In a study of head injury against the instrument panel of a car during sudden or crash

For what acceleration a of the frame will the uniform slender rod maintain the orientation shown in the figure? Neglect the friction and mass of the small rollers at A and B. Problem 6/1

The right-angle bar with equal legs weighs 6 lb and is freely hinged to the vertical plate at C. The bar is prevented from rotating by the two pegs A and B fixed to the plate. Determine the acceleration a of the plate for which no force is exerted on the bar by either peg A or B. Problem 6/2

In Prob. if the plate is given a horizontal acceleration calculate the force exerted on the bar by either peg A or B.

The uniform slender bar of mass m is freely pivoted at point O of the frame of mass M. Determine the force P required to maintain the bar perpendicular to the incline of angle as the system accelerates in translation down the incline. The coefficient of kinetic friction between the frame and the incline is Problem 6/4

What acceleration a of the collar along the horizontal guide will result in a steady-state deflection of the pendulum from the vertical? The slender rod of length l and the particle each have mass m. Friction at the pivot P is negligible. Problem 6/5

The uniform box of mass m slides down the rough incline. Determine the location d of the effective normal force N. The effective normal force is located at the centroid of the nonuniform pressure distribution which the incline exerts on the bottom surface of the block. Problem 6/6

The homogeneous create of mass m is mounted on small wheels as shown. Determine the maximum force P which can be applied without overturning the crate about its lower front edge with and its lower back edge with Problem 6/7

Determine the value of P which will cause the homogeneous cylinder to begin to roll up out of its rectangular recess. The mass of the cylinder is m and that of the cart is M. The cart wheels have negligible mass and friction. Problem 6/8 P m M

Determine the acceleration of the initially stationary 20-kg body when the 50-N force P is applied as shown. The small wheels at B are ideal, and the feet at A are small. Problem 6/9

Repeat the previous problem for the case when the wheels and feet have been reversed as shown in the figure for this problem. Compare your answer to the stated result for the previous problem. Problem 6/10

The uniform 30-kg bar OB is secured to the accelerating frame in the position from the horizontal by the hinge at O and roller at A. If the horizontal acceleration of the frame is compute the force on the roller and the x- and y-components of the force supported by the pin at O. Problem 6/11 30 3000 mm 1000 mm B y x A a O

The rear-wheel-drive lawn mower, when placed into gear while at rest, is observed to momentarily spin its rear tires as it accelerates. If the coefficients of friction between the rear tires and the ground are and determine the forward acceleration a of the mower. The mass of the mower and attached bag is 50 kg with center of mass at G. Assume that the operator does not push on the handle, so that Problem 6/12

The 6-kg frame AC and 4-kg uniform slender bar AB of length l slide with negligible friction along the fixed horizontal rod under the action of the 80-N force. Calculate the tension T in wire BC and the x- and y-components of the force exerted on the bar by the pin at A. The x-y plane is vertical. Problem 6/13

The mass center of the rear-engine 3200-lb car is at G. Determine the normal forces and exerted by the road on the front and rear pairs of tires for the conditions of (a) being stationary and (b) braking from a forward velocity v with all wheels locked. The coefficient of kinetic friction is 0.90 at all tire/road interfaces. Express all answers in terms of pounds and as percentages of the vehicle weight.

Repeat the questions of the previous problem for the 3200-lb front-engine car shown, and compare your answers with those listed for the previous problem. Problem 6/15

The uniform 4-m boom has a mass of 60 kg and is pivoted to the back of a truck at A and secured by a cable at C. Calculate the magnitude of the total force supported by the connection at A if the truck starts from rest with an acceleration of Problem 6/16 A

The loaded trailer has a mass of 900 kg with center of mass at G and is attached at A to a rear-bumper hitch. If the car and trailer reach a velocity of on a level road in a distance of from rest with constant acceleration, compute the vertical component of the force supported by the hitch at A. Neglect the small friction force exerted on the relatively light wheels. Problem 6/17

Arm AB of a classifying accelerometer has a weight of 0.25 lb with mass center at G and is pivoted freely to the frame F at A. The torsional spring at A is set to preload the arm with an applied clockwise moment of 2 lb-in. Determine the downward acceleration a of the frame at which the contact at B will separate and break the electrical circuit. Problem 6/18

The uniform 60-lb log is supported by the two cables and used as a battering ram. If the log is released from rest in the position shown, calculate the initial tension induced in each cable immediately after release and the corresponding angular acceleration of the cables.

Determine the magnitude P and direction of the force required to impart a rearward acceleration to the loaded wheelbarrow with no rotation from the position shown. The combined weight of the wheelbarrow and its load is 500 lb with center of gravity at G. Compare the normal force at B under acceleration with that for static equilibrium in the position shown. Neglect the friction and mass of the wheel. Problem 6/20

Solid homogeneous cylinders 400 mm high and 250 mm in diameter are supported by a flat conveyor belt which moves horizontally. If the speed of the belt increases according to where t is the time in seconds measured from the instant the increase begins, calculate the value of t for which the cylinders begin to tip over. Cleats on the belt prevent the cylinders from slipping. Problem 6/21

The block A and attached rod have a combined mass of 60 kg and are confined to move along the guide under the action of the 800-N applied force. The uniform horizontal rod has a mass of 20 kg and is welded to the block at B. Friction in the guide is negligible. Compute the bending moment M exerted by the weld on the rod at B. Problem 6/22

The parallelogram linkage shown moves in the vertical plane with the uniform 8-kg bar EF attached to the plate at E by a pin which is welded both to the plate and to the bar. A torque (not shown) is applied to link AB through its lower pin to drive the links in a clockwise direction. When reaches the links have an angular acceleration and an angular velocity of and respectively. For this instant calculate the magnitudes of the force F and torque M supported by the pin at E. Problem 6/23

The riding power mower has a mass of 140 kg with center of mass at The operator has a mass of 90 kg with center of mass at Calculate the minimum effective coefficient of friction which will permit the front wheels of the mower to lift off the ground as the mower starts to move forward. Problem 6/24

The 25-kg bar BD is attached to the two light links AB and CD and moves in the vertical plane. The lower link is subjected to a clockwise torque applied through its shaft at A. If each link has an angular velocity as it passes the horizontal position, calculate the force which the upper link exerts on the bar at D at this instant. Also find the angular acceleration of the links at this position. Problem 6/25

jet transport with a landing speed of reduces its speed to with a negative thrust R from its jet thrust reversers in a distance of 425 m along the runway with constant deceleration. The total mass of the aircraft is 140 Mg with mass center at G. Compute the reaction N under the nose wheel B toward the end of the braking interval and prior to the application of mechanical braking. At the lower speed, aerodynamic forces on the aircraft are small and may be neglected. Problem 6/26

The uniform L-shaped bar pivots freely at point P of the slider, which moves along the horizontal rod. Determine the steady-state value of the angle if (a) and (b) For what value of a would the steady-state value of be zero? Problem 6/27

The van seen from the rear is traveling at a speed v around a turn of mean radius r banked inward at an angle . The effective coefficient of friction between the tires and the road is . Determine (a) the proper bank angle for a given v to eliminate any tendency to slip or tip, and (b) the maximum speed v before the van tips or slips for a given . Note that the forces and the acceleration lie in the plane of the figure so that the problem may be treated as one of plane motion even though the velocity is normal to this plane.

The parallelogram linkage is used to transfer crates from platform A to platform B and is hydraulically operated. The oil pressure in the cylinder is programmed to provide a smooth transition of motion from to rad given by where t is in seconds. Determine the force at D on the pin (a) just after the start of the motion with and t essentially zero and (b) when The crate and platform have a combined mass of 200 kg with mass center at G. The mass of each link is small and may be neglected. Problem 6/29 600 mm

The 1800-kg rear-wheel-drive car accelerates forward at a rate of If the modulus of each of the rear and front springs is estimate the resulting momentary nose-up pitch angle . (This upward pitch angle during acceleration is called squat, while the downward pitch during braking is called dive!) Neglect the unsprung mass of the wheels and tires. (Hint: Begin by assuming a rigid vehicle.) Problem 6/30

The two wheels of the vehicle are connected by a 20-kg link AB with center of mass at G. The link is pinned to the wheel at B, and the pin at A fits into a smooth horizontal slot in the link. If the vehicle has a constant speed of determine the magnitude of the force supported by the pin at B for the position  30.

