Eng3934 - Problem Sets 1

1 Problem Set 1

1. (12-82 Hibbeler, 9e) The center of the wheel istravelling at 60 ft= s. If it encounters the tran-sition of two rails such that there is a drop of0:25 in, at the joint between the rails, determinethe distance s to point A where the wheel strikesthe next rail. (s = 2:16 ft)

2. (12-96 Hibbeler, 9e) The missile at A takes o¤from rest and rises vertically to B, where its fuelruns out, in 8 s. If the acceleration varies withtime as shown, determine the height and speed ofthe missile at B, i.e. hB and vB . If by internalcontrols the missile is suddenly pointed 45 � asshown, and allowed to travel in free �ight, deter-mine the maximum height attained, hC , and therange R to where it crashes. (hB = 426:67m,vB = 160m= s, hC = 1079:1m, R = 2:98 km)

3. (12-125 Hibbeler, 9e) The race car travels aroundthe circular track with a speed of 16m= s. Whenit reaches point A it increases its speed at _v =43v1=4m= s2, where v is in m= s. Determine

the velocity and acceleration of the car whenit reaches point B. Also, how much time isrequired for the car to travel from A to B?(vB = 47:55m= s, aB = 11:83m= s2, t = 10:11 s)

4. (12-130 Hibbeler, 9e) The ball is kicked with aninitial speed vA = 8m= s at an angle � = 40 �

with the horizontal. Find the equation of thepath, y = f (x), and then determine the veloc-ity of the ball and the normal and tangentialcomponents of its acceleration when t = 0:25 s.(~v = 6:13{̂ + 2:70|̂m= s, at = �3:96m= s2, an =8:98m= s2)

5. (12-155 Hibbeler, 9e) For a short time the rollercoaster travels along the track de�ned by theequation r = 200 cos (2�) ft, where � is in ra-dians. If the angular speed of the radial co-ordinate line is _� = 0:008t2 rad= s, where t isin seconds, determine the radial and transversecomponents of the car�s velocity and accelerationat the instant � = 30 �. Note: when t = 0,� = 0 �. (vr = �93:53 ft= s, v� = 27 ft= s,ar = �68:67 ft= s2, a� = �41:21 ft= s2)

Eng3934 - Problem Sets 2

6. (12-156 Hibbeler, 9e) The motion of the particleB is controlled by the rotation of the groovedlink OA. If the link is rotating at a constantangular rate _� = 6 rad= s, determine the mag-nitudes of the velocity and acceleration of B atthe instant � = �=2 rad. The spiral path is de-�ned by the equation r = 40�mm, where � is inradians. (v = 446:9mm= s, a = 3:66m= s2)

7. (12-167 Hibbeler, 9e) The car travels along aroad which for a short distance is de�ned byr = 200=� ft, where � is in radians. If it main-tains a constant speed of v = 35 ft= s, determinethe radial and transverse components of its ve-locity when � = �=3 rad. (vr = �24:2 ft= s,v� = 25:3 ft= s)

2 Problem Set 2

1. (12-175 Hibbeler, 9e) The mine car C is beingpulled up the incline using the motor M and therope and pulley arrangement shown. Determinethe speed vP at which a point P on the cablemust be travelling toward the motor to move thecar up the plane with a constant speed of v =2m= s. (vP = 6m= s)

2. (12-184 Hibbeler, 9e) If motors A and B drawin their attached cables with an acceleration ofa = 0:2tm= s2, where t is in seconds, determinethe speed of the block when it reaches a heighth = 4m, starting from rest. Also, how muchtime does it take to reach this height? (t =5:43 s, vC = 2:21m= s)

Eng3934 - Problem Sets 3

3. (12-194 Hibbeler, 9e) The motor C pulls in thecable with an acceleration aC = 3t2m= s2, wheret is in seconds. The motor D draws in its ca-ble at aD = 5m= s2. If both motors start atthe same instant from rest when d = 3m, de-termine (a) the time needed for d = 0, and(b) the relative velocity of block A with respectto block B when this occurs. (t = 1:07 s,~vA=B = 5:93m= s!)

