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engineering
engineering mechanics dynamics

Engineering Mechanics Dynamics 8th Edition James L. Meriam, L. G. Kraige, J. N. Bolton - Solutions

Car A is traveling along the straightaway with constant speed v. Car B is moving along the circular on-ramp with constant speed v/2. Determine the velocity and acceleration of car A as seen by an observer fixed to car B. Use the values v = 60 mi/hr and R = 200 ft, and utilize the x-y coordinates
For the instant represented, link CB is rotating counterclockwise at a constant rate N = 4 rad/s, and its pin A causes a clockwise rotation of the slotted member ODE. Determine the angular velocity w and angular acceleration a of ODE for this instant. 120 mm イN E 120 mm 45° D B.
The disk rotates about a fixed axis through point O with a clockwise angular velocity w0 = 20 rad/s and a counterclockwise angular acceleration a0 = 5 rad/s2 at the instant under consideration. The value of r is 200 mm. Pin A is fixed to the disk but slides freely within the slotted member BC.
All conditions of the previous problem remain the same, except now, rather than rotating about a fixed center, the disk rolls without slipping on the horizontal surface. If the disk has a clockwise angular velocity of 20 rad/s and a counterclockwise angular acceleration of 5 rad/s2, determine the
The space shuttle A is in an equatorial circular orbit of 240-km altitude and is moving from west to east. Determine the velocity and acceleration which it appears to have to an observer B fixed to and rotating with the earth at the equator as the shuttle passes overhead. Use R = 6378 km for the
Determine the angular acceleration of link EC in the position shown, where w = β˙ = 2 rad/sec and β¨ = 6 rad/sec2 when θ = β = 60°. Pin A is fixed to link EC. The circular slot in link DO has a radius of curvature of 6 in. In the position shown, the tangent to the slot at the point of
One wheel of an experimental vehicle F, which has a constant velocity v = 36 km / h, is shown. The wheel rolls without slipping and causes an oscillation of the slotted arm through the action of its pin A. Control rod DB, in turn, moves back and forth relative to the vehicle by virtue of the motion
The circular disk of radius r rotates about a fixed axis through point O with the angular properties indicated in the figure. Determine the instantaneous velocity and acceleration of point A. Take all given quantities to be positive. の A y Lーーズ 3
The circular disk rotates about its z-axis with an angular velocity w = 2 rad/s. A point P located on the rim has a velocity given by v = −0.8i − 0.6j m/s. Determine the coordinates of P and the radius r of the disk.
The frictional resistance to the rotation of a flywheel consists of a retardation due to air friction which varies as the square of the angular velocity and a constant frictional retardation in the bearing. As a result the angular acceleration of the flywheel while it is allowed to coast is given
What angular velocity w of bar AC will result in point B having zero velocity? What would be the corresponding velocity of point C? Take the length L of the bar and the velocity v of the collar as given quantities. A L/2 В L/2 C
The rectangular plate rotates about its fixed z-axis. At the instant considered its angular velocity is w = 3 rad/s and is decreasing at the rate of 6 rad/s per second. For this instant write the vector expressions for the velocity of P and its normal and tangential components of acceleration. P
Roller B of the linkage has a velocity of 0.75 m/s to the right as the angle θ passes 60° and bar AB also makes an angle of 60° with the horizontal. Locate the instantaneous center of zero velocity for bar AB and determine its angular velocity wAB. A 360 mm 540 mm BO -0.75 m/s
Rotation of the slotted bar OA is controlled by the lead screw that imparts a horizontal velocity v to collar C. Pin P is attached to the collar. Determine the angular velocity w of bar OA in terms of v and the displacement x. h P v C
The helicopter is flying in the horizontal x-direction with a velocity v = 120 mi/hr, and the plane of rotation of the 26-ft-diameter rotor is tilted 10° from the horizontal x-y plane. The rotor blades rotate with an angular velocity Ω = 800 rev/min. For the instant represented write the vector
The large power-cable reel is rolled up the incline by the vehicle as shown. The vehicle starts from rest with x = 0 for the reel and accelerates at the constant rate of 2 ft/sec2. For the instant when x = 6 ft, calculate the magnitude of the acceleration of point P on the reel in the position
The isosceles triangular plate is guided by the two vertex rollers A and B which are confined to move in the perpendicular slots. The control rod gives A a constant velocity vA to the left for an interval of motion. Determine the value of θ for which the horizontal component of the velocity of C
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. 8". a 8" A B
The driver of a pickup truck accelerates from rest to a speed of 45 mi / hr over a horizontal distance of 225 ft with constant acceleration. The truck is hauling an empty 500-lb trailer with a uniform 60-lb gate hinged at O and held in the slightly tilted position by two pegs, one on each side of
A passenger car of an overhead monorail system is driven by one of its two small wheels A or B. Select the one for which the car can be given the greater acceleration without slipping the driving wheel and compute the maximum acceleration if the effective coefficient of friction is limited to 0.25