The uniform 200-kg bar AB is raised in the vertical plane by the application of a constant couple applied to the link at C. The mass of the links is small and may be neglected. If the bar starts from rest at determine the magnitude of the force supported by the pin at A as the position is passed. Problem 6/32

The uniform 20-kg slender bar is pivoted at O and swings freely in the vertical plane. If the bar is released from rest in the horizontal position, calculate the initial value of the force R exerted by the bearing on the bar an instant after release. Problem 6/33

The 20-kg uniform steel plate is freely hinged about the z-axis as shown. Calculate the force supported by each of the bearings at A and B an instant after the plate is released from rest in the horizontal y-z plane. Problem 6/34

The uniform 100-kg beam is freely hinged about its upper end A and is initially at rest in the vertical position with Determine the initial angular acceleration of the beam and the magnitude of the force supported by the pin at A due to the application of a force on the attached cable.

The automotive dynamometer is able to simulate road conditions for an acceleration of 0.5g for the loaded pickup truck with a gross weight of 5200 lb. Calculate the required moment of inertia of the dynamometer drum about its center O assuming that the drum turns freely during the acceleration phase of the test. Problem 6/36

A momentum wheel for dynamics-class demonstrations is shown. It is basically a bicycle wheel modified with rim band-weighting, handles, and a pulley for cord startup. The heavy rim band causes the radius of gyration of the 7-lb wheel to be 11 in. If a steady 10-lb pull T is applied to the cord, determine the angular acceleration of the wheel. Neglect bearing friction. Problem 6/37

Determine the angular acceleration and the force on the bearing at O for (a) the narrow ring of mass m and (b) the flat circular disk of mass m immediately after each is released from rest in the vertical plane with OC horizontal. Problem 6/38

The 30-in. slender bar weighs 20 lb and is mounted on a vertical shaft at O. If a torque lb-in. is applied to the bar through its shaft, calculate the horizontal force R on the bearing as the bar starts to rotate.

The uniform slender bar AB has a mass of 8 kg and swings in a vertical plane about the pivot at A. If when compute the force supported by the pin at A at that instant. Problem 6/40

The uniform quarter-circular sector of mass m is released from rest with one straight edge vertical as shown. Determine the initial angular acceleration and the horizontal and vertical components of the reaction at the ideal pivot at O. Problem 6/41 O b m Vertical

The circular sector of uniform thickness and mass m is released from rest when one of its straight edges is vertical as shown. Determine the initial angular acceleration about the ideal pivot at O. Evaluate your general expression for and Compare your results to the stated answer for the previous problem. Problem 6/42

The square frame is composed of four equal lengths of uniform slender rod, and the ball attachment at O is suspended in a socket (not shown). Beginning from the position shown, the assembly is rotated about axis A-A and released. Determine the initial angular acceleration of the frame. Repeat for a rotation about axis B-B. Neglect the small mass, offset, and friction of the ball. Problem 6/43

If the system is released from rest while in the horizontal position shown, determine the angular acceleration of the lightweight right-angle shaft. The sphere of radius r has mass m. Neglect friction at the bearing O.

If the slender-bar assembly is released from rest while in the horizontal position shown, determine its angular acceleration. The mass per unit length of the bar is . Neglect friction at the bearing O. Problem 6/45

An air table is used to study the elastic motion of flexible spacecraft models. Pressurized air escaping from numerous small holes in the horizontal surface provides a supporting air cushion which largely eliminates friction. The model shown consists of a cylindrical hub of radius r and four appendages of length l and small thickness t. The hub and the four appendages all have the same depth d and are constructed of the same material of density . Assume that the spacecraft is rigid and determine the moment M which must be applied to the hub to spin the model from rest to an angular velocity in a time period of seconds. (Note that for a spacecraft with highly flexible appendages, the moment must be judiciously applied to the rigid hub to avoid undesirable large elastic deflections of the appendages.) Problem 6/46

The narrow ring of mass m is free to rotate in the vertical plane about O. If the ring is released from rest at determine expressions for the n- and t-components of the force at O in terms of . Problem 6/47

Determine the angular acceleration of the uniform disk if (a) the rotational inertia of the disk is ignored and (b) the inertia of the disk is considered. The system is released from rest, the cord does not slip on the disk, and bearing friction at O may be neglected. Problem 6/48

The solid homogeneous cylinder weighs 300 lb and is free to rotate about the horizontal axis O-O. If the cylinder, initially at rest, is acted upon by the 100-lb force shown, calculate the horizontal component R of the force supported by each of the two symmetrically placed bearings when the 100-lb force is first applied.

The solid cylindrical rotor B has a mass of 43 kg and is mounted on its central axis C-C. The frame A rotates about the fixed vertical axis O-O under the applied torque The rotor may be unlocked from the frame by withdrawing the locking pin P. Calculate the angular acceleration of the frame A if the locking pin is (a) in place and (b) withdrawn. Neglect all friction and the mass of the frame. Problem 6/50

The uniform 40-lb bar is released from rest in the horizontal position shown and strikes the fixed corner B at the center of percussion of the bar. Determine the t-component of the force exerted by the bearing O on the bar just prior to impact, during impact, and just after impact. Problem 6/51

Each of the two grinding wheels has a diameter of 6 in., a thickness of in., and a specific weight of When switched on, the machine accelerates from rest to its operating speed of 3450 rev/min in 5 sec. When switched off, it comes to rest in 35 sec. Determine the motor torque and frictional moment, assuming that each is constant. Neglect the effects of the inertia of the rotating motor armature. Problem 6/52

The bar A of mass m is formed into a circular arc of radius r and is attached to the hub by the light rods. The curved bar oscillates about the vertical axis under the action of a torsional spring B. At the instant under consideration, the angular velocity is and the angular acceleration is . Write expressions for the moment M exerted by the spring on the hub and the horizontal force R exerted by the shaft on the hub. Problem 6/53

The uniform slender bar is released from rest in the horizontal position shown. Determine the value of x for which the angular acceleration is a maximum, and determine the corresponding angular acceleration . Problem 6/54

The uniform rectangular slab is released from rest in the position shown. Determine the value of x for which the angular acceleration is a maximum, and determine the corresponding angular acceleration. Compare your answers with those listed for Prob. 6/54. Problem 6/55

The spring is uncompressed when the uniform slender bar is in the vertical position shown. Determine the initial angular acceleration of the bar when it is released from rest in a position where the bar has been rotated clockwise from the position shown. Neglect any sag of the spring, whose mass is negligible. Problem 6/56

A gimbal pedestal supports a payload in the space shuttle and deploys it when the doors of the cargo bay are opened in orbit. The payload is modeled as a homogeneous rectangular block with a mass of 6000 kg. The torque on the gimbal axis O-O is 30 supplied by a d-c brushless motor. With the shuttle orbiting in a weightless condition, determine the time t required to bring the payload from its stowed position at to its deployed position at if the torque is applied for the first of travel and then reversed for the remaining to bring the payload to a stop ( 0). 45 45  0  90 N m l k A G O B m l 4 l 4 l 4 l 4 30 x G b O b Problem 6/57

A uniform slender bar of mass m and length 2b is mounted in a right-angle frame of negligible mass. The bar and frame rotate in the vertical plane about a fixed axis at O. If the bar is released from rest in the vertical position derive an expression for the magnitude of the force exerted by the bearing at O on the frame as a function of . Problem 6/58

The uniform semicircular bar of mass m and radius r is hinged freely about a horizontal axis through A. If the bar is released from rest in the position shown, where AB is horizontal, determine the initial angular acceleration of the bar and the expression for the force exerted on the bar by the pin at A. (Note carefully that the initial tangential acceleration of the mass center is not vertical.) Problem 6/59

A device for impact testing consists of a 34-kg pendulum with mass center at G and with radius of gyration about O of 620 mm. The distance b for the pendulum is selected so that the force on the bearing at O has the least possible value during impact with the specimen at the bottom of the swing. Determine b and calculate the magnitude of the total force R on the bearing O an instant after release from rest at Problem 6/60 600 mm Specimen b O G

The 12-kg cylinder supported by the bearing brackets at A and B has a moment of inertia about the vertical through its mass center G equal to The disk and brackets have a moment of inertia about the vertical z-axis of rotation equal to If a torque is applied to the disk through its shaft with the disk initially at rest, calculate the horizontal x-components of force supported by the bearings at A and B. Problem 6/61

The 24-kg uniform slender bar AB is mounted on end rollers of negligible mass and rotates about the fixed point O as it follows the circular path in the vertical plane. The bar is released from a position which gives it an angular velocity as it passes the position Calculate the forces and exerted by the guide on the rollers for this instant. Problem 6/62

