coordinate Equation: Using law of cosines.(r1 r2)2 x2 r12 2r1 x cos (θ )x r1 cos ( θ ) r1 cos ( θ ) 2r1 r2 r22220 2x x' 2r1 x' cos ( θ ) 2r1 x sin ( θ ) θ'419
Engineering Mechanics - Dynamicsx' Chapter 16 r1 x sin ( θ ) θ' v r1 sin ( θ ) ω 1 x r1 cos ( θ ) r1 cos ( θ ) 222 r1 cos ( θ ) 2r1 r2 r2 Problem 16-38The crankshaft AB is rotating at constant angular velocity ω. Determine the velocity of thepiston P for the given θ.Given:radsω 150θ 30 dega 0.2 ftb 0.75 ftSolution:x ( a)cos ( θ ) b a sin ( θ )x' ( a) sin ( θ ) θ' 22a cos ( θ ) sin ( θ ) θ'2b a sin ( θ )2v ( a) sin ( θ ) ω 222a cos ( θ ) sin ( θ ) ω2b a sin ( θ )22v 18.502ftsProblem 16-39At the instant θ θ1 the slotted guide is moving upward with acceleration a and velocity v.Determine the angular acceleration and angular velocity of link AB at this instant. Note: Theupward motion of the guide is in the negative y direction.Given:θ 1 50 deg v 2a 3m2msL 300 mms420
Engineering Mechanics - DynamicsChapter 16Solution:y L cos ( θ )y' L sin ( θ ) θ'y'' L sin ( θ ) θ'' L cos ( θ ) θ'ω α vL sin ( θ 1 )ω 8.70a L cos ( θ 1 ) ω2rads2L sin ( θ 1 )α 50.50rad2s*Problem 16-40Determine the velocity of the rod R for any angle θ of cam C as the cam rotates with aconstant angular velocity ω. The pin connection at O does not cause an interference with themotion of plate A on C.Solution:x r r cos ( θ )x' r sin ( θ ) θ'v r ω sin ( θ )Problem 16-41The end A of the bar is moving downward along the slotted guide with a constant velocity vA.Determine the angular velocity ω and angular acceleration a of the bar as a function of its position y.421
Engineering Mechanics - Dynamicsy' vASolution:Chapter 16y'' 0r y sin ( θ )0 y' sin ( θ ) y cos ( θ ) θ'0 y'' sin ( θ ) 2y' cos ( θ ) θ' y sin ( θ ) θ' y cos ( θ ) θ''2 vA tan ( θ ) y ω ω vA r y22 y r vA 2 ω tan ( θ ) ω y α 2 2 3vA rr α 2 2 3y y2 r2 222y (y r )vA22α (22r vA 2y r2y)( y2 r2)3Problem 16-42The inclined plate moves to the left with a constant velocity v. Determine the angular velocity andangular acceleration of the slender rod of length l. The rod pivots about the step at C as it slides onthe plate.x' vSolution:xsin ( φ θ ) lsin ( 180 deg φ ) lsin ( φ )x sin ( φ ) l sin ( φ θ )x' sin ( φ ) l cos ( φ θ ) θ'Thus,ω v sin ( φ )l cos ( φ θ )x'' sin ( φ ) l cos ( φ θ ) θ'' l sin ( φ θ ) θ'0 cos ( φ θ ) α sin ( φ θ ) ωα 2() 2cos φ θ l cos ( φ θ ) sin ( φ θ ) 2v sinφ222 v sin ( φ ) sin ( φ θ )2α 4222l cos ( φ θ )23