4. (12-200 Hibbeler, 9e) Two planes A and Bare �ying side by side at a constant speed of900 km=h. Maintaining this speed, plane A be-gins to travel along a spiral path r = 1500� km,where � is in radians, whereas plane B contin-ues to �y in a straight line. Determine thespeed of plane A with respect to plane B whenr = 750 km. (~vA=B = 834\117:6 �m= s)

5. (13-11 Hibbeler, 9e) The two boxcars A and Bhave a weight of 20,000 lb and 30,000 lb, respec-tively. If they are freely coasting down the in-cline when the brakes are applied to, and lock-ing, all the wheels of car A, determine the forcein the coupling C between the two cars. The co-e¢ cient of kinetic friction between the wheels ofA and the tracks is �k = 0:5. The wheels of carB are free to roll. Neglect the mass of the car Bwheels in the calculation. Suggestion: Solve theproblem by representing single resultant normalforces acting on A and B, respectively. (5976 lbf,compression)

6. (13-26 Hibbeler, 9e) A freight elevator, includingits load, has mass 500 kg. It is prevented fromrotating by using the track and wheels mountedalong its sides. Starting from rest, in t = 2 s,the motor M draws in the cable with a speedof 6m= s, measured with respect to the elevator.Determine the constant acceleration and tensionin the cable. Neglect the mass of the pulleysand cables. (~a = 0:75m= s2 ", T = 1320 lbf)

Eng3934 - Problem Sets 4

7. (13-27 Hibbeler, 9e) The safe S has a weight of200 lb and is supported by the rope and pulleyarrangement shown. If the end of the rope isgiven to a boy of weight 90 lb, determine hisacceleration if in the confusion he doesn�t let goof the rope. Neglect the mass of the pulleys andrope. (~aB = 2:30m= s2 ")

3 Problem Set 3

1. (13-35 Hibbeler, 9e) The 30 lb crate is beinghoisted upward with a constant acceleration of6 ft= s2. If the uniform beam AB has a weightof 200 lb, determine the components of reactionat A. Neglect the size and mass of the pulley atB. Hint: First �nd the tension in the cable, thenanalyze the forces on the beam using statics.

2. (13-47 Hibbeler, 9e) The tractor is used to lift the150 kg load with B the 24m long rope, boom,and pulley system. If the tractor is travellingto the right with an acceleration of 3m= s2 andhas a velocity of 4m= s at the instant sA = 5m,determine the tension in the rope at this instant.When sA = 0, sB = 0. (T = 1802N)

3. (13-48 Hibbeler, 9e) Cylinder B has mass mand is hoisted using the cord and pulley systemshown. Determine the magnitude of force F asa function of the cylinder�s vertical position y sothat when F is applied the cylinder rises with aconstant acceleration aB . Neglect the mass ofthe cord, pulleys, hook and chain.

Eng3934 - Problem Sets 5

4. (13-61 Hibbeler, 9e) An acrobat has weight 150lb and is sitting on a chair which is perchedon top of a pole as shown. If by a mechan-ical drive the pole rotates downward at a con-stant rate from � = 0 �, such that the acrobat�scenter of mass G maintains a constant speed ofva = 10 ft= s, determine the angle � at which hebegins to "�y" out of the chair. Neglect frictionand assume that the distance from the pivot Oto G is � = 15 ft. (� = 78:05 �)

5. (13-75 Hibbeler, 9e) The 10-lb suitcase slidesdown the curved ramp for which the coe¢ cientof kinetic friction is �k = 0:2. If at the instantit reaches point A it has speed of 5 ft= s, deter-mine the normal force on the suitcase and therate of increase of its speed. (N = 5:88 lbf,at = 23:0 ft= s

2)

6. (13-98 Hibbeler, 9e) The particle has mass 1 kgand is con�ned to move along the smooth verticalslot due to the rotation of the smooth arm OA.Determine the force of the rod on the particleand the normal force of the slot on the particlewhen � = 30 �. The arm has angular acceler-ation of �� = 2 rad= s2 and angular velocity _� =3 rad= s when � = 30 �. (~FOA = 20:9\60 �N,~N = 10:4N � )