The mass of gear A is 20 kg and its centroidal radius of gyration is 150 mm. The mass of gear B is 10 kg and its centroidal radius of gyration is 100 mm. Calculate the angular acceleration of gear B when a torque of 12 is applied to the shaft of gear A. Neglect friction. Problem 6/63

Prior to deployment of its two instrument arms AB, the spacecraft shown in the upper view is spinning at the constant rate of 1 revolution per second. Each instrument arm, shown in the lower view, has a mass of 10 kg with mass center at G. Calculate the tension T in the deployment cable prior to release. Also find the magnitude of the force on the pin at A. Neglect any acceleration of the center O of the spacecraft. Problem 6/64

Disk B weighs 50 lb and has a centroidal radius of gyration of 8 in. The power unit C consists of a motor M and a disk A, which is driven at a constant angular speed of 1600 rev/min. The coefficients of static and kinetic friction between the two disks are and respectively. Disk B is initially stationary when contact with disk A is established by application of the constant force Determine the angular acceleration of B and the time t required for B to reach its steady-state speed. Problem 6/65

Two slender bars AB, each of mass m and length l, are pivoted at A to the plate. The plate rotates in the horizontal plane about a fixed vertical axis through its center O and is given a constant angular acceleration . (a) Determine the force F exerted on each of the two rollers as the assembly starts to rotate. (b) Find the total force on the pin at A and show that it remains constant as long as (c) Determine the angular velocity at which contact with the rollers ceases. Problem 6/66 l 2 l 2 l 2 l 2 A B B A

The robotic device consists of the stationary pedestal OA, arm AB pivoted at A, and arm BC pivoted at B. The rotation axes are normal to the plane of the figure. Estimate (a) the moment applied to arm AB required to rotate it about joint A at counterclockwise from the position shown with joint B locked and (b) the moment applied to arm BC required to rotate it about joint B at the same rate with joint A locked. The mass of arm AB is 25 kg and that of BC is 4 kg, with the stationary portion of joint A excluded entirely and the mass of joint B divided equally between the two arms. Assume that the centers of mass and are in the geometric centers of the arms and model the arms as slender rods. Problem 6/67

Each of the two uniform slender bars OA and BC has a mass of 8 kg. The bars are welded at A to form a T-shaped member and are rotating freely about a horizontal axis through O. If the bars have an angular velocity of 4 rad/s as OA passes the horizontal position shown, calculate the total force R supported by the bearing at O. Problem 6/68

The uniform slender bar of mass m and length l is released from rest in the vertical position and pivots on its square end about the corner at O. (a) If the bar is observed to slip when find the coeffi- cient of static friction between the bar and the corner. (b) If the end of the bar is notched so that it cannot slip, find the angle at which contact between the bar and the corner ceases. Problem 6/69

The uniform rectangular block is released from rest with essentially zero and pivots in the vertical plane about the center A of its lower face on the fixed corner. (a) If the block is observed to slip when find the coefficient of static friction between the block and the corner. (b) If the bottom face of the block is notched so that it cannot slip, find the angle at which contact between the block and the corner ceases. Problem 6/70

The uniform square plate of mass m is lying motionless on the horizontal surface when the force P is applied at A as shown. Determine the resulting initial acceleration of point B. Friction is negligible. Problem 6/71

The L-shaped bar of mass m is lying motionless on the horizontal surface when the force P is applied at A as shown. Determine the initial acceleration of point A. Neglect friction and the thickness of the bar. Problem 6/72

The body consists of a uniform slender bar and a uniform disk, each of mass It rests on a smooth surface. Determine the angular acceleration and the acceleration of the mass center of the body when the force is applied as shown. The value of the mass m of the entire body is 1.2 kg

Repeat Prob. 6/73, except now the location of force P has been changed. The value of the mass m of the entire body is 1.2 kg. Problem 6/74

Above the earths atmosphere at an altitude of 400 km where the acceleration due to gravity is a certain rocket has a total remaining mass of 300 kg and is directed 30 from the vertical. If the thrust T from the rocket motor is 4 kN and if the rocket nozzle is tilted through an angle of as shown, calculate the angular acceleration of the rocket and the x- and y-components of the acceleration of its mass center G. The rocket has a centroidal radius of gyration of 1.5 m. Problem 6/75 G 3 m 1 T

The 10-kg wheel with a radius of gyration of 180 mm m2 8 kg about its center O is released from rest on the incline and slips as it rolls. If the coefficient of kinetic friction is calculate the acceleration of the center O of the wheel and its angular acceleration . Problem 6/76

How large would the coefficient of static friction have to be in order that the wheel of Prob. 6/76 not slip as it rolls?

The solid homogeneous cylinder is released from rest on the ramp. If and determine the acceleration of the mass center G and the friction force exerted by the ramp on the cylinder. Problem 6/78

Repeat Prob. 6/78, except let and k 0.10.

The uniform disk of mass pivots freely on the cart of mass Determine the acceleration of the assembly and the angular acceleration of the disk under the action of the force applied to a cord wrapped securely around the disk. Problem 6/80

The fairing which covers the spacecraft package in the nose of the booster rocket is jettisoned when the rocket is in space where gravitational attraction is negligible. A mechanical actuator moves the two halves slowly from the closed position I to position II at which point the fairings are released to rotate freely about their hinges at O under the influence of the constant acceleration a of the rocket. When position III is reached, the hinge at O is released and the fairings drift away from the rocket. Determine the angular velocity of the fairing at the position. The mass of each fairing is m with center of mass at G and radius of gyration about O. Problem 6/81

Determine the angular acceleration of each of the two wheels as they roll without slipping down the inclines. For wheel A investigate the case where the mass of the rim and spokes is negligible and the mass of the bar is concentrated along its centerline. For wheel B assume that the thickness of the rim is negligible compared with its radius so that all of the mass is concentrated in the rim. Also specify the minimum coefficient of static friction required to prevent each wheel from slipping. Problem 6/82

A uniform slender rod of length l and mass m is secured to a circular hoop of radius l as shown. The mass of the hoop is negligible. If the rod and hoop are released from rest on a horizontal surface in the position illustrated, determine the initial values of the friction force F and normal force N under the hoop if friction is sufficient to prevent slipping

The uniform 12-kg square panel is suspended from point C by the two wires at A and B. If the wire at B suddenly breaks, calculate the tension T in the wire at A an instant after the break occurs. Problem 6/84

The uniform steel beam of mass m and length l is suspended by the two cables at A and B. If the cable at B suddenly breaks, determine the tension T in the cable at A immediately after the break occurs. Treat the beam as a slender rod and show that the result is independent of the length of the beam. Problem 6/85

The circular disk of mass m and radius r is rolling through the bottom of the circular path of radius R. If the disk has an angular velocity , determine the force N exerted by the path on the disk. Problem 6/86

The system is released from rest with the cable taut, and the homogeneous cylinder does not slip on the rough incline. Determine the angular acceleration of the cylinder and the minimum coefficient of friction for which the cylinder will not slip. Problem 6/87

The circular disk of 200-mm radius has a mass of 25 kg with centroidal radius of gyration mm and has a concentric circular groove of 75-mm radius cut into it. A steady force T is applied at an angle to a cord wrapped around the groove as shown. If and determine the angular acceleration of the disk, the acceleration a of its mass center G, and the friction force F which the surface exerts on the disk.

Repeat Prob. 6/88, except let and 30.