Engineering Mechanics - DynamicsChapter 16Problem 16-43The bar remains in contact with the floor and with point A. If point B moves to the rightwith a constant velocity vB, determine the angular velocity and angular acceleration of thebar as a function of x.x' vBSolution:x'' 0x h tan ( θ )x' h sec ( θ ) θ'2x'' h sec ( θ ) θ'' 2h sec ( θ ) tan ( θ ) θ'22h vB22h xhsec ( θ ) ω 22h xtan ( θ ) α 2h x vB2xh2(h2 x2)2*Problem 16-44The crate is transported on a platform which rests on rollers, each having a radius r. If the rollersdo not slip, determine their angular velocity if the platform moves forward with a velocity v.Solution:Position coordinate equation: sG rθ. Using similar triangles sA 2sG 2rθs'A v 2rθ'ω where θ' ωv2r423
Engineering Mechanics - DynamicsChapter 16Problem 16-45Bar AB rotates uniformly about the fixed pin A with a constant angular velocity ω. Determine thevelocity and acceleration of block C when θ θ1.Given:L 1mθ 1 60 degω 2radsα 0rad2sSolution:θ θ1Guessesθ' ωφ 60 degsC 1 mθ'' αφ' 1radsvC 1msφ'' 1rad2saC 2m2sGivenL cos ( θ ) L cos ( φ ) Lsin ( θ ) θ' sin ( φ ) φ' 0cos ( θ ) θ' sin ( θ ) θ'' sin ( φ ) φ'' cos ( φ ) φ' 022sC L sin ( φ ) L sin ( θ )vC L cos ( φ ) φ' L cos ( θ ) θ'aC L sin ( φ ) φ' L cos ( φ ) φ'' L sin ( θ ) θ' L cos ( θ ) θ''2 φ φ' φ'' s Find ( φ , φ' , φ'' , sC , vC , aC) C vC aC 2φ 60.00 degsC 0.00 m424φ' 2.00vC 2.00radsφ'' 4.62msaC 2.31rad2sm2s
Engineering Mechanics - DynamicsChapter 16Problem 16-46The bar is confined to move along the vertical and inclined planes. If the velocity of the roller at A isvA downward when θ θ1. determine the bar's angular velocity and the velocity of roller B at thisinstant.Given:vA 6ftsθ 1 45 degφ 30 degL 5 ftSolution:θ θ1GuessessA 1 ftsB 1 ftω 1radsvB 1ftsGivenL sin ( θ ) sB cos ( φ )L cos ( θ ) ω vB cos ( φ )L cos ( θ ) sA sB sin ( φ ) L sin ( θ ) ω vA vB sin ( φ ) sA sB Find ( s , s , ω , v )A BB ω vB sA 1.49 ft sB 4.08 ω 1.08radsvB 4.39ftsProblem 16-47When the bar is at the angle θ the rod is rotating clockwise at ω and has an angular acceleration α.Determine the velocity and acceleration of the weight A at this instant. The cord is of length L.Given:L 20 fta 10 ft425
Engineering Mechanics - DynamicsChapter 16b 10 ftθ 30 degω 3radsα 5rad2sθ' ωSolution:θ'' αa b 2a b cos ( θ )2sA L 2 a b sin ( θ ) θ'vA a b 2a b cos ( θ )22 a b sin ( θ ) θ'' a b cos ( θ ) θ'aA 22a bsA 14.82 ft 2a b cos ( θ )vA 29.0fts2 ( a b sin ( θ )θ' )2(a2 b2 2a b cos (θ))3aA 59.9The slotted yoke is pinned at A while end B is used tomove the ram R horizontally. If the disk rotates witha constant angular velocity ω, determine the velocityand acceleration of the ram. The crank pin C is fixedto the disk and turns with it. The length of AB is L.Solution:x L sin ( φ )d r 2r d cos ( θ )22s sin ( φ ) r sin ( θ )Thusx L r sin ( θ )d r 2r d cos ( θ )22s*Problem 16-48s ft2426