7. (13-99 Hibbeler, 9e) For a short time, the 250 kgroller coaster car is travelling along the spi-ral track such that its position measured fromthe top of the track has components r = 8m,� = (0:1t+ 0:5) rad, and z = �0:2tm, wheret is in seconds. Determine the magnitudes ofthe components of force which the track exertson the car in the r; �; z directions at the instantt = 2 s. Neglect the size of the car. (FR = 20N,F� = 0, Fz = 2453N)

Eng3934 - Problem Sets 6

8. (13-110 Hibbeler, 9e) The pilot of an airplaneexecutes a vertical loop which in part follows thepath of a cardoid, r = 600 (1 + cos �) ft, where� is in radians. If his speed at A (� = 0 �) isa constant vP = 80 ft= s, determine the verticalforce the seat belt must exert on him to hold himto his seat when the plane is upside down at A.He weighs 150 lb. (N = 112:7N)

4 Problem Set 4

1. (16-21 Hibbeler, 9e) The aerobic machine man-ufactured by Precor, Inc. transfers pedallingpower to a �ywheel F, which develops resistanceusing an AC-powered electromagnet. If the op-erator initially drives the pedals at 12 rev/min,and then begins an angular acceleration of 8rev/min2, determine the angular velocity of the�ywheel after 2 revolutions of the pedal arm.Note that the pedal arm is �xed-connected to thechain wheel A, which in turn drives the sheave Busing the �xed-connection clutch gear D. Thepoly-V belt wrapped around the sheave, thendrives the pulley E and �xed-connected �ywheel.(~! = 484 rpm �)

2. (16-27 Hibbeler 9e) Determine the distance theload W is lifted in t = 5 s using the hoist. Theshaft of the motor M turns with an angular ve-locity ! = 100 (4 + t) rad= s, where t is in sec-onds. (4s = 2:9m)

3. (16-54 Hibbeler, 9e) The gear rests in a �xedhorizontal rack. A cord is wrapped around theinner core of the gear so that it remains horizon-tally tangent to the inner core at A. If the cordis pulled to the right with a constant velocity of2 ft= s, determine the velocity of the center of thegear, C. (~vC = 1:33m= s;!)

Eng3934 - Problem Sets 7

4. (16-70 Hibbeler, 9e) At the instant shown, thetruck is travelling to the right at 8m= s. Ifthe pipe does not slip at B, determine its an-gular velocity if its mass center G appears toan observer on the ground to remain stationary.(~! = 5:33 rad= s )

5. (16-125 Hibbeler, 9e) A single pulley having bothan inner and outer rim is pin-connected to theblock at A. As cord CF unwinds from the innerrim of the pulley with the motion shown, cordDE unwinds from the outer rim. Determinethe angular acceleration of the pulley and theacceleration of the block at the instant shown.

6. (16-133 Hibbeler, 9e) The man stands at theplatform at O and runs out toward the edgesuch that when he is at A, y = 5 ft, his masscenter has a velocity of 2 ft= s and an accelera-tion of 3 ft= s2, both measured with respect tothe platform and directed along the y axis. Ifthe platform has the angular motions shown, de-termine the velocity and acceleration of his masscenter at this instant. (~vA = �2:5{̂ + 2|̂ ft= s,~aA = �3{̂+ 1:75|̂ ft= s2)

5 Problem Set 5

1. (17-26 Hibbeler, 9e) The machine has a mass of1500 kg and rests on the bed of the truck and onthe smooth surface at B. If it does not slip atA, determine the maximum acceleration of thetruck so that the machine will not move relativeto the truck. Also, what are the horizontal andvertical components of reaction at A when thisoccurs? (aG = 4:905m= s2, Ax = 7357:5N !,Ay = 14:72N ")

2. (17-36 Hibbeler, 9e) The pipe has a length of3m and a mass of 500 kg. It is attached tothe back of the truck using a 0:6m long chainAB. If the coe¢ cient of kinetic friction at C is�k = 0:4, determine the acceleration of the truckif the angle � = 10 � with the road as shown.(aG = 2:33m= s2)