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. h m N P-
The uniform slender bar of mass m and length L is held in the position shown by the stop at A. What acceleration a will cause the normal force acting on the roller at B to become (a) One-half of the static value.(b) One-fourth of the static value.(c) Zero? B a т L A
The homogeneous crate of mass m is mounted on small wheels as shown. Determine the maximum force P which can be applied without overturning the crate about(a) Its lower front edge with h = b. (b) Its lower back edge with h = 0. P> h B A
The frame is made from uniform rod which has a mass ρ per unit length. A smooth recessed slot constrains the small rollers at A and B to travel horizontally. Force P is applied to the frame through a cable attached to an adjustable collar C. Determine the magnitudes and directions of the normal
A uniform slender rod rests on a car seat as shown. Determine the deceleration a for which the rod will begin to tip forward. Assume that friction at B is sufficient to prevent slipping. Vertical А 300/ 90° В 7° Horizontal a
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. m Go r/2 r/2 P M
The uniform 5-kg bar AB is suspended in a vertical position from an accelerating vehicle and restrained by the wire BC. If the acceleration is a = 0.6g, determine the tension T in the wire and the magnitude of the total force supported by the pin at A. -300 mm a C A 400 mm
If the collar P is given a constant acceleration a = 3g to the right, the pendulum will assume a steady-state deflection θ = 30°. Determine the stiffness kT of the torsional spring which will allow this to happen. The torsional spring is undeformed when the pendulum is in the vertical position.
The uniform 30-kg bar OB is secured to the accelerating frame in the 30° position from the horizontal by the hinge at O and roller at A. If the horizontal acceleration of the frame is a = 20 m/s2, compute the force FA on the roller and the x- and y-components of the force supported by the pin at
The bicyclist applies the brakes as he descends the 10° incline. What deceleration a would cause the dangerous condition of tipping about the front wheel A? The combined center of mass of the rider and bicycle is at G. G. 36" B 25" 15"- - A' 10°
The right-angle bar acts to control the maximum acceleration of an experimental vehicle by depressing the spring-loaded limit switch A with a vertical force. If a 60-percent reduction in the force applied to the button at A (relative to the static value) results in the switch being tripped,
The 1650-kg car has its mass center at G. Calculate the normal forces NA and NB between the road and the front and rear pairs of wheels under conditions of maximum acceleration. The mass of the wheels is small compared with the total mass of the car. The coefficient of static friction between the
The four-wheel-drive all-terrain vehicle has a mass of 300 kg with center of mass G2. The driver has a mass of 85 kg with center of mass G1. If all four wheels are observed to spin momentarily as the driver attempts to go forward, what is the forward acceleration of the driver and ATV? The
A cleated conveyor belt transports solid homogeneous cylinders up a 15° incline. The diameter of each cylinder is half its height. Determine the maximum acceleration which the belt may have without tipping the cylinders as it starts. 15°
The thin hoop of negligible mass and radius r contains a homogeneous semicylinder of mass m which is rigidly attached to the hoop and positioned such that its diametral face is vertical. The assembly is centered on the top of a cart of mass M which rolls freely on the horizontal surface. If the
Determine the magnitude P and direction θ of the force required to impart a rearward acceleration a = 5 ft/sec2 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
The mine skip has a loaded mass of 2000 kg and is attached to the towing vehicle by the light hinged link CD. If the towing vehicle has an acceleration of 3 m/s2, calculate the corresponding reactions under the small wheels at A and B. a D G C 400 800 400 B -1000→1000- 600 2000- Dimensions in
The block A and attached rod have a combined mass of 60 kg and are confined to move along the 60° 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
The homogeneous rectangular plate weighs 40 lb and is supported in the vertical plane by the light parallel links shown. If a couple M = 80 lb-ft is applied to the end of link AB with the system initially at rest, calculate the force supported by the pin at C as the plate lifts off its support with
A jet transport with a landing speed of 200 km/h reduces its speed to 60 km/h 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
The 1300-lb homogeneous plate is suspended from the overhead carriage by the two parallel steel cables. What acceleration a will cause the tensions in the two cables to be equal? What is the resulting steady-state deflection θ of the cables from the vertical? Evaluate your results for the case
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 θ if. (a) A = 0 (b) A = g/2. For what value of a would the steady-state value of θ be zero? a 1. 21
The 30,000-lb concrete pipe section is being transported on a flatbed truck. Five inextensible cables are passed across the top of the pipe and tightened securely to the flatbed with an initial tension of 2000 lb. What is the maximum deceleration a which the truck can experience if the pipe is to