Engineering Mechanics - DynamicsChapter 16L r cos ( θ ) ωv 22d ra 22d r2 2r d cos ( θ ) L r sin ( θ ) ωd L r sin ( θ ) ω (d2 r2 2r d cos (θ))33d L r sin ( θ ) cos ( θ ) ω2 2r d cos ( θ )2 (d22 r 2r d cos ( θ )2)3d L r sin ( θ ) ω2 332(d2 r2 2r d cos (θ))5Problem 16-49The Geneva wheel A provides intermittent rotary motion ωA for continuous motion ωD of disk D. Bychoosing d 2 r, the wheel has zero angular velocity at the instant pin B enters or leaves one ofthe four slots. Determine the magnitude of the angular velocity ωΑ of the Geneva wheel when θ θ1so that pin B is in contact with the slot.Given:ωD 2radsr 100 mmθ 1 30 degSolution:θ θ1Guessesφ 10 degωA 1radssBA 10 mm s'BA 10Givenr cos ( θ ) sBA cos ( φ ) mms2r r sin ( θ ) ωD s'BA cos ( φ ) sBA sin ( φ ) ωA 0r sin ( θ ) sBA sin ( φ )r cos ( θ ) ωD s'BA sin ( φ ) sBA cos ( φ ) ωA427
Engineering Mechanics - DynamicsChapter 16 φ ω A Find ( φ , ω , s , s' )A BA BA sBA s'BA radsφ 42.37 degωA 0.816sBA 74.20 mms'BA 190.60The general solution ismms 2 cos ( θ ) 1 3 2 2 cos ( θ ) ωA ωD Problem 16-50If h and θ are known, and the speed of A and B is vA vB v, determine the angular velocityω of the body and the direction φ of vB.Solution:vB vA ω rBA v cos ( φ ) i v sin ( φ ) j v cos ( θ ) i v sin ( θ ) j ( ω k) ( h j) v cos ( φ ) v cos ( θ ) ωh( 1)v sin ( φ ) v sin ( θ )( 2)From Eq. (2),φ θFrom Eq. (1),ω 2vcos ( θ )h428
Engineering Mechanics - DynamicsChapter 16Problem 16-51The wheel is rotating with an angular velocity ω. Determine the velocity of the collar A for thegiven values of θ and φ.Given:θ 30 degφ 60 degω 8radsrA 500 mmrB 150 mmvB 1.2msSolution:GuessesGivenωAB 1radsvA 1ms 0 rB cos ( θ ) 0 rA cos ( φ ) vA 0 r sin ( θ ) 0 r sin ( φ ) 0 A B ω ω 0 AB 0 0 ωAB Find ( ωAB , vA)vA ωAB 4.16radsvA 2.40ms*Problem 16-52The pinion gear A rolls on the fixed gear rack B with angular velocity ω. Determine the velocityof the gear rack C.Given: ω 4Solution:radsr 0.3 ftvC vB vCBvC ω( 2r)vC 6.56fts429
Engineering Mechanics - DynamicsChapter 16Problem 16-53The pinion gear rolls on the gear racks. If B is moving to the right at speed vB and C is moving tothe left at speed vC determine the angular velocity of the pinion gear and the velocity of its center A.Given:vB 8ftsvC 4ftsr 0.3 ftSolution:vC vB vCB vC vB ( 2r) ωω vC vB2rω 20.00radsvA vB vABvA vB ( ω) rvA 2ftsProblem 16-54The shaper mechanism is designed to give a slow cutting strokeand a quick return to a blade attached to the slider at C.Determine the velocity of the slider block C at the instantshown, if link AB is rotating at angular velocity ωAB.Given:θ 60 degφ 45 degωAB 4radsa 300 mmb 125 mm430