The uniform rectangular panel of mass m is moving to the right when wheel B drops off the horizontal support rail. Determine the resulting angular acceleration and the force in the strap at A immediately after wheel B rolls off the rail. Neglect friction and the mass of the small straps and wheels. Problem 6/90

The uniform slender bar AB has a mass of 0.8 kg and is driven by crank OA and constrained by link CB of negligible mass. If OA has an angular acceleration and an angular velocity when both OA and CB are normal to AB, calculate the force in CB for this instant. (Suggestion: Consider the use of Eq. 6/3 with A as a moment center.) Problem 6/91

The crank OA rotates in the vertical plane with a constant clockwise angular velocity of For the position where OA is horizontal, calculate the force under the light roller B of the 10-kg slender bar AB. Problem 6/92

Each of the two solid disk wheels weighs 20 lb, and the inner solid cylinder weighs 16 lb. The disk wheels and the inner disk are mounted on the small central shaft O-O and can rotate independently of each other. Friction in the shaft bearings is negligible, whereas friction between the incline and the large disk wheels is sufficient to prevent slippage of the wheels. Determine the acceleration of the center O after the assembly is released on the incline. The cord wrapped securely around the inner cylinder is fastened to point A. Problem 6/93

The robotic device of Prob. 6/67 is repeated here. Member AB is rotating about joint A with a counterclockwise angular velocity of and this rate is increasing at Determine the moment exerted by arm AB on arm BC if joint B is held in a locked condition. The mass of arm BC is 4 kg, and the arm may be treated as a uniform slender rod. Problem 6/94

The uniform slender rod of mass m and length L is released from rest in the inverted vertical position shown. Neglect friction and the mass of the small end roller and find the initial acceleration of A. Evaluate your result for Problem 6/95 L

In an investigation of whiplash resulting from rearend collision, sudden rotation of the head is modeled by using a homogeneous solid sphere of mass m and radius r pivoted about a tangent axis (at the neck) to represent the head. If the axis at O is given a constant acceleration a with the head initially at rest, determine expressions for the initial angular acceleration of the head and its angular velocity as a function of the angle of rotation. Assume that the neck is relaxed so that no moment is applied to the head at O. Problem 6/96

The uniform 15-kg bar is supported on the horizontal surface at A by a small roller of negligible mass. If the coefficient of kinetic friction between end B and the vertical surface is 0.30, calculate the initial acceleration of end A as the bar is released from rest in the position shown. Problem 6/97

The assembly consisting of a uniform slender bar (mass m/5) and a rigidly attached uniform disk (mass 4m/5) is freely pinned to point O on the collar that in turn slides on the fixed horizontal guide. The assembly is at rest when the collar is given a sudden acceleration a to the left as shown. Determine the initial angular acceleration of the assembly. Problem 6/98

The uniform 12-ft pole is hinged to the truck bed and released from the vertical position as the truck starts from rest with an acceleration of If the acceleration remains constant during the motion of the pole, calculate the angular velocity of the pole as it reaches the horizontal position. Problem 6/99 12

The T-shaped body of mass m is composed of two identical slender bars welded together. If the body is released from rest in the vertical plane in the position shown, determine the initial acceleration of point A. Neglect the small mass and friction of the roller. Problem 6/100

A bowling ball with a circumference of 27 in. weighs 14 lb and has a radius of gyration of 3.28 in. If the ball is released with a velocity of but with no angular velocity as it touches the alley floor, compute the distance traveled by the ball before it begins to roll without slipping. The coefficient of friction between the ball and the floor is 0.20. Problem 6/101

The compound pendulum of mass m and radius of gyration about O is freely hinged to the trolley, which is given a constant horizontal acceleration a from rest with the pendulum initially at rest with Determine an expression for the angular acceleration and the n- and t-components of the force at O as functions of . Calculate the maximum value reached by if Problem 6/102

In a study of head injury against the instrument panel of a car during sudden or crash stops where lap belts without shoulder straps or airbags are used, the segmented human model shown in the figure is analyzed. The hip joint O is assumed to remain fixed relative to the car, and the torso above the hip is treated as a rigid body of mass m freely pivoted at O. The center of mass of the torso is at G with the initial position of OG taken as vertical. The radius of gyration of the torso about O is If the car is brought to a sudden stop with a constant deceleration a, determine the velocity v relative to the car with which the models head strikes the instrument panel. Substitute the values and and compute v. Problem 6/103

The uniform slender bar of mass m and length L with small end rollers is released from rest in the position shown with the lower roller in contact with the horizontal plane. Determine the normal force N under the lower roller and the angular acceleration of the bar immediately after release. Problem 6/104 L

The connecting rod AB of a certain internalcombustion engine weighs 1.2 lb with mass center at G and has a radius of gyration about G of 1.12 in. The piston and piston pin A together weigh 1.80 lb. The engine is running at a constant speed of 3000 so that the angular velocity of the crank is Neglect the weights of the components and the force exerted by the gas in the cylinder compared with the dynamic forces generated and calculate the magnitude of the force on the piston pin A for the crank angle (Suggestion: Use the alternative moment relation, Eq. 6/3, with B as the moment center.) Problem 6/105

A particle of mass m is embedded at the periphery of the otherwise uniform disk of mass M and radius r as shown. The disk starts from rest and does not slip on the rough incline. (a) For the position shown, what condition on m will cause the disk to begin to roll up the incline? (b) If determine the initial angular acceleration of the disk and the minimum value of the coefficient of static friction required for the no-slip condition.

The small rollers at the ends of the uniform slender bar are confined to the circular slot in the vertical surface. If the bar is released from rest in the position shown, determine the initial angular acceleration . Neglect the mass and friction of the rollers. Problem 6/107

The small end rollers of the 8-lb uniform slender bar are constrained to move in the slots, which lie in a vertical plane. At the instant when , the angular velocity of the bar is counterclockwise. Determine the angular acceleration of the bar, the reactions at A and B, and the accelerations of points A and B under the action of the 6-lb force P. Neglect the friction and mass of the small rollers. 2 rad/sec  30 Problem 6/108 105 B P

The uniform slender bar of mass m and length L is released from rest when in the horizontal position shown. Determine its angular velocity and masscenter speed as it passes the vertical position. Problem 6/109

The slender rod (mass m, length L) has a particle (mass 2m) attached to one end. If the body is nudged away from the vertical equilibrium position shown, determine its angular speed after it has rotated 180 . Problem 6/110

The 32.2-lb wheel is released from rest and rolls on its hubs without slipping. Calculate the velocity v of the center O of the wheel after it has moved a distance ft down the incline. The radius of gyration of the wheel about O is 5 in.

The uniform quarter-circular sector is released from rest with one edge vertical as shown. Determine its subsequent maximum angular velocity. Problem 6/112

The velocity of the 8-kg cylinder is 0.3 m/s at a certain instant. What is its speed v after dropping an additional 1.5 m? The mass of the grooved drum is 12 kg, its centroidal radius of gyration is and the radius of its groove is The frictional moment at O is a constant 3 . Problem 6/113

The log is suspended by the two parallel 5-m cables and used as a battering ram. At what angle should the log be released from rest in order to strike the object to be smashed with a velocity of 4 m/s? Problem 6/114

The T-shaped body of total mass m is constructed of uniform rod. If it is released from rest while in the position shown, determine the vertical force reaction at O as it passes the vertical position (120 after release). Problem 6/115

The two wheels of Prob. 6/82, shown again here, represent two extreme conditions of distribution of mass. For case A all of the mass m is assumed to be concentrated in the center of the hoop in the axial bar of negligible diameter. For case B all of the mass m is assumed to be concentrated in the rim. Determine the velocity of the center of each hoop after it has traveled a distance x down the incline from rest. The hoops roll without slipping.

The 15-kg slender bar OA is released from rest in the vertical position and compresses the spring of stiffness as the horizontal position is passed. Determine the proper setting of the spring, by specifying the distance h, which will result in the bar having an angular velocity as it crosses the horizontal position. What is the effect of x on the dynamics of the problem? Problem 6/117 h A O x = 400 mm

The wheel is composed of a 10-kg hoop stiffened by four thin spokes, each with a mass of 2 kg. A horizontal force of 40 N is applied to the wheel initially at rest. Calculate the angular velocity of the wheel after its center has moved 3 m. Friction is sufficient to prevent slipping. Problem 6/118

A steady 5-lb force is applied normal to the handle of the hand-operated grinder. The gear inside the housing with its shaft and attached handle together weigh 3.94 lb and have a radius of gyration about their axis of 2.85 in. The grinding wheel with its attached shaft and pinion (inside housing) together weigh 1.22 lb and have a radius of gyration of 2.14 in. If the gear ratio between gear and pinion is 4:1, calculate the speed N of the grinding wheel after 6 complete revolutions of the handle starting from rest. Problem 6/119

The 1.2-kg uniform slender bar rotates freely about a horizontal axis through O. The system is released from rest when it is in the horizontal position where the spring is unstretched. If the bar is observed to momentarily stop in the position determine the spring constant k. For your computed value of k, what is the angular velocity of the bar when Problem 6/120

Specify the unstretched length l0 of the spring of stiffness which will result in a velocity of for the contact at A if the toggle is given a slight nudge from its null position at The toggle has a mass of 1.5 kg and a radius of gyration about O of 55 mm. Motion occurs in the horizontal plane. Problem 6/121

The 50-kg flywheel has a radius of gyration about its shaft axis and is subjected to the torque where is in radians. If the flywheel is at rest when determine its angular velocity after 5 revolutions. Problem 6/122