Determine the maximum counterweight W for which the loaded 4000-lb coal car will not overturn about the rear wheels B. Neglect the mass of all pulleys and wheels. 12" 3 B 30° W
The 1800-kg rear-wheel-drive car accelerates forward at a rate of g/2. If the modulus of each of the rear and front springs is 35 kN/m, estimate the resulting momentary nose-up pitch angle θ. (This upward pitch angle during acceleration is called squat, while the downward pitch angle during
Two pulleys are fastened together to form an integral unit. At a certain instant, the indicated belt tensions act on the unit and the unit is turning counterclockwise. Determine the angular acceleration of the unit for this instant if the moment due to friction in the bearing at O is 2.5 N ∙ m.
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. 1.6 m
The figure shows an overhead view of a hydraulically-operated gate. As fluid enters the piston side of the cylinder near A, the rod at B extends causing the gate to rotate about a vertical axis through O. For a 2-in.-diameter piston, what fluid pressure p will give the gate an initial
The motor M is used to hoist the 12,000-lb stadium panel (centroidal radius of gyration k̅ = 6.5 ft) into position by pivoting the panel about its corner A. If the motor is capable of producing 5000 lb-ft of torque, what pulley diameter d will give the panel an initial counterclockwise angular
Each of the two drums and connected hubs of 8-in. radius weighs 200 lb and has a radius of gyration about its center of 15 in. Calculate the angular acceleration of each drum. Friction in each bearing is negligible. 30 lb 30 lb (a) (b)
Determine the angular acceleration and the force on the bearing at O for (a) The narrow ring of mass m. (b) The flat circular disk of mass m immediately after each is released from rest in the vertical plane with OC horizontal. to (a) (b)
The uniform 5-kg portion of a circular hoop is released from rest while in the position shown where the torsional spring of stiffness kT = 15 N ∙ m/rad has been twisted 90° clockwise from its undeformed position. Determine the magnitude of the pin force at O at the instant of release. Motion
The 30-in. slender bar weighs 20 lb and is mounted on a vertical shaft at O. If a torque M = 100 lb-in. is applied to the bar through its shaft, calculate the horizontal force R on the bearing as the bar starts to rotate. 12" 18" M
The half ring of mass m and radius r is welded to a small horizontal shaft mounted in a bearing as shown. Neglect the mass of the shaft and determine the angular acceleration of the ring when a torque M is applied to the shaft. M
The uniform plate of mass m is released from rest while in the position shown. Determine the initial angular acceleration a of the plate and the magnitude of the force supported by the pin at O. The axis of rotation is horizontal. 26 b
The uniform slender bar AB has a mass of 8 kg and swings in a vertical plane about the pivot at A. If θ˙ = 2 rad/s when θ = 30°, compute the force supported by the pin at A at that instant. Horizontal 900 mm
The uniform 16.1-lb slender bar is hinged about a horizontal axis through O and released from rest in the horizontal position. Determine the distance b from the mass center to O which will result in an initial angular acceleration of 16.1 rad/sec2, and find the force R on the bar at O just after
The 15-kg uniform steel plate is freely hinged about the horizontal z-axis. Calculate the force supported by each of the bearings at A and B an instant after the plate is released from rest while in the horizontal position shown. 80 mm 250 mm 80 mm 400 mm
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 45° about axis A-A and released. Determine the initial angular acceleration of the frame. Repeat
A uniform torus and a cylindrical ring, each solid and of mass m, are released from rest in the positions shown. Determine the magnitude of the pin reaction at O and the angular acceleration of each body an instant after release. Neglect friction in the pivot at O for each case. FR R. (a)
A reel of flexible power cable is mounted on the dolly, which is fixed in position. There are 200 ft of cable weighing 0.436 lb per foot of length wound on the reel at a radius of 15 in. The empty spool weighs 62 lb and has a radius of gyration about its axis of 12 in. A tension T of 20 lb is
The uniform bar of mass m is supported by the smooth pin at O and is connected to the cylinder of mass m1 by the light cable which passes over the light pulley at C. If the system is released from rest while in the position shown, determine the tension in the cable. Use the values m = 30 kg, m1 =
A vibration test is run to check the design adequacy of bearings A and B. The unbalanced rotor and attached shaft have a combined mass of 2.8 kg. To locate the mass center, a torque of 0.660 N ∙ m is applied to the shaft to hold it in equilibrium in a position rotated 90° from that shown. A
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 M = 30 N∙m. The rotor may be unlocked from the frame by withdrawing the locking pin P. Calculate the angular acceleration a of
The right-angle body is made of uniform slender bar of mass m and length L. It is released from rest while in the position shown. Determine the initial angular acceleration a of the body and the magnitude of the force supported by the pivot at O. L 4 m 3L 4
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 a.