Eng3934 - Problem Sets 8

3. (17-48 Hibbeler, 9e) The handcart has a massof 200 kg and center of mass at G. Determinethe magnitude of the largest force P that can beapplied to the handle so that the wheels at A orB continue to maintain contact with the ground.Neglect the mass of the wheels. (P = 1998N)

4. (17-42 Hibbeler, 12e) The uniform crate has amass of 50 kg and rests on the cart having aninclined surface. Determine the smallest accel-eration that will cause the crate to either slip ortip relative to the cart. What is the magnitudeof this acceleration? the coe¢ cient of static fric-tion between the crate and the cart is �s = 0:5.(a = 2:01m= s2)

5. (17-48 Hibbeler, 12e) Determine the greatestacceleration with which the 1000 kg forklift canraise the 750 kg crate, without causing the wheelsat B to leave the ground. The centers of massfor the forklift and crate are located at G1 andG2, respectively. (a = 4:72m= s2)

6. (17-52 Hibbeler, 12e) The 50 kg uniform craterests on the platform for which the coe¢ cient ofstatic friction is �s = 0:5. If the supportinglinks have an angular velocity ~! = 1 rad= s ,determine the greatest angular acceleration ~� they can have so that the crate does not slip ortip at the instant � = 30 �. (� = 0:59 rad= s2)

6 Problem Set 6

1. (17-54 Hibbeler, 9e) The pendulum consists of a20-lb sphere and a 5-lb slender rod. Determinethe reaction at the pin O just after the pendulumis released from the position shown. (Ox = 0,Oy = 1:94 lbf ")

2. (17-59,60 Hibbeler, 9e) A motor supplies a con-stant torque M = 2Nm to a 50mm diametershaft O connected to the center of the 30 kg �y-wheel. The resultant bearing friction F, which

Eng3934 - Problem Sets 9

the pin exerts on the shaft acts tangent to thesupporting shaft and has magnitude 50N. De-termine how long the torque must be applied tothe shaft to increase the rotational speed of the�ywheel from 4 rad= s to 15 rad= s. The �ywheelhas a radius of gyration of kO = 0:15m about itscenter. If the motor is disengaged from the shaftonce the �ywheel is rotating at 15 rad= s, so thatM = 0, determine how long it will take beforethe resultant bearing frictional force F = 50Nstops the �ywheel from rotating. (t12 = 9:90 s,t23 = 8:10 s)

3. (17-67 Hibbeler, 9e) The device acts as a pop-upbarrier to prevent the passage of a vehicle. Itconsists of a 100 kg steel plate AC and two 100 kgcounterweight solid concrete blocks located oneach side of the plate. Determine the momentof inertia of the plate and block about the hingedaxis through A. Negect the mass of the support-ing arms AB. Also, determine the initial angularacceleration of the assembly when it is releasedfrom rest at � = 45 �. (IA = 174:75 kg � m2,~� = 1:25 rad=s2 �)

4. (17-91 Hibbeler, 9e) Two men exert constantforces of 40 lb and 30 lb at the ends A and Bof the uniform plank which has a weight of 50lb. If the plank is originally at rest, determinethe acceleration of its center and its angular ac-celeration. Asume the plank to be a slender rod.(aG = 12:9 ft=s2; ~� = 2:58 rad=s2 �)

5. (17-96 Hibbeler, 9e) The spool has a mass of100 kg and a radius of gyration of kG = 0:3m.If the coe¢ cients of static and kinetic frictionat A are �s = 0:2 and �k = 0:15, determinethe angular acceleration of the spool if the cordand force P = 50N are directed vertically up-ward (i.e. rotate the cord and 90 � counter-clockwise from the position shown in the �gure).(~� = 0:55 rad=s2 �)

6. (17-101 Hibbeler, 9e) A uniform rod having aweight of 10 lb is pin-supported at A from aslider block of negigible mass which slides on ahorizontal track. Note: Replace the roller shownin the �gure with a block. If the rod is initiallyat rest, and a horizontal force F = 15 lbf is ap-plied to the block, determine the acceleration ofthe block. The coe¢ cient of kinetic friction be-tween the block and the track is �k = 0:2. Ne-glect the dimension d and the size of the blockin the computations. (~aA = 167:5m= s2)