The 12-lb lever OA with 10-in. radius of gyration about O is initially at rest in the vertical position where the attached spring of stiffness is unstretched. Calculate the constant moment M applied to the lever through its shaft at O which will give the lever an angular velocity as the lever reaches the horizontal position Problem 6/123

The two identical links, each of length b and mass m, may be treated as uniform slender bars. If they are released from rest in the position shown with end A constrained by the smooth vertical guide, determine the velocity v with which A reaches O with essentially zero. Problem 6/124

The torsional spring has a stiffness of 30 m/rad and is undeflected when the 6-kg uniform slender bar is in the upright position. If the bar is released from rest in the horizontal position shown, determine its angular velocity as it passes the vertical position. Friction is negligible. Problem 6/125

The wheel consists of a 4-kg rim of 250-mm radius with hub and spokes of negligible mass. The wheel is mounted on the 3-kg yoke OA with mass center at G and with a radius of gyration about O of 350 mm. If the assembly is released from rest in the horizontal position shown and if the wheel rolls on the circular surface without slipping, compute the velocity of point A when it reaches . Problem 6/126

The uniform slender bar ABC weighs 6 lb and is initially at rest with end A bearing against the stop in the horizontal guide. When a constant couple -in. is applied to end C, the bar rotates causing end A to strike the side of the vertical guide with a velocity of 10 ft/sec. Calculate the loss of energy E due to friction in the guides and rollers. The mass of the rollers may be neglected. Problem 6/127

The center of the 200-lb wheel with centroidal radius of gyration of 4 in. has a velocity of 2 ft/sec down the incline in the position shown. Calculate the normal reaction N under the wheel as it rolls past position A. Assume that no slipping occurs. Problem 6/128

The uniform slender bar of mass m pivots freely about a horizontal axis through O. If the bar is released from rest in the horizontal position shown where the spring is unstretched, it is observed to rotate a maximum of 30 clockwise. The spring constant and the distance Determine (a) the mass m of the bar and (b) the angular velocity of the bar when the angular displacement is 15 clockwise from the release position. Problem 6/129 b 4 3b 4 b 4 O m A

The system is released from rest when the angle Determine the angular velocity of the uniform slender bar when equals 60 . Use the values Problem 6/130

The two identical steel frames with the dimensions shown are fabricated from the same bar stock and are hinged at the midpoints A and B of their sides. If the frame is resting in the position shown on a horizontal surface with negligible friction, determine the velocity v with which each of the upper ends of the frame hits the horizontal surface if the cord at C is cut. Problem 6/131 b b b b c c

The electric motor shown is delivering 4 kW at 1725 rev/min to a pump which it drives. Calculate the angle through which the motor deflects under load if the stiffness of each of its four spring mounts is 15 kN/m. In what direction does the motor shaft turn? Problem 6/132

The two uniform right-angle bars are released from rest when in the position at which the spring of modulus is unstretched. The bars then rotate in a vertical plane about the fixed centers of the attached light gears, thus maintaining the same angle for both bars. Determine the angular speed of the bars as the position is passed. Problem 6/133

A lid-support mechanism is being designed for a storage chest to limit the angular velocity of the 10-lb uniform lid to 1.5 rad/sec for when it is released from rest with essentially equal to 90 . Two identical mechanisms are included as indicated on the pictorial sketch. Specify the necessary stiffness k of each of the two springs, which are compressed 2 in. upon closure. Neglect the weight of the links and any friction in the sliding collars C. Also, the thickness of the lid is small compared with its other dimensions. Problem 6/134

Each of the two links has a mass of 2 kg and a centroidal radius of gyration of 60 mm. The slider at B has a mass of 3 kg and moves freely in the vertical guide. The spring has a stiffness of 6 kN/m. If a constant torque is applied to link OA through its shaft at O starting from the rest position at determine the angular velocity of OA when  0.

The system is at rest with the spring unstretched when The 5-kg uniform slender bar is then given a slight clockwise nudge. The value of b is 0.4 m. (a) If the bar comes to momentary rest when determine the spring constant k. (b) For the value find the angular velocity of the bar when Problem 6/136 k A

The body shown is constructed of uniform slender rod and consists of a ring of radius r attached to a straight section of length 2r. The body pivots freely about a ball-and-socket joint at O. If the body is at rest in the vertical position shown and is given a slight nudge, compute its angular velocity after a 90 rotation about (a) axis A-A and (b) axis B-B. Problem 6/137

A facility for testing the performance of motorized golf carts consists of an endless belt where the angle can be adjusted. The cart of mass m is slowly brought up to its rated ground speed v with the braking torque M on the upper pulley constantly adjusted so that the cart remains in a fixed position A on the test stand. With no cart on the belt, a torque is required on the pulley to overcome friction and turn the pulleys regardless of speed. Friction is sufficient to prevent the wheels from slipping on the belt. Determine an expression for the power P absorbed by the braking torque M. Do the static friction forces between the wheels and the belt do work? Problem 6/138 r v v A M

The 8-kg crank OA, with mass center at G and radius of gyration about O of 0.22 m, is connected to the 12-kg uniform slender bar AB. If the linkage is released from rest in the position shown, compute the velocity v of end B as OA swings through the vertical. Problem 6/139

The figure shows the cross section of a uniform 200-lb ventilator door hinged about its upper horizontal edge at O. The door is controlled by the spring-loaded cable which passes over the small pulley at A. The spring has a stiffness of 15 lb per foot of stretch and is undeformed when If the door is released from rest in the horizontal position, determine the maximum angular velocity reached by the door and the corresponding angle . Problem 6/140

Motive power for the experimental 10-Mg bus comes from the energy stored in a rotating flywheel which it carries. The flywheel has a mass of 1500 kg and a radius of gyration of 500 mm and is brought up to a maximum speed of 4000 rev/min. If the bus starts from rest and acquires a speed of 72 km/h at the top of a hill 20 m above the starting position, compute the reduced speed N of the flywheel. Assume that 10 percent of the energy taken from the flywheel is lost. Neglect the rotational energy of the wheels of the bus. The 10-Mg mass includes the flywheel. Problem 6/141

The two identical uniform bars are released from rest from the position shown in the vertical plane. Determine the angular velocity of AB when the bars become collinear. Problem 6/142

The figure shows the cross section of a garage door which is a uniform rectangular panel 8 by 8 ft and weighing 200 lb. The door carries two spring assemblies, one on each side of the door, like the one shown. Each spring has a stiffness of 50 lb/ft and is unstretched when the door is in the open position shown. If the door is released from rest in this position, calculate the velocity of the edge at A as it strikes the garage floor.

The uniform slender rod of length l is released from rest in the dashed vertical position. With what speed does end A strike the 30 incline? Neglect the small mass and friction of the end rollers. Problem 6/144

The 10-kg double wheel with radius of gyration of 125 mm about O is connected to the spring of stiffness by a cord which is wrapped securely around the inner hub. If the wheel is released from rest on the incline with the spring stretched 225 mm, calculate the maximum velocity v of its center O during the ensuing motion. The wheel rolls without slipping. Problem 6/145

Motion of the 600-mm slender bar of mass 4 kg is controlled by the constrained movement of its small rollers A and B of negligible mass and friction. The bar starts from rest in the horizontal position with and moves in the vertical plane under the action of the constant force applied normal to the bar at end C. Calculate the velocity v with which roller A strikes the wall of the vertical guide at Problem 6/146

The position of the horizontal platform of mass is controlled by the parallel slender links of masses m and 2m. Determine the initial angular acceleration of the links as they start from their supported position shown under the action of a force P applied normal to AB at its end. Problem 6/147

The uniform slender bar of mass m is shown in its equilibrium position in the vertical plane before the couple M is applied to the end of the bar. Determine the initial angular acceleration of the bar upon application of M. The mass of each guide roller is negligible. Problem 6/148

The two uniform slender bars are hinged at O and supported on the horizontal surface by their end rollers of negligible mass. If the bars are released from rest in the position shown, determine their initial angular acceleration as they collapse in the vertical plane. (Suggestion: Make use of the instantaneous center of zero velocity in writing the expression for dT.)