Engineering Mechanics - DynamicsChapter 16Solution:GuessesωBC 1radsvC 1ms 0 cos ( θ ) 0 cos ( φ ) vC 0 a sin ( θ ) 0 b sin ( φ ) 0 ωAB 0 ωBC 0 0 Given ωBC Find ( ωBC , vC)vC ωBC 6.79radsvC 1.64msProblem 16-55The shaper mechanism is designed to give a slow cutting stroke and a quick return to a bladeattached to the slider at C. Determine the velocity of the slider block C at the instant shown, iflink AB is rotating at angular velocity ωAB.Given:θ 45 degφ 45 degωAB 4radsa 300 mmb 125 mmSolution:GuessesGivenωBC 1radsvC 1ms 0 cos ( θ ) 0 cos ( φ ) vC 0 a sin ( θ ) 0 b sin ( φ ) 0 ωAB 0 ωBC 0 0 ωBC Find ( ωBC , vC) vC ωBC 9.60rads431vC 1.70ms
Engineering Mechanics - DynamicsChapter 16Problem 16-56The velocity of the slider block C is vC up the inclined groove. Determine the angular velocity oflinks AB and BC and the velocity of point B at the instant shown.Given:vC 4ftsL 1 ftGuessesvBx 1ωAB 1GivenftsradsvBy 1ωBC 1ftsrads 0 0 vBx 0 L vBy ωAB 0 0 vBx 0 L vC cos ( 45 deg) vBy 0 0 vC sin ( 45 deg) 0 ωBC 0 0 vBx vBy Find ( vBx , vBy , ωAB , ωBC)ωAB ωBC ωAB 2.83 rad ωBC 2.83 s vBx 2.83 ft vBy 0.00 s vBx ft 2.83s vBy Problem 16-57Rod AB is rotating with an angular velocity ωAB. Determine the velocity of the collar C for the givenangles θ and φ.Given:ωAB 5vB 10radsfts432
Engineering Mechanics - DynamicsChapter 16a 2 ftb 2.5 ftθ 60 degφ 45 degSolution:GuessesvC 4ftsωCB 7radsGiven 0 ( a)cos ( φ ) 0 b cos ( θ ) vC 0 ( a)sin ( φ ) 0 b sin ( θ ) 0 ωAB 0 ωCB 0 0 vC Find ( vC , ωCB) ωCB ωCB 5.66radsvC 5.18ftsProblem 16-58If rod CD is rotating with an angular velocity ωDC, determine the angular velocities of rods ABand BC at the instant shown.Given:ωDC 8radsθ 1 45 degθ 2 30 degrAB 150 mmrBC 400 mmrCD 200 mmSolution:Guesses θ 3 20 deg ωAB 1radsωBC 1433rads
Engineering Mechanics - DynamicsChapter 16GivenrAB sin ( θ 1 ) rBC sin ( θ 3 ) rCD sin ( θ 2 ) 0 0 rCD cos ( θ 2) 0 rBC cos ( θ 3 ) 0 rAB cos ( θ 1) 0 rCD sin ( θ 2) 0 rBC sin ( θ 3 ) 0 rAB sin ( θ 1) 0 ωDC ωBC ωAB 0 0 0 θ3 ωAB Find ( θ 3 , ωAB , ωBC) ω BC θ 3 31.01 deg ωAB 9.615 rad ωBC 1.067 sPositive means CCWNegative means CWProblem 16-59The angular velocity of link AB is ωAB.Determine the velocity of the collar atC and the angular velocity of link CBfor the given angles θ and φ. Link CBis horizontal at this instant.Given:ωAB 4radsφ 45 degrAB 500 mmθ 60 degrBC 350 mmθ 1 30 degSolution:GuessesvC 1msωCB 1radsGiven 0 rAB cos ( θ ) 0 rBC vC cos ( φ ) 0 rAB sin ( θ ) 0 0 vC sin ( φ ) ωAB ωCB 0 0 0 vC
Engineering Mechanics - Dynamics Chapter 16 θ 1 2 αAt 2 θ 90.00rad Problem 16-7 The mechanism for a car window winder is shown in the figure. Here the handle turns the small cog C, which rotates the spur gear S, thereby rotating the fixed-connected lever AB which raises track D in which the window rests. The window is free to slide on the .