The right-angle plate is formed from a flat plate having a mass p per unit area and is welded to the horizontal shaft mounted in the bearing at O. If the shaft is free to rotate, determine the initial angular acceleration a of the plate when it is released from rest with the upper surface in the
The semicircular disk of mass m and radius r is released from rest at θ = 0 and rotates freely in the vertical plane about its fixed bearing at O. Derive expressions for the n- and t-components of the force F on the bearing as functions of θ. to
The uniform steel I-beam has a mass of 300 kg and is supported in the vertical plane as shown. Calculate the force R supported by the pin at O for the condition immediately after the support at B is suddenly removed. The mass of the bracket on the left end is small and may be neglected. Also treat
The gear train shown operates in a horizontal plane and is used to transmit motion to the rack D of mass mD. If an input torque M is applied to gear A, what will be the resulting acceleration a of the unloaded rack? (The mechanism which it normally drives has been disengaged.) Gear C is keyed to
The uniform semicircular ring of mass m = 2.5 kg and mean radius r = 200 mm is mounted on spokes of negligible mass and pivoted about a horizontal axis through O. If the ring is released from rest in the position θ = 30°, determine the force R supported by the bearing O just after release.
The link B weighs 0.80 lb with center of mass 2.20 in. from O-O and has a radius of gyration about O-O of 2.76 in. The link is welded to the steel tube and is free to rotate about the fixed horizontal shaft at O-O. The tube weighs 1.84 lb. If the tube is released from rest with the link in the
A flexible cable 60 meters long with a mass of 0.160 kg per meter of length is wound around the reel. With y = 0, the weight of the 4-kg cylinder is required to start turning the reel to overcome friction in its bearings. Determine the downward acceleration a in meters per second squared of the
The uniform 72-ft mast weighs 600 lb and is hinged at its lower end to a fixed support at O. If the winch C develops a starting torque of 900 lb-ft, calculate the total force supported by the pin at O as the mast begins to lift off its support at B. Also fi nd the corresponding angular acceleration
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 MA applied to arm AB required to rotate it about joint A at 4 rad/s2 counterclockwise from the position
The uniform parabolic plate has a mass per unit area of 225 kg/m2. If the plate is released from rest while in the horizontal position shown, determine the initial angular acceleration a about the bearing axis AB. 150 mm- 200 mm 400 mm 150 mm - y = k1 – k,x² A
The 64.4-lb solid circular disk is initially at rest on the horizontal surface when a 3-lb force P, constant in magnitude and direction, is applied to the cord wrapped securely around its periphery. Friction between the disk and the surface is negligible. Calculate the angular velocity w of the
A long cable of length L and mass ρ per unit length is wrapped around the periphery of a spool of negligible mass. One end of the cable is fixed, and the spool is released from rest in the position shown. Find the initial acceleration a of the center of the spool.