Links A and B each weigh 8 lb, and bar C weighs 12 lb. Calculate the angle  assumed by the links if the body to which they are pinned is given a steady horizontal acceleration a of 4 ft/sec2 . Problem 6/150

The mechanism shown moves in the vertical plane. The vertical bar AB weighs 10 lb, and each of the two links weighs 6 lb with mass center at G and with a radius of gyration of 10 in. about its bearing (O or C). The spring has a stiffness of 15 lb/ft and an unstretched length of 18 in. If the support at D is suddenly withdrawn, determine the initial angular acceleration of the links. Problem 6/151

The load of mass m is given an upward acceleration a from its supported rest position by the application of the forces P. Neglect the mass of the links compared with m and determine the initial acceleration a. Problem 6/152

The cargo box of the food-delivery truck for aircraft servicing has a loaded mass m and is elevated by the application of a couple M on the lower end of the link which is hinged to the truck frame. The horizontal slots allow the linkage to unfold as the cargo box is elevated. Determine the upward acceleration of the box in terms of h for a given value of M. Neglect the mass of the links. Problem 6/153

The box and load of the dump truck have a mass m with mass center at G and a moment of inertia IA about the pivot at A. Determine the angular acceleration of the box when it is started from rest in the position shown under the application of the couple M to link CD. Neglect the mass of the links. The figure ABDC is a parallelogram. Problem 6/154

Each of the uniform bars OA and OB has a mass of 2 kg and is freely hinged at O to the vertical shaft, which is given an upward acceleration a g/2. The links which connect the light collar C to the bars have negligible mass, and the collar slides freely on the shaft. The spring has a stiffness k 130 N/m and is uncompressed for the position equivalent to  0. Calculate the angle  assumed by the bars under conditions of steady acceleration. Problem 6/155

The linkage consists of the two slender bars and moves in the horizontal plane under the influence of force P. Link OC has a mass m and link AC has a mass 2m. The sliding block at B has negligible mass. Without dismembering the system, determine the initial angular acceleration of the links as P is applied at A with the links initially at rest. (Suggestion: Replace P by its equivalent force-couple system.) Problem 6/156

The portable work platform is elevated by means of the two hydraulic cylinders articulated at points C. The pressure in each cylinder produces a force F. The platform, man, and load have a combined mass m, and the mass of the linkage is small and may be neglected. Determine the upward acceleration a of the platform and show that it is independent of both b and . Problem 6/157

Each of the three identical uniform panels of a segmented industrial door has mass m and is guided in the tracks (one shown dashed). Determine the horizontal acceleration a of the upper panel under the action of the force P. Neglect any friction in the guide rollers. Problem 6/158

The mechanical tachometer measures the rotational speed N of the shaft by the horizontal motion of the collar B along the rotating shaft. This movement is caused by the centrifugal action of the two 12-oz weights A, which rotate with the shaft. Collar C is fixed to the shaft. Determine the rotational speed N of the shaft for a reading  15 . The stiffness of the spring is 5 lb/in., and it is uncompressed when  0 and  0. Neglect the weights of the links. Problem 6/159

A planetary gear system is shown, where the gear teeth are omitted from the figure. Each of the three identical planet gears A, B, and C has a mass of 0.8 kg, a radius r 50 mm, and a radius of gyration of 30 mm about its center. The spider E has a mass of 1.2 kg and a radius of gyration about O of 60 mm. The ring gear D has a radius R 150 mm and is fixed. If a torque M 5 is applied to the shaft of the spider at O, determine the initial angular acceleration of the spider. Problem 6/160

The sector and attached wheels are released from rest in the position shown in the vertical plane. Each wheel is a solid circular disk weighing 12 lb and rolls on the fixed circular path without slipping. The sector weighs 18 lb and is closely approximated by one-fourth of a solid circular disk of 16-in. radius. Determine the initial angular acceleration of the sector. Problem 6/161

The aerial tower shown is designed to elevate a workman in a vertical direction. An internal mechanism at B maintains the angle between AB and BC at twice the angle  between BC and the ground. If the combined mass of the man and the cab is 200 kg and if all other masses are neglected, determine the torque M applied to BC at C and the torque MB in the joint at B required to give the cab an initial vertical acceleration of 1.2 m/s2 when it is started from rest in the position  30. Problem 6/162

The uniform arm OA has a mass of 4 kg, and the gear D has a mass of 5 kg with a radius of gyration about its center of 64 mm. The large gear B is fixed and cannot rotate. If the arm and small gear are released from rest in the position shown in the vertical plane, calculate the initial angular acceleration of OA. Problem 6/163 B O A D 200 mm 100 mm

The vehicle is used to transport supplies to and from the bottom of the 25-percent grade. Each pair of wheels, one at A and the other at B, has a mass of 140 kg with a radius of gyration of 150 mm. The drum C has a mass of 40 kg and a radius of gyration of 100 mm. The total mass of the vehicle is 520 kg. The vehicle is released from rest with a restraining force T of 500 N in the control cable which passes around the drum and is secured at D. Determine the initial acceleration a of the vehicle. The wheels roll without slipping. Problem 6/164

The mass center G of the slender bar of mass 0.8 kg and length 0.4 m is falling vertically with a velocity at the instant depicted. Calculate the angular momentum of the bar about point O if the angular velocity of the bar is (a) clockwise and (b) counterclockwise. Problem 6/165

The grooved drums in the two systems shown are identical. In both cases, (a) and (b), the system is at rest at time Determine the angular velocity of each grooved drum at time Neglect friction at the pivot O. Problem 6/166 m m = 14 kg, k = 225 mm ro = 325 mm, ri = 215 mm

The 75-kg flywheel has a radius of gyration about its shaft axis of and is subjected to the torque where t is in seconds. If the flywheel is at rest at time determine its angular velocity at Problem 6/167

The constant tensions of 200 N and 160 N are applied to the hoisting cable as shown. If the velocity v of the load is 2 m/s down and the angular velocity of the pulley is 8 rad/s counterclockwise at time determine v and after the cable tensions have been applied for 5 s. Note the independence of the results. Problem 6/168

Determine the angular momentum of the earth about the center of the sun. Assume a homogeneous earth and a circular earth orbit of radius 149.6 consult Table D/2 for other needed information. Comment on the relative contributions of the terms and Problem 6/169

The constant 9-lb force is applied to the 80-lb stepped cylinder as shown. The centroidal radius of gyration of the cylinder is and it rolls on the incline without slipping. If the cylinder is at rest when the force is first applied, determine its angular velocity eight seconds later. Problem 6/170

The frictional moment acting on a rotating turbine disk and its shaft is given by where is the angular velocity of the turbine. If the source of power is cut off while the turbine is running with an angular velocity determine the time t for the speed of the turbine to drop to half of its initial value. The moment of inertia of the turbine disk and shaft is I.

The man is walking with speed to the right when he trips over a small floor discontinuity. Estimate his angular velocity just after the impact. His mass is 76 kg with center-of-mass height and his mass moment of inertia about the ankle joint O is 66 kg where all are properties of the portion of his body above O; i.e., both the mass and moment of inertia do not include the foot. Problem 6/172

Repeat the previous problem, only now the man carries a 10-kg backpack as shown. Develop a general expression for the angular velocity of the man just after impact with the small step. Evaluate your expression for the backpack center-of-mass positions (a) and and (b) Case (b) is the condition of a beltpack. The mass conditions for the man remain unchanged from the previous problem. State any assumptions and compare your results with those from the previous problem. Problem 6/173 v1 h d hB O G d 0.2 m hB 0.3 m d hB 0

A uniform slender bar of mass M and length L is translating on the smooth horizontal x-y plane with a velocity when a particle of mass m traveling with a velocity as shown strikes and becomes embedded in the bar. Determine the final linear and angular velocities of the bar with its embedded particle. Problem 6/174

The initially stationary uniform disk of mass and radius b is allowed to drop onto the moving belt from a very small elevation. Determine the time t required for the disk to acquire its steadystate angular speed. The belt drive pulley rotates with a constant counterclockwise velocity . Problem 6/175

Repeat the previous problem if the belt drive pulley rotates clockwise with a constant angular velocity .

The wad of clay of mass m is initially moving with a horizontal velocity when it strikes and sticks to the initially stationary uniform slender bar of mass M and length L. Determine the final angular velocity of the combined body and the x-component of the linear impulse applied to the body by the pivot O during the impact. Problem 6/177

The uniform rectangular panel is falling vertically with speed when its small peg A engages in the receptacle. Determine the angular velocity of the body as well as the x- and y-components of its masscenter velocity just after the impact. Problem 6/178

Just after leaving the platform, the divers fully extended 80-kg body has a rotational speed of 0.3 rev/s about an axis normal to the plane of the trajectory. Estimate the angular velocity N later in the dive when the diver has assumed the tuck position. Make reasonable assumptions concerning the mass moment of inertia of the body in each configuration. Problem 6/179

The slender rod of mass and length L has a movable slider of mass which can be tightened at any location x along the rod. The assembly is initially falling in translation with speed A small peg on the left end of the rod becomes engaged in the receptacle. Determine the angular velocity of the body just after impact. For the condition determine the maximum value of and the corresponding value of x. Plot versus x/L for this mass condition. Problem 6/180

A cylindrical shell of 400-mm diameter and mass m is rotating about its central horizontal axis with an angular velocity when it is released onto a horizontal surface with no velocity of its center If slipping between the shell and the surface occurs for 1.5 s, calculate the coefficient of kinetic friction and the maximum velocity v reached by the center of the shell.