Eng3934 - Problem Sets 10

7 Problem Set 7

1. (14-17 Hibbeler, 11e) The 100 lb block slidesdown the incline plane for which the coe¢ cient ofkinetic friction is �k = 0:25. If it is moving at10 ft= s when it reaches point A, determine themaximum deformation of the spring needed tomomentarily arrest the motion. (s = 2:565 ft)

2. (14-20 Hibbeler, 11e) The motion of a truck is ar-rested using a bed of loose stones AB and a set ofcrash barrels BC. If experiments show that thestones provide a rolling resistance of 160 lb perwheel and the crash barrels provide a resistanceas shown in the graph, determine the distancex the 4500 lb truck penetrates the barrels if thetruck is coasting at 60 ft= s when it approachesA. Take s = 50 ft and neglect the size of thetruck. (x = 5:44 ft)

3. (14-34 Hibbeler, 11e) The 10 lb block is pressedagainst the spring so as to compress it 2 ft whenit is at A. If the plane is smooth, determine thedistance d, measured from the wall, to where theblock strikes the ground. Neglect the size of theblock. (d = 36:28 ft)

4. (14-38 Hibbeler, 11e) The spring has sti¤nessk = 50 lbf= ft and an unstretched length of 2 ft.As shown, it is con�ned by the plate and wallusing cables so that its length is 1:5 ft. A 4 lbblock is given a speed vA when it is at A, andit slides down the incline having a coe¢ cient ofkinetic friction �k = 0:2. If it strikes the plateand pushes it forward 0:25 ft before stopping, de-termine its speed at A. Neglect the mass of theplate and spring. (vA = 5:80 ft= s)

5. (14-55 Hibbeler, 11e) The elevator E and itsfreight have a total mass of 400 kg. Hoisting isprovided by the motor M and the 60 kg block C.If the motor has an e¢ ciency � = 0:6, determinethe power that must be supplied to the motorwhen the elevator is hoisted upward at a con-stant speed of vE = 4m= s. (Pin = 22:24 kW)

6. (14-56 Hibbeler, 11e) The 50 kg crate is hoistedup the 30 � incline by the pulley system and themotor M. If the crate starts from rest and byconstant acceleration attains a speed of 4m= safter travelling 8m along the plane, determinethe power that must be supplied to the motor atthis instant. Neglect friction along the plane.The motor has an e¢ ciency of � = 0:74. (Pin =1595:9 kW)

Eng3934 - Problem Sets 11

7. (14-83 Hibbeler, 11e) Just for fun, two 150 lbengineering students A and B intend to jumpo¤ the bridge from rest using an elastic cord(bungee cord) having a sti¤ness of k = 80 lbf= ft.They wish to just reach the surface of the river,when A, attached to the cord, lets go of B atthe instant they touch the water. Determinethe proper unstretched length of the cord to dothe stunt, and calculate the maximum acceler-ation of student A and the maximum heighthe reaches above the water after the rebound.From your results, comment on the feasibility ofdoing this stunt. (L = 90 ft, amax = 483 ft= s2!,ymax = 218:9 ft!, Stoopid!)

8. (14-88 Hibbeler, 11e) The Raptor is an outsideloop roller coaster in which riders are belted intoseats resembling ski-lift chairs. If the cars travelat vo = 4m= s when they are at the top of thehill, determine their speed when they are the topof the loop, and the reaction of the 70 kg passen-ger on his seat at this instant. The car has amass of 50 kg. Neglect friction and the size of thecar and passenger. (v = 7:43m= s, N = 86:6N)

9. (14-94 Hibbeler, 11e) A tank car is stopped bytwo spring bumpers A and B, having sti¤fness ofkA = 15 � 103 lbf= ft and kB = 20 � 103 lbf= ft,respectively. Bumper A is attached to the car,whereas bumper B is attached to the wall. If thecar has weight 25� 103 lb and is freely coastingat 3 ft= s, determine the maximum de�ection ofeach spring at the instant the bumper stops thecar. (sA = 0:516 ft, sB = 0:387 ft)