The uniform semicircular rod of mass m is lying motionless on the smooth horizontal surface when the force F is applied at B as shown. Determine the resulting initial acceleration of point A. m F В A
The spacecraft is spinning with a constant angular velocity w about the z-axis at the same time that its mass center O is traveling with a velocity vO in the y-direction. If a tangential hydrogen-peroxide jet is fi red when the craft is in the position shown, determine the expression for the
The body consists of a uniform slender bar and a uniform disk, each of mass m/2. It rests on a smooth surface. Determine the angular acceleration a and the acceleration of the mass center of the body when the force P = 6 N is applied as shown. The value of the mass m of the entire body is 1.2 kg.
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 solid homogeneous cylinder is released from rest on the ramp. If θ = 40°, μs = 0.30, and μk = 0.20, determine the acceleration of the mass center G and the friction force exerted by the ramp on the cylinder. W = 8 lb 6" G Hs
The 30-kg spool of outer radius ro = 450 mm has a centroidal radius of gyration k̅ = 275 mm and a central shaft of radius ri = 200 mm. The spool is at rest on the incline when a tension T = 300 N is applied to the end of a cable which is wrapped securely around the central shaft as shown.
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
The uniform heavy bar AB of mass m is moving on its light end rollers along the horizontal with a velocity v when end A passes point C and begins to move on the curved portion of the path with radius r. Determine the force exerted by the path on roller A immediately after it passes C. B A C
The uniform slender bar of mass m and total length L is released from rest in the position shown. Determine the force supported by the small roller at A and the acceleration of roller A along the smooth guide. Evaluate your results for θ = 15°. В L A 3L 4
During a test, the car travels in a horizontal circle of radius R and has a forward tangential acceleration a. Determine the lateral reactions at the front and rear wheel pairs if (a) The car speed v = 0. (b) The speed v ≠ 0. The car mass is m and its polar moment of inertia (about a
The 9-ft steel beam weighs 280 lb and is hoisted from rest where the tension in each of the cables is 140 lb. If the hoisting drums are given initial angular accelerations a1 = 4 rad/sec2 and a2 = 6 rad/sec2, calculate the corresponding tensions TA and TB in the cables. The beam may be treated as a
The mass center G of the 20-lb wheel is off center by 0.50 in. If G is in the position shown as the wheel rolls without slipping through the bottom of the circular path of 6-ft radius with an angular velocity w of 10 rad/sec, compute the force P exerted by the path on the wheel. (Be careful to use
End A of the uniform 5-kg bar is pinned freely to the collar, which has an acceleration a = 4 m/s2 along the fixed horizontal shaft. If the bar has a clockwise angular velocity w = 2 rad/s as it swings past the vertical, determine the components of the force on the bar at A for this instant.
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 TA in the strap at A immediately after wheel B rolls off the rail. Neglect friction and the mass of the small straps and
The truck, initially at rest with a solid cylindrical roll of paper in the position shown, moves forward with a constant acceleration a. Find the distances which the truck goes before the paper rolls off the edge of its horizontal bed. Friction is sufficient to prevent slipping.
The uniform rectangular 300-lb plate is held in the horizontal position by two cables each of length L = 3 ft. If the cable at A suddenly breaks, calculate the tension TB in the cable at B an instant after the break occurs. L 60° A. 60° B 2' 4'
Small ball-bearing rollers mounted on the ends of the slender bar of mass m and length l constrain the motion of the bar in the horizontal x-y slots. If a couple M is applied to the bar initially at rest at θ = 45°, determine the forces exerted on the rollers at A and B as the bar starts to move.
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.
The yo-yo has a mass m and a radius of gyration k about its center O. The cord has a maximum length y = L and is wound around the small inner hub of radius r with its end secured to a point on the hub. If the yo-yo is released from the position y = 0 with a downward velocity vO of its center O,
The uniform bar of mass m is constrained by the light rollers which move in the smooth guide, which lies in a vertical plane. If the bar is released from rest, what is the force at each roller an instant after release? Use the values m = 18 kg and r = 150 mm. 60° В 45° 2r
A bowling ball with a circumference of 27 in. weighs 14 lb and has a centroidal radius of gyration of 3.28 in. If the ball is released with a velocity of 20 ft/sec but with no angular velocity as it touches the alley floor, compute the distance traveled by the ball before it begins to roll without
The figure shows the edge view of a uniform concrete slab with a mass of 12 Mg. The slab is being hoisted slowly by the winch D with cable attached to the dolly. At the position θ = 60°, the distance x from the fixed ground position to the dolly is equal to the length L = 4 m of the slab. If the

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