Two small variable-thrust jets are actuated to keep the spacecraft angular velocity about the z-axis constant at as the two telescoping booms are extended from at a constant rate over a 2-min period. Determine the necessary thrust T for each jet as a function of time where is the time when the telescoping action is begun. The small 10-kg experiment modules at the ends of the booms may be treated as particles, and the mass of the rigid booms is negligible. Problem 6/182

With the gears initially at rest and the couple M equal to zero, the forces exerted by the frame on the shafts of the gears at A and B are 30 and 16 lb, respectively, both upward to support the weights of the two gears. A couple is now applied to the larger gear through its shaft at A. After 4 sec the larger gear has a clockwise angular momentum of 12 ft-lb-sec, and the smaller gear has a counterclockwise angular momentum of 4 ft-lb-sec. Calculate the new values of the forces and exerted by the frame on the shafts during the 4-sec interval. Isolate the two gears together as the system. Problem 6/183

The phenomenon of vehicle tripping is investigated here. The sport-utility vehicle is sliding sideways with speed and no angular velocity when it strikes a small curb. Assume no rebound of the right-side tires and estimate the minimum speed which will cause the vehicle to roll completely over to its right side. The mass of the SUV is 2300 kg and its mass moment of inertia about a longitudinal axis through the mass center G is Problem 6/184

The base B has a mass of 5 kg and a radius of gyration of 80 mm about the central vertical axis shown. Each plate P has a mass of 3 kg. If the system is freely rotating about the vertical axis with an angular speed with the plates in the vertical position, estimate the angular speed when the plates have moved to the horizontal positions indicated. Neglect friction. Problem 6/185

In the initial position shown, the disk of axial mass moment of inertia rotates freely with angular velocity relative to the lightweight frame. The turntable of axial mass moment of inertia rotates freely with angular velocity Then the axis of the disk is turned through an angular displacement What is the resulting angular velocity of the turntable? Assume that the thickness of the disk is sufficiently small so that its axial moment of inertia can be approximated by twice its transverse moment of inertia.

The system is initially rotating freely with angular velocity rad/s when the inner rod A is centered lengthwise within the hollow cylinder B as shown in the figure. Determine the angular velocity of the system (a) if the inner rod A has moved so that a length b/2 is protruding from the cylinder, (b) just before the rod leaves the cylinder, and (c) just after the rod leaves the cylinder. Neglect the moment of inertia of the vertical support shafts and friction in the two bearings. Both bodies are constructed of the same uniform material. Use the values and and refer to the results of Prob. B/37 as needed. Problem 6/187

The homogeneous sphere of mass m and radius r is projected along the incline of angle with an initial speed and no angular velocity If the coefficient of kinetic friction is determine the time duration t of the period of slipping. In addition, state the velocity v of the mass center G and the angular velocity at the end of the period of slipping. Problem 6/188

The homogeneous sphere of Prob. 6/188 is placed on the incline with a clockwise angular velocity but no linear velocity of its center Determine the time duration t of the period of slipping. In addition, state the velocity v and angular velocity at the end of the period of slipping

The 165-lb ice skater with arms extended horizontally spins about a vertical axis with a rotational speed of 1 rev/sec. Estimate his rotational speed N if he fully retracts his arms, bringing his hands very close to the centerline of his body. As a reasonable approximation, model the extended arms as uniform slender rods, each of which is 27 in. long and weighs 15 lb. Model the torso as a solid 135-lb cylinder 13 in. in diameter. Treat the man with arms retracted as a solid 165-lb cylinder of 13-in. diameter. Neglect friction at the skateice interface. Problem 6/190 27

The elements of a spacecraft with axial mass symmetry and a reaction-wheel control system are shown in the figure. When the motor exerts a torque on the reaction wheel, an equal and opposite torque is exerted on the spacecraft, thereby changing its angular momentum in the z-direction. If all system elements start from rest and the motor exerts a constant torque M for a time period t, determine the final angular velocity of (a) the spacecraft and (b) the wheel relative to the spacecraft. The mass moment of inertia about the z-axis of the entire spacecraft, including the wheel, is I and that of the wheel alone is The spin axis of the wheel is coincident with the z-axis of symmetry of the spacecraft. Problem 6/191

The body of the spacecraft weighs 322 lb on earth and has a radius of gyration about its z-axis of 1.5 ft. Each of the two solar panels may be treated as a uniform flat plate weighing 16.1 lb. If the spacecraft is rotating about its z-axis at the angular rate of 1.0 rad/sec with determine the angular rate after the panels are rotated to the position by an internal mechanism. Neglect the small momentum change of the body about the y-axis.

A 55-kg dynamics instructor is demonstrating the principles of angular momentum to her class. She stands on a freely rotating platform with her body aligned with the vertical platform axis. With the platform not rotating, she holds a modified bicycle wheel so that its axis is vertical. She then turns the wheel axis to a horizontal orientation without changing the 600-mm distance from the centerline of her body to the wheel center, and her students observe a platform rotation rate of 30 rev/min. If the rim-weighted wheel has a mass of 10 kg and a centroidal radius of gyration and is spinning at a fairly constant rate of 250 rev/min, estimate the mass moment of inertia I of the instructor (in the posture shown) about the vertical platform axis. Problem 6/193

If the dynamics instructor of Prob. 6/193 reorients the wheel axis by 180 with respect to its initial vertical position, what rotational speed N will her students observe? All the given information and the result of Prob. 6/193 may be utilized.

The slotted circular disk whose mass is 6 kg has a radius of gyration about O of 175 mm. The disk carries the four steel balls, each of mass 0.15 kg and located as shown, and rotates freely about a vertical axis through O with an angular speed of 120 rev/min. Each of the small balls is held in place by a latching device not shown. If the balls are released while the disk is rotating and come to rest relative to the disk at the outer ends of the slots, compute the new angular velocity of the disk. Also find the magnitude of the energy loss due to the impact of the balls with the ends of the slots. Neglect the diameter of the balls and discuss this approximation. Problem 6/195

A uniform pole of length L, inclined at an angle with the vertical, is dropped and both ends have a velocity v as end A hits the ground. If end A pivots about its contact point during the remainder of the motion, determine the velocity with which end B hits the ground. Problem 6/196

The 17.5-Mg lunar landing module with center of mass at G has a radius of gyration of 1.8 m about G. The module is designed to contact the lunar surface with a vertical free-fall velocity of 8 km/h. If one of the four legs hits the lunar surface on a small incline and suffers no rebound, compute the angular velocity of the module immediately after impact as it pivots about the contact point. The 9-m dimension is the distance across the diagonal of the square formed by the four feet as corners. Problem 6/197

A uniform circular disk which rolls with a velocity v without slipping encounters an abrupt change in the direction of its motion as it rolls onto the incline . Determine the new velocity of the center of the disk as it starts up the incline, and find the fraction n of the initial energy which is lost because of impact with the incline if Problem 6/198

Determine the minimum velocity v which the wheel must have to just roll over the obstruction. The centroidal radius of gyration of the wheel is k, and it is assumed that the wheel does not slip. Problem 6/199

A frozen-juice can rests on the horizontal rack of a freezer door as shown. With what maximum angular velocity can the door be slammed shut against its seal and not dislodge the can? Assume that the can rolls without slipping on the corner of the rack, and neglect the dimension d compared with the 500-mm distance. Problem 6/200

The force P is applied to the homogeneous crate of mass m. If the coefficient of kinetic friction between the crate and the horizontal platform is determine the limiting values of h so that the crate will slide without tipping about either the front edge or the rear edge. Problem 6/201

A person who walks through the revolving door exerts a 90-N horizontal force on one of the four door panels. If each panel is modeled by a 60-kg uniform rectangular plate which is 1.2 m in length as viewed from above, determine the angular acceleration of the door unit. Neglect friction. Problem 6/202

The preliminary design of a unit for automatically reducing the speed of a freely rotating assembly is shown. Initially the unit is rotating freely about a vertical axis through O at a speed of 600 rev/min with the arms secured in the positions shown by AB. When the arms are released, they swing outward and become latched in the dashed positions shown. The disk has a mass of 30 kg with a radius of gyration of 90 mm about O. Each arm has a length of 160 mm and a mass of 0.84 kg and may be treated as a uniform slender rod. Determine the new speed N of rotation and calculate the loss of energy of the system. Would the results be affected by either the direction of rotation or the sequence of release of the rods?

Each of the solid circular disk wheels has a mass of 2 kg, and the inner solid cylinder has a mass of 3 kg. The disks and cylinder are mounted on the small central shaft so that each can rotate independently of the other with negligible friction in the bearings. Calculate the acceleration of the center of the wheels when the 20-N force is applied as shown. The coeffi- cients of friction between the wheels and the horizontal surface are Problem 6/204

A slender rod of mass and length l is welded at its midpoint A to the rim of the solid circular disk of mass m and radius r. The center of the disk, which rolls without slipping, has a velocity v at the instant when A is at the top of the disk with the rod parallel to the ground. For this instant determine the angular momentum of the combined body about O. Problem 6/205

The uniform slender rod of mass m and length l is freely hinged about a horizontal axis through its end O and is given an initial angular velocity as it crosses the vertical position where If the rod swings through a maximum angle derive an expression in integral form for the time t from release at is reached. (Express in terms of ) Problem 6/206

The uniform rectangular block with the given dimensions is dropped from rest from the position shown. Corner A strikes the ledge at B and becomes latched to it. Determine the angular velocity of the block immediately after it becomes attached to B. Also find the percentage n of energy loss during the corner attachment for the case Problem 6/207

Four identical slender rods each of mass m are welded at their ends to form a square, and the corners are then welded to a light metal hoop of radius r. If the rigid assembly of rods and hoop is allowed to roll down the incline, determine the minimum value of the coefficient of static friction which will prevent slipping. Problem 6/208

A couple is applied at C to the springtoggle mechanism, which is released from rest in the position In this position the spring, which has a stiffness of 140 N/m, is stretched 150 mm. Bar AB has a mass of 3 kg and BC a mass of 6 kg. Calculate the angular velocity of BC as it crosses the position Motion is in the vertical plane, and friction is negligible. Problem 6/209

The link OA and pivoted circular disk are released from rest in the position shown and swing in the vertical plane about the fixed bearing at O. The 6-kg link OA has a radius of gyration about O of 375 mm. The disk has a mass of 8 kg. The two bearings are assumed to be frictionless. Find the force exerted at O on the link (a) just after release and (b) as OA swings through the vertical position Problem 6/210

The small block of mass m slides along the radial slot of the disk while the disk rotates in the horizontal plane about its center O. The block is released from rest relative to the disk and moves outward with an increasing velocity along the slot as the disk turns. Determine the expression in terms of r and for the torque M that must be applied to the disk to maintain a constant angular velocity of the disk. Problem 6/211

The forklift truck with center of mass at has a weight of 3200 lb including the vertical mast. The fork and load have a combined weight of 1800 lb with center of mass at The roller guide at B is capable of supporting horizontal force only, whereas the connection at C, in addition to supporting horizontal force, also transmits the vertical elevating force. If the fork is given an upward acceleration which is sufficient to reduce the force under the rear wheels at A to zero, calculate the corresponding reaction at B. Problem 6/212

A space telescope is shown in the figure. One of the reaction wheels of its attitude-control system is spinning as shown at 10 rad/s, and at this speed the friction in the wheel bearing causes an internal moment of Both the wheel speed and the friction moment may be considered constant over a time span of several hours. If the mass moment of inertia of the entire spacecraft about the x-axis is determine how much time passes before the line of sight of the initially stationary spacecraft drifts by 1 arc-second, which is 1/3600 degree. All other elements are fixed relative to the spacecraft, and no torquing of the reaction wheel shown is performed to correct the attitude drift. Neglect external torques. Problem 6/213

Each of the solid square blocks is allowed to fall by rotating clockwise from the rest positions shown. The support at in case is a hinge and in case is a small roller. Determine the angular velocity of each block as edge becomes horizontal just before striking the supporting surface. Problem 6/214

The mechanical flyball governor operates with a vertical shaft As the shaft speed is increased, the rotational radius of the two balls tends to increase, and the weight A is lifted up by the collar Determine the steady-state value of for a rotational speed of 150 rev/min. Neglect the mass of the arms and collar. Problem 6/215

In an acrobatic stunt, man of mass drops from a raised platform onto the end of the light but strong beam with a velocity The boy of mass is propelled upward with a velocity For a given ratio determine in terms of to maximize the upward velocity of the boy. Assume that both man and boy act as rigid bodies. Problem 6/216

The small block of mass slides in the smooth radial slot of the disk, which turns freely in its bearing. If the block is displaced slightly from the center position when the angular velocity of the disk is determine its radical velocity as a function of the radical distance The mass moment of inertia of the disk about its axis of rotation is Problem 6/217

The pendulum with mass center at is pivoted at A to the fixed support It has a radius of gyration of 17 in. about and swings through an amplitude For the instant when the pendulum is in the extreme position, calculate the moments applied by the base support to the column at Problem 6/218

Before it hits the ground a falling chimney, such as the one shown, will usually crack at the point where the bending moment is greatest. Show that the position of maximum moment occurs at the center of percussion relative to the upper end for a slender chimney of constant cross section. Neglect any restraining moment at the base. Problem 6/219

The two slender bars, each having a mass of 4 kg, are hinged at and pivoted at If a horizontal impulse is applied to the end of the lower bar during an interval of during which the bars are still essentially in their vertical rest positions, compute the angular velocity of the upper bar immediately after the impulse. Problem 6/220 C B F A 1.2 m

The system of Prob. 6/120 is repeated here. If the uniform slender bar is released from rest in the position where the spring is unstretched, determine and plot its angular velocity as a function of over the range where is the value of at which the bar momentarily comes to rest. The value of the spring constant k is 100 N/m, and friction can be neglected. State the maximum angular speed and the value of at which it occurs. Problem 6/221

The crate slides down the incline with velocity and its corner strikes a small obstacle at Determine the minimum required velocity if the crate is to rotate about A so that it travels on the conveyor belt on its side as indicated in the figure. Plot the variation of with for Problem 6/222

The uniform 4-ft slender bar with light end rollers is released from rest in the vertical plane with essentially zero. Determine and graph the velocity of A as a function of and find the maximum velocity of A and the corresponding angle .

The cart moves to the right with acceleration If and determine the steady-state angular deflection of the uniform slender rod of mass Treat the small end sphere of mass as a particle. The spring, which exerts a moment of magnitude on the rod, is undeformed when the rod is vertical. Problem 6/224 l a B O K

The steel I-beam is to be transported by the overhead trolley to which it is hinged at If the trolley starts from rest with and is given a constant horizontal acceleration find the maximum values of and The magnitude of the initial swing would constitute a shop safety consideration. Problem 6/225

The uniform power pole of mass and length is hoisted into a vertical position with its lower end supported by a fixed pivot at The guy wires supporting the pole are accidentally released, and the pole falls to the ground. Plot the x- and y-components of the force exerted on the pole at in terms of from Can you explain why increases again after going to zero? Problem 6/226

The uniform slender bar has a particle attached to its end. The spring constant is and the distance If the bar is released from rest in the horizontal position shown where the spring is unstretched, determine the maximum angular deflection of the bar. Also determine the value of the angular velocity at Neglect friction. Problem 6/227

The uniform 100-kg beam is hanging initially at rest with when the constant force is applied to the cable. Determine (a) the maximum angular velocity reached by the beam with the corresponding angle and the maximum angle reached by the beam. Problem 6/228 3 m

The 30-kg slender bar has an initial angular velocity in the vertical position, where the spring is unstretched. Determine the minimum angular velocity reached by the bar and the corresponding angle Also find the angular velocity of the bar as it strikes the horizontal surface. Problem 6/229 1.2 m k = 3 kN/m 1.2 m 1.2 m A O 0

The 60-ft telephone pole of essentially uniform diameter is being hoisted into the vertical position by two cables attached at as shown. The end rests on a fixed support and cannot slip. When the pole is nearly vertical, the fitting at suddenly breaks, releasing both cables. When the angle reaches the speed of the upper end A of the pole is 4.5 ft/sec. From this point, calculate the time which the workman would have to get out of the way before the pole hits the ground. With what speed does end A hit the ground? Problem 6/230 O x y z A 60